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Words 140657

Pages 563

College of Computing Georgia Institute of Technology

Copyright c 2001

Approximation Algorithms

Springer

Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Singapore Tokyo

To my parents

Preface

Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872–1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed conjecture that P = NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientiﬁc inquiry in computer science and mathematics. This book presents the theory of approximation algorithms as it stands today. It is reasonable to expect the picture to change with time. The book is divided into three parts. In Part I we cover a combinatorial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected – nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly categorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them. In Part II, we present linear programming based algorithms. These are categorized under two fundamental techniques: rounding and the primal– dual schema. But once again, the exact approximation guarantee obtainable depends on the speciﬁc LP-relaxation used, and there is no ﬁxed recipe for discovering good relaxations, just as there is no ﬁxed recipe for proving a theorem in mathematics (readers familiar with complexity theory will recognize this as the philosophical point behind the P = NP question). Part III covers four important topics. The ﬁrst is the problem of ﬁnding a shortest vector in a lattice which, for several reasons, deserves individual treatment (see Chapter 27). The second topic is the approximability of counting, as opposed to optimization, problems (counting the number of solutions to a given instance). The counting versions of all known NP-complete problems are #Pcomplete1 . Interestingly enough, other than a handful of exceptions, this is true of problems in P as well. An impressive theory has been built for ob1

However, there is no theorem to this eﬀect yet.

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taining eﬃcient approximate counting algorithms for this latter class of problems. Most of these algorithms are based on the Markov chain Monte Carlo (MCMC) method, a topic that deserves a book by itself and is therefore not treated here. In Chapter 28 we present combinatorial algorithms, not using the MCMC method, for two fundamental counting problems. The third topic is centered around recent breakthrough results, establishing hardness of approximation for many key problems, and giving new legitimacy to approximation algorithms as a deep theory. An overview of these results is presented in Chapter 29, assuming the main technical theorem, the PCP Theorem. The latter theorem, unfortunately, does not have a simple proof at present. The fourth topic consists of the numerous open problems of this young ﬁeld. The list presented should by no means be considered exhaustive, and is moreover centered around problems and issues currently in vogue. Exact algorithms have been studied intensively for over four decades, and yet basic insights are still being obtained. Considering the fact that among natural computational problems, polynomial time solvability is the exception rather than the rule, it is only reasonable to expect the theory of approximation algorithms to grow considerably over the years. The set cover problem occupies a special place, not only in the theory of approximation algorithms, but also in this book. It oﬀers a particularly simple setting for introducing key concepts as well as some of the basic algorithm design techniques of Part I and Part II. In order to give a complete treatment for this central problem, in Part III we give a hardness result for it, even though the proof is quite elaborate. The hardness result essentially matches the guarantee of the best algorithm known – this being another reason for presenting this rather diﬃcult proof. Our philosophy on the design and exposition of algorithms is nicely illustrated by the following analogy with an aspect of Michelangelo’s art. A major part of his eﬀort involved looking for interesting pieces of stone in the quarry and staring at them for long hours to determine the form they naturally wanted to take. The chisel work exposed, in a minimalistic manner, this form. By analogy, we would like to start with a clean, simply stated problem (perhaps a simpliﬁed version of the problem we actually want to solve in practice). Most of the algorithm design eﬀort actually goes into understanding the algorithmically relevant combinatorial structure of the problem. The algorithm exploits this structure in a minimalistic manner. The exposition of algorithms in this book will also follow this analogy, with emphasis on stating the structure oﬀered by problems, and keeping the algorithms minimalistic. An attempt has been made to keep individual chapters short and simple, often presenting only the key result. Generalizations and related results are relegated to exercises. The exercises also cover other important results which could not be covered in detail due to logistic constraints. Hints have been

Preface

IX

provided for some of the exercises; however, there is no correlation between the degree of diﬃculty of an exercise and whether a hint is provided for it. This book is suitable for use in advanced undergraduate and graduate level courses on approximation algorithms. It has more than twice the material that can be covered in a semester long course, thereby leaving plenty of room for an instructor to choose topics. An undergraduate course in algorithms and the theory of NP-completeness should suﬃce as a prerequisite for most of the chapters. For completeness, we have provided background information on several topics: complexity theory in Appendix A, probability theory in Appendix B, linear programming in Chapter 12, semideﬁnite programming in Chapter 26, and lattices in Chapter 27. (A disproportionate amount of space has been devoted to the notion of self-reducibility in Appendix A because this notion has been quite sparsely treated in other sources.) This book can also be used is as supplementary text in basic undergraduate and graduate algorithms courses. The ﬁrst few chapters of Part I and Part II are suitable for this purpose. The ordering of chapters in both these parts is roughly by increasing diﬃculty. In anticipation of this wide audience, we decided not to publish this book in any of Springer’s series – even its prestigious Yellow Series. (However, we could not resist spattering a patch of yellow on the cover!) The following translations are currently planned: French by Claire Kenyon, Japanese by Takao Asano, and Romanian by Ion M˘ndoiu. Corrections and comments a from readers are welcome. We have set up a special email address for this purpose: approx@cc.gatech.edu. Finally, a word about practical impact. With practitioners looking for high performance algorithms having error within 2% or 5% of the optimal, what good are algorithms that come within a factor of 2, or even worse, O(log n), of the optimal? Further, by this token, what is the usefulness of improving the approximation guarantee from, say, factor 2 to 3/2? Let us address both issues and point out some fallacies in these assertions. The approximation guarantee only reﬂects the performance of the algorithm on the most pathological instances. Perhaps it is more appropriate to view the approximation guarantee as a measure that forces us to explore deeper into the combinatorial structure of the problem and discover more powerful tools for exploiting this structure. It has been observed that the diﬃculty of constructing tight examples increases considerably as one obtains algorithms with better guarantees. Indeed, for some recent algorithms, obtaining a tight example has been a paper by itself (e.g., see Section 26.7). Experiments have conﬁrmed that these and other sophisticated algorithms do have error bounds of the desired magnitude, 2% to 5%, on typical instances, even though their worst case error bounds are much higher. Additionally, the theoretically proven algorithm should be viewed as a core algorithmic idea that needs to be ﬁne tuned to the types of instances arising in speciﬁc applications.

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Preface

We hope that this book will serve as a catalyst in helping this theory grow and have practical impact.

Acknowledgments

This book is based on courses taught at the Indian Institute of Technology, Delhi in Spring 1992 and Spring 1993, at Georgia Tech in Spring 1997, Spring 1999, and Spring 2000, and at DIMACS in Fall 1998. The Spring 1992 course resulted in the ﬁrst set of class notes on this topic. It is interesting to note that more than half of this book is based on subsequent research results. Numerous friends – and family members – have helped make this book a reality. First, I would like to thank Naveen Garg, Kamal Jain, Ion M˘ndoiu, a Sridhar Rajagopalan, Huzur Saran, and Mihalis Yannakakis – my extensive collaborations with them helped shape many of the ideas presented in this book. I was fortunate to get Ion M˘ndoiu’s help and advice on numerous a matters – his elegant eye for layout and ﬁgures helped shape the presentation. A special thanks, Ion! I would like to express my gratitude to numerous experts in the ﬁeld for generous help on tasks ranging all the way from deciding the contents and its organization, providing feedback on the writeup, ensuring correctness and completeness of references to designing exercises and helping list open problems. Thanks to Sanjeev Arora, Alan Frieze, Naveen Garg, Michel Goemans, Mark Jerrum, Claire Kenyon, Samir Khuller, Daniele Micciancio, Yuval Rabani, Sridhar Rajagopalan, Dana Randall, Tim Roughgarden, Amin Saberi, Leonard Schulman, Amin Shokrollahi, and Mihalis Yannakakis, with special ´ thanks to Kamal Jain, Eva Tardos, and Luca Trevisan. Numerous other people helped with valuable comments and discussions. In particular, I would like to thank Sarmad Abbasi, Cristina Bazgan, Rogerio Brito Gruia Calinescu, Amit Chakrabarti, Mosses Charikar, Joseph Cheriyan, Vasek Chv´tal, Uri Feige, Cristina Fernandes, Ashish Goel, Parikshit Gopalan, a Mike Grigoriadis, Sudipto Guha, Dorit Hochbaum, Howard Karloﬀ, Leonid Khachian, Stavros Kolliopoulos, Jan van Leeuwen, Nati Lenial, George Leuker, Vangelis Markakis, Aranyak Mehta, Rajeev Motwani, Prabhakar Raghavan, Satish Rao, Miklos Santha, Jiri Sgall, David Shmoys, Alistair Sinclair, Prasad Tetali, Pete Veinott, Ramarathnam Venkatesan, Nisheeth Vishnoi, and David Williamson. I am sure I am missing several names – my apologies and thanks to these people as well. A special role was played by the numerous students who took my courses on this topic and scribed notes. It will be impossible to individually remember their names. I would like to express my gratitude collectively to them. I would like to thank IIT Delhi – with special thanks to Shachin Maheshwari – Georgia Tech, and DIMACS for providing pleasant, supportive and academically rich environments. Thanks to NSF for support under grants CCR-9627308 and CCR-9820896.

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XI

It was a pleasure to work with Hans W¨ssner on editorial matters. The o personal care with which he handled all such matters and his sensitivity to an author’s unique point of view were especially impressive. Thanks also to A Frank Holzwarth for sharing his expertise with L TEX. A project of this magnitude would be hard to pull oﬀ without wholehearted support from family members. Fortunately, in my case, some of them are also fellow researchers – my wife, Milena Mihail, and my brother, Umesh Vazirani. Little Michel’s arrival, halfway through this project, brought new joys and energies, though made the end even more challenging! Above all, I would like to thank my parents for their unwavering support and inspiration – my father, a distinguished author of several Civil Engineering books, and my mother, with her deep understanding of Indian Classical Music. This book is dedicated to them. Atlanta, Georgia, May 2001 Vijay Vazirani

Table of Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Lower bounding OPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 An approximation algorithm for cardinality vertex cover 3 1.1.2 Can the approximation guarantee be improved? . . . . . . 3 1.2 Well-characterized problems and min–max relations . . . . . . . . . 5 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Part I. Combinatorial Algorithms 2 Set 2.1 2.2 2.3 2.4 2.5 Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application to shortest superstring . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 17 19 22 26 27 27 28 30 31 32 33 37 38 38 40 44 46

3

Steiner Tree and TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Metric Steiner tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 MST-based algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Metric TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 A simple factor 2 algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Improving the factor to 3/2 . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiway Cut and k-Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The multiway cut problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The minimum k-cut problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

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5

k-Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Parametric pruning applied to metric k-center . . . . . . . . . . . . . . 5.2 The weighted version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Vertex Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Cyclomatic weighted graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Layering applied to feedback vertex set . . . . . . . . . . . . . . . . . . . . 6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shortest Superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A factor 4 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Improving to factor 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Achieving half the optimal compression . . . . . . . . . . . . . 7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Knapsack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 A pseudo-polynomial time algorithm for knapsack . . . . . . . . . . 8.2 An FPTAS for knapsack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Strong NP-hardness and the existence of FPTAS’s . . . . . . . . . 8.3.1 Is an FPTAS the most desirable approximation algorithm? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bin 9.1 9.2 9.3 Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An asymptotic PTAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 52 53 54 54 57 60 60 61 61 64 66 66 67 68 69 69 71 72 72 73 74 74 77 78 79 79 80 81 81 83 83 84 84 87 89 89

6

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9

10 Minimum Makespan Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Factor 2 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A PTAS for minimum makespan . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Bin packing with ﬁxed number of object sizes . . . . . . . . 10.2.2 Reducing makespan to restricted bin packing . . . . . . . . 10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Euclidean TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Proof of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II. LP-Based Algorithms 12 Introduction to LP-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The LP-duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Min–max relations and LP-duality . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Two fundamental algorithm design techniques . . . . . . . . . . . . . . 12.3.1 A comparison of the techniques and the notion of integrality gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Set Cover via Dual Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Dual-ﬁtting-based analysis for the greedy set cover algorithm 13.1.1 Can the approximation guarantee be improved? . . . . . . 13.2 Generalizations of set cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Dual ﬁtting applied to constrained set multicover . . . . . 13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Rounding Applied to Set Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 A simple rounding algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Randomized rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Half-integrality of vertex cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Set Cover via the Primal–Dual Schema . . . . . . . . . . . . . . . . . . . 15.1 Overview of the schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Primal–dual schema applied to set cover . . . . . . . . . . . . . . . . . . . 15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Maximum Satisﬁability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Dealing with large clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Derandomizing via the method of conditional expectation . . . 16.3 Dealing with small clauses via LP-rounding . . . . . . . . . . . . . . . . 16.4 A 3/4 factor algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Scheduling on Unrelated Parallel Machines . . . . . . . . . . . . . . . 17.1 Parametric pruning in an LP setting . . . . . . . . . . . . . . . . . . . . . . 17.2 Properties of extreme point solutions . . . . . . . . . . . . . . . . . . . . . . 17.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 93 97 100 101 103 107 108 108 111 112 112 116 118 119 119 120 122 123 124 125 125 127 129 129 131 132 132 134 136 137 139 140 140 141 142

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17.4 Additional properties of extreme point solutions . . . . . . . . . . . . 143 17.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 17.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 18 Multicut and Integer Multicommodity Flow in Trees . . . . . 18.1 The problems and their LP-relaxations . . . . . . . . . . . . . . . . . . . . 18.2 Primal–dual schema based algorithm . . . . . . . . . . . . . . . . . . . . . . 18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Multiway Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 An interesting LP-relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Randomized rounding algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Half-integrality of node multiway cut . . . . . . . . . . . . . . . . . . . . . 19.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Multicut in General Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Sum multicommodity ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 LP-rounding-based algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Growing a region: the continuous process . . . . . . . . . . . . 20.2.2 The discrete process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.3 Finding successive regions . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 A tight example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Some applications of multicut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Sparsest Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Demands multicommodity ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Linear programming formulation . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Metrics, cut packings, and 1 -embeddability . . . . . . . . . . . . . . . . 21.3.1 Cut packings for metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 1 -embeddability of metrics . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Low distortion 1 -embeddings for metrics . . . . . . . . . . . . . . . . . . 21.4.1 Ensuring that a single edge is not overshrunk . . . . . . . . 21.4.2 Ensuring that no edge is overshrunk . . . . . . . . . . . . . . . . 21.5 LP-rounding-based algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.1 Edge expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.2 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.3 Balanced cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.4 Minimum cut linear arrangement . . . . . . . . . . . . . . . . . . . 21.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 146 149 152 154 155 155 157 160 163 167 168 168 170 171 172 173 175 176 177 179 180 180 181 183 183 185 186 187 190 191 192 192 192 193 194 195 197

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22 Steiner Forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 LP-relaxation and dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Primal–dual schema with synchronization . . . . . . . . . . . . . . . . . 22.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Steiner Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 The LP-relaxation and half-integrality . . . . . . . . . . . . . . . . . . . . 23.2 The technique of iterated rounding . . . . . . . . . . . . . . . . . . . . . . . 23.3 Characterizing extreme point solutions . . . . . . . . . . . . . . . . . . . . 23.4 A counting argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Facility Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1 An intuitive understanding of the dual . . . . . . . . . . . . . . . . . . . . 24.2 Relaxing primal complementary slackness conditions . . . . . . . . 24.3 Primal–dual schema based algorithm . . . . . . . . . . . . . . . . . . . . . . 24.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.1 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.2 Tight example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 k-Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1 LP-relaxation and dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 The high-level idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Randomized rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.1 Derandomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.2 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.3 Tight example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.4 Integrality gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 A Lagrangian relaxation technique for approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Semideﬁnite Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1 Strict quadratic programs and vector programs . . . . . . . . . . . . . 26.2 Properties of positive semideﬁnite matrices . . . . . . . . . . . . . . . . 26.3 The semideﬁnite programming problem . . . . . . . . . . . . . . . . . . . 26.4 Randomized rounding algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Improving the guarantee for MAX-2SAT . . . . . . . . . . . . . . . . . . 26.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

198 198 199 204 207 212 213 213 217 219 221 224 231 232 233 234 235 236 238 238 239 242 243 243 244 247 248 249 249 250 250 251 254 255 255 257 258 260 263 265 268

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Part III. Other Topics 27 Shortest Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1 Bases, determinants, and orthogonality defect . . . . . . . . . . . . . . 27.2 The algorithms of Euclid and Gauss . . . . . . . . . . . . . . . . . . . . . . 27.3 Lower bounding OPT using Gram–Schmidt orthogonalization 27.4 Extension to n dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.5 The dual lattice and its algorithmic use . . . . . . . . . . . . . . . . . . . 27.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Counting Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1 Counting DNF solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Network reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1 Upperbounding the number of near-minimum cuts . . . . 28.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Hardness of Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1 Reductions, gaps, and hardness factors . . . . . . . . . . . . . . . . . . . . 29.2 The PCP theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.3 Hardness of MAX-3SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Hardness of MAX-3SAT with bounded occurrence of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5 Hardness of vertex cover and Steiner tree . . . . . . . . . . . . . . . . . . 29.6 Hardness of clique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.7 Hardness of set cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.7.1 The two-prover one-round characterization of NP . . . . 29.7.2 The gadget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.7.3 Reducing error probability by parallel repetition . . . . . . 29.7.4 The reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1 Problems having constant factor algorithms . . . . . . . . . . . . . . . . 30.2 Other optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Counting problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 274 276 278 280 284 288 292 294 295 297 298 300 302 305 306 306 309 311 313 316 318 322 322 324 325 326 329 332 334 334 336 338

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Appendix A An Overview of Complexity Theory for the Algorithm Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Certiﬁcates and the class NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Reductions and NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . A.3 NP-optimization problems and approximation algorithms . . . A.3.1 Approximation factor preserving reductions . . . . . . . . . . A.4 Randomized complexity classes . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Self-reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Facts from Probability Theory . . . . . . . . . . . . . . . . . . . . . . B.1 Expectation and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Deviations from the mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Basic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 343 344 345 347 347 348 351 352 352 353 354 354

B

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Problem Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

1 Introduction

NP-hard optimization problems exhibit a rich set of possibilities, all the way from allowing approximability to any required degree, to essentially not allowing approximability at all. Despite this diversity, underlying the process of design of approximation algorithms are some common principles. We will explore these in the current chapter. An optimization problem is polynomial time solvable only if it has the algorithmically relevant combinatorial structure that can be used as “footholds” to eﬃciently home in on an optimal solution. The process of designing an exact polynomial time algorithm is a two-pronged attack: unraveling this structure in the problem and ﬁnding algorithmic techniques that can exploit this structure. Although NP-hard optimization problems do not oﬀer footholds for ﬁnding optimal solutions eﬃciently, they may still oﬀer footholds for ﬁnding near-optimal solutions eﬃciently. So, at a high level, the process of design of approximation algorithms is not very diﬀerent from that of design of exact algorithms. It still involves unraveling the relevant structure and ﬁnding algorithmic techniques to exploit it. Typically, the structure turns out to be more elaborate, and often the algorithmic techniques result from generalizing and extending some of the powerful algorithmic tools developed in the study of exact algorithms. On the other hand, looking at the process of designing approximation algorithms a little more closely, one can see that it has its own general principles. We illustrate some of these principles in Section 1.1, using the following simple setting. Problem 1.1 (Vertex cover) Given an undirected graph G = (V, E), and a cost function on vertices c : V → Q+ , ﬁnd a minimum cost vertex cover, i.e., a set V ⊆ V such that every edge has at least one endpoint incident at V . The special case, in which all vertices are of unit cost, will be called the cardinality vertex cover problem. Since the design of an approximation algorithm involves delicately attacking NP-hardness and salvaging from it an eﬃcient approximate solution, it will be useful for the reader to review some key concepts from complexity theory. Appendix A and some exercises in Section 1.3 have been provided for this purpose.

2

1

Introduction

It is important to give precise deﬁnitions of an NP-optimization problem and an approximation algorithm for it (e.g., see Exercises 1.9 and 1.10). Since these deﬁnitions are quite technical, we have moved them to Appendix A. We provide essentials below to quickly get started. An NP-optimization problem Π is either a minimization or a maximization problem. Each valid instance I of Π comes with a nonempty set of feasible solutions, each of which is assigned a nonnegative rational number called its objective function value. There exist polynomial time algorithms for determining validity, feasibility, and the objective function value. A feasible solution that achieves the optimal objective function value is called an optimal solution. OPTΠ (I) will denote the objective function value of an optimal solution to instance I. We will shorten this to OPT when there is no ambiguity. For the problems studied in this book, computing OPTΠ (I) is NP-hard. For example, valid instances of the vertex cover problem consist of an undirected graph G = (V, E) and a cost function on vertices. A feasible solution is a set S ⊆ V that is a cover for G. Its objective function value is the sum of costs of all vertices in S. A minimum cost such set is an optimal solution. An approximation algorithm, A, for Π produces, in polynomial time, a feasible solution whose objective function value is “close” to the optimal; by “close” we mean within a guaranteed factor of the optimal. In the next section, we will present a factor 2 approximation algorithm for the cardinality vertex cover problem, i.e., an algorithm that ﬁnds a cover of cost ≤ 2 · OPT in time polynomial in |V |.

1.1 Lower bounding OPT

When designing an approximation algorithm for an NP-hard NP-optimization problem, one is immediately faced with the following dilemma. In order to establish the approximation guarantee, the cost of the solution produced by the algorithm needs to be compared with the cost of an optimal solution. However, for such problems, not only is it NP-hard to ﬁnd an optimal solution, but it is also NP-hard to compute the cost of an optimal solution (see Appendix A). In fact, in Section A.5 we show that computing the cost of an optimal solution (or even solving its decision version) is precisely the diﬃcult core of such problems. So, how do we establish the approximation guarantee? Interestingly enough, the answer to this question provides a key step in the design of approximation algorithms. Let us demonstrate this in the context of the cardinality vertex cover problem. We will get around the diﬃculty mentioned above by coming up with a “good” polynomial time computable lower bound on the size of the optimal cover.

1.1

Lower bounding OPT

3

1.1.1

An approximation algorithm for cardinality vertex cover

We provide some deﬁnitions ﬁrst. Given a graph H = (U, F ), a subset of the edges M ⊆ F is said to be a matching if no two edges of M share an endpoint. A matching of maximum cardinality in H is called a maximum matching, and a matching that is maximal under inclusion is called a maximal matching. A maximal matching can clearly be computed in polynomial time by simply greedily picking edges and removing endpoints of picked edges. More sophisticated means lead to polynomial time algorithms for ﬁnding a maximum matching as well. Let us observe that the size of a maximal matching in G provides a lower bound. This is so because any vertex cover has to pick at least one endpoint of each matched edge. This lower bounding scheme immediately suggests the following simple algorithm: Algorithm 1.2 (Cardinality vertex cover) Find a maximal matching in G and output the set of matched vertices.

Theorem 1.3 Algorithm 1.2 is a factor 2 approximation algorithm for the cardinality vertex cover problem. Proof: No edge can be left uncovered by the set of vertices picked – otherwise such an edge could have been added to the matching, contradicting its maximality. Let M be the matching picked. As argued above, |M | ≤ OPT. The approximation factor follows from the observation that the cover picked by the algorithm has cardinality 2 |M |, which is at most 2 · OPT. ✷ Observe that the approximation algorithm for vertex cover was very much related to, and followed naturally from, the lower bounding scheme. This is in fact typical in the design of approximation algorithms. In Part II of this book, we show how linear programming provides a uniﬁed way of obtaining lower bounds for several fundamental problems. The algorithm itself is designed around the LP that provides the lower bound. 1.1.2 Can the approximation guarantee be improved?

The following questions arise in the context of improving the approximation guarantee for cardinality vertex cover: 1. Can the approximation guarantee of Algorithm 1.2 be improved by a better analysis?

4

1

Introduction

2. Can an approximation algorithm with a better guarantee be designed using the lower bounding scheme of Algorithm 1.2, i.e., size of a maximal matching in G? 3. Is there some other lower bounding method that can lead to an improved approximation guarantee for vertex cover? Example 1.4 shows that the answer to the ﬁrst question is “no”, i.e., the analysis presented above for Algorithm 1.2 is tight. It gives an inﬁnite family of instances in which the solution produced by Algorithm 1.2 is twice the optimal. An inﬁnite family of instances of this kind, showing that the analysis of an approximation algorithm is tight, will be referred to as a tight example. The importance of ﬁnding tight examples for an approximation algorithm one has designed cannot be overemphasized. They give critical insight into the functioning of the algorithm and have often led to ideas for obtaining algorithms with improved guarantees. (The reader is advised to run algorithms on the tight examples presented in this book.) Example 1.4 Consider the inﬁnite family of instances given by the complete bipartite graphs Kn,n . r ✏r ◗ ❙ ◗ ✏✏✏✑ ✑ ❙ ◗✏ ✏✏◗✑✑ r r ❙ ✏ ✑◗ ✏✏ ❅ ✏✏ ✑ ◗ ❙ ✑ ❙ ❅✏✏◗ ✏ r ✏❅ ◗r ✑ ◗ ✑ ❙ ❙ ◗ ❅ ✑ . . ◗ ❅✑ . . ◗✑ ❙ . . ✑◗❅❙ ◗❅ ✑ ✑ ◗ ❙ ❅r ❙ ◗ r ✑ When run on Kn,n , Algorithm 1.2 will pick all 2n vertices, whereas picking one side of the bipartition gives a cover of size n. ✷ Let us assume that we will establish the approximation factor for an algorithm by simply comparing the cost of the solution it ﬁnds with the lower bound. Indeed, almost all known approximation algorithms operate in this manner. Under this assumption, the answer to the second question is also “no”. This is established in Example 1.5, which gives an inﬁnite family of instances on which the lower bound, of size of a maximal matching, is in fact half the size of an optimal vertex cover. In the case of linear-programmingbased approximation algorithms, the analogous question will be answered by determining a fundamental quantity associated with the linear programming relaxation – its integrality gap (see Chapter 12). The third question, of improving the approximation guarantee for vertex cover, is currently a central open problem in the ﬁeld of approximation algorithms (see Section 30.1).

1.2

Well-characterized problems and min–max relations

5

Example 1.5 The lower bound, of size of a maximal matching, is half the size of an optimal vertex cover for the following inﬁnite family of instances. Consider the complete graph Kn , where n is odd. The size of any maximal matching is (n − 1)/2, whereas the size of an optimal cover is n − 1. ✷

1.2 Well-characterized problems and min–max relations

Consider decision versions of the cardinality vertex cover and maximum matching problems. • Is the size of the minimum vertex cover in G at most k? • Is the size of the maximum matching in G at least l? Both these decision problems are in NP and therefore have Yes certiﬁcates (see Appendix A for deﬁnitions). Do these problems also have No certiﬁcates? We have already observed that the size of a maximum matching is a lower bound on the size of a minimum vertex cover. If G is bipartite, then in fact equality holds; this is the classic K¨nig-Egerv´ry theorem. o a Theorem 1.6 In any bipartite graph, matching M

max

|M | =

vertex cover U

min

|U |.

Therefore, if the answer to the ﬁrst decision problem is “no”, there must be a matching of size k + 1 in G that suﬃces as a certiﬁcate. Similarly, a vertex cover of size l −1 must exist in G if the answer to the second decision problem is “no”. Hence, when restricted to bipartite graphs, both vertex cover and maximum matching problems have No certiﬁcates and are in co-NP. In fact, both problems are also in P under this restriction. It is easy to see that any problem in P trivially has Yes as well as No certiﬁcates (the empty string suﬃces). This is equivalent to the statement that P ⊆ NP ∩ co-NP. It is widely believed that the containment is strict; the conjectured status of these classes is depicted below.

NP

co-NP

P

6

1

Introduction

Problems that have Yes and No certiﬁcates, i.e., are in NP ∩ co-NP, are said to be well-characterized. The importance of this notion can be gauged from the fact that the quest for a polynomial time algorithm for matching started with the observation that it is well-characterized. Min–max relations of the kind given above provide proof that a problem is well-characterized. Such relations are some of the most powerful and beautiful results in combinatorics, and some of the most fundamental polynomial time algorithms (exact) have been designed around such relations. Most of these min–max relations are actually special cases of the LP-duality theorem (see Section 12.2). As pointed out above, LP-duality theory plays a vital role in the design of approximation algorithms as well. What if G is not restricted to be bipartite? In this case, a maximum matching may be strictly smaller than a minimum vertex cover. For instance, if G is simply an odd length cycle on 2p + 1 vertices, then the size of a maximum matching is p, whereas the size of a minimum vertex cover is p + 1. This may happen even for graphs having a perfect matching, for instance, the Petersen graph: r ✚ ✚ ✚ r ✚ ✂❇ r ✚ r r ✥r ✥ ❇ ✂ ❇ ✚ ✂ ✂ ❇ ✂ ❇ ✚ ✂ ✚❇ ✂ ❇ ❇ ✚ r ✂ r✚ ✂ ❇ ❏✂ ❏r ❇r ✡ ✡ This graph has a perfect matching of cardinality 5; however, the minimum vertex cover has cardinality 6. One can show that there is no vertex cover of size 5 by observing that any vertex cover must pick at least p + 1 vertices from an odd cycle of length 2p + 1, just to cover all the edges of the cycle, and the Petersen graph has two disjoint cycles of length 5. Under the widely believed assumption that NP = co-NP, NP-hard problems do not have No certiﬁcates. Thus, the minimum vertex cover problem in general graphs, which is NP-hard, does not have a No certiﬁcate, assuming NP = co-NP. The maximum matching problem in general graphs is in P. However, the No certiﬁcate for this problem is not a vertex cover, but a more general structure: an odd set cover. An odd set cover C in a graph G = (V, E) is a collection of disjoint odd cardinality subsets of V , S1 , . . . , Sk , and a collection v1 , . . . , vl of vertices such that each edge of G is either incident at one of the vertices vi or has both endpoints in one of the sets Si . The weight of this cover C is deﬁned to be k w(C) = l + i=1 (|Si | − 1)/2. The following min–max relation holds.

1.3

Exercises

7

Theorem 1.7 In any graph,

matching M

max

|M | =

odd set cover C

min

w(C).

As shown above, in general graphs a maximum matching can be smaller than a minimum vertex cover. Can it be arbitrarily smaller? The answer is “no”. A corollary of Theorem 1.3 is that in any graph, the size of a maximum matching is at least half the size of a minimum vertex cover. More precisely, Theorem 1.3 gives, as a corollary, the following approximate min–max relation. Approximation algorithms frequently yield such approximate min–max relations, which are of independent interest. Corollary 1.8 In any graph, matching M

max

|M | ≤

vertex cover U

min

|U | ≤ 2 ·

matching M

max

|M | .

Although the vertex cover problem does not have No certiﬁcate under the assumption NP = co-NP, surely there ought to be a way of certifying that (G, k) is a “no” instance for small enough values of k. Algorithm 1.2 (more precisely, the lower bounding scheme behind this approximation algorithm) provides such a method. Let A(G) denote the size of vertex cover output by Algorithm 1.2. Then, OPT(G) ≤ A(G) ≤ 2 · OPT(G). If k < A(G)/2 then k < OPT(G), and therefore (G, k) must be a “no” instance. Furthermore, if k < OPT(G)/2 then k < A(G)/2. Hence, Algorithm 1.2 provides a No certiﬁcate for all instances (G, k) such that k < OPT(G)/2. A No certiﬁcate for instances (I, B) of a minimization problem Π satisfying B < OPT(I)/α is called a factor α approximate No certiﬁcate. As in the case of normal Yes and No certiﬁcates, we do not require that this certiﬁcate be polynomial time computable. An α factor approximation algorithm A for Π provides such a certiﬁcate. Since A has polynomial running time, this certiﬁcate is polynomial time computable. In Chapter 27 we will show an intriguing result – that the shortest vector problem has a factor n approximate No certiﬁcate; however, a polynomial time algorithm for constructing such a certiﬁcate is not known.

1.3 Exercises

1.1 Give a factor 1/2 algorithm for the following. Problem 1.9 (Acyclic subgraph) Given a directed graph G = (V, E), pick a maximum cardinality set of edges from E so that the resulting subgraph is acyclic. Hint: Arbitrarily number the vertices and pick the bigger of the two sets, the forward-going edges and the backward-going edges. What scheme are you using for upper bounding OPT?

8

1

Introduction

1.2 Design a factor 2 approximation algorithm for the problem of ﬁnding a minimum cardinality maximal matching in an undirected graph. Hint: Use the fact that any maximal matching is at least half the maximum matching. 1.3 (R. Bar-Yehuda) Consider the following factor 2 approximation algorithm for the cardinality vertex cover problem. Find a depth ﬁrst search tree in the given graph, G, and output the set, say S, of all the nonleaf vertices of this tree. Show that S is indeed a vertex cover for G and |S| ≤ 2 · OPT. Hint: Show that G has a matching of size |S|. 1.4 Perhaps the ﬁrst strategy one tries when designing an algorithm for an optimization problem is the greedy strategy. For the vertex cover problem, this would involve iteratively picking a maximum degree vertex and removing it, together with edges incident at it, until there are no edges left. Show that this algorithm achieves an approximation guarantee of O(log n). Give a tight example for this algorithm. Hint: The analysis is similar to that in Theorem 2.4. 1.5 A maximal matching can be found via a greedy algorithm: pick an edge, remove its two endpoints, and iterate until there are no edges left. Does this make Algorithm 1.2 a greedy algorithm? 1.6 Give a lower bounding scheme for the arbitrary cost version of the vertex cover problem. Hint: Not easy if you don’t use LP-duality. 1.7 Let A = {a1 , . . . , an } be a ﬁnite set, and let “≤” be a relation on A that is reﬂexive, antisymmetric, and transitive. Such a relation is called a partial ordering of A. Two elements ai , aj ∈ A are said to be comparable if ai ≤ aj or aj ≤ ai . Two elements that are not comparable are said to be incomparable. A subset S ⊆ A is a chain if its elements are pairwise comparable. If the elements of S are pairwise incomparable, then it is an antichain. A chain (antichain) cover is a collection of chains (antichains) that are pairwise disjoint and cover A. The size of such a cover is the number of chains (antichains) in it. Prove the following min–max result. The size of a longest chain equals the size of a smallest antichain cover. Hint: Let the size of the longest chain be m. For a ∈ A, let φ(a) denote the size of the longest chain in which a is the smallest element. Now, consider the partition of A, Ai = {a ∈ A | φ(a) = i}, for 1 ≤ i ≤ m. 1.8 (Dilworth’s theorem, see [195]) Prove that in any ﬁnite partial order, the size of a largest antichain equals the size of a smallest chain cover. Hint: Derive from the K¨nig-Egerv´ry Theorem. Given a partial order on no a element set A, consider the bipartite graph G = (U, V, E) with |U | = |V | = n and (ui , vj ) ∈ E iﬀ ai ≤ aj .

1.3

Exercises

9

The next ten exercises are based on Appendix A. 1.9 Is the following an NP-optimization problem? Given an undirected graph G = (V, E), a cost function on vertices c : V → Q+ , and a positive integer k, ﬁnd a minimum cost vertex cover for G containing at most k vertices. Hint: Can valid instances be recognized in polynomial time (such an instance must have at least one feasible solution)? 1.10 Let A be an algorithm for a minimization NP-optimization problem Π such that the expected cost of the solution produced by A is ≤ αOPT, for a constant α > 1. What is the best approximation guarantee you can establish for Π using algorithm A? Hint: A guarantee of 2α − 1 follows easily. For guarantees arbitrarily close to α, run the algorithm polynomially many times and pick the best solution. Apply Chernoﬀ’s bound. 1.11 Show that if SAT has been proven NP-hard, and SAT has been reduced, via a polynomial time reduction, to the decision version of vertex cover, then the latter is also NP-hard. Hint: Show that the composition of two polynomial time reductions is also a polynomial time reduction. 1.12 Show that if the vertex cover problem is in co-NP, then NP = co-NP. 1.13 (Pratt [222]) Let L be the language consisting of all prime numbers. Show that L ∈ NP. ∗ Hint: Consider the multiplicative group modn, Zn = {a ∈ Z+ | 1 ≤ a < ∗ ∗ n and (a, n) = 1}. Clearly, |Zn | ≤ n − 1. Use the fact that |Zn | = n − 1 iﬀ ∗ n is prime, and that Zn is cyclic if n is prime. The Yes certiﬁcate consists ∗ of a primitive root of Zn , the prime factorization of n − 1, and, recursively, similar information about each prime factor of n − 1. 1.14 Give proofs of self-reducibility for the optimization problems discussed later in this book, in particular, maximum matching, MAX-SAT (Problem 16.1), clique (Problem 29.15), shortest superstring (Problem 2.9), and Minimum makespan scheduling (Problem 10.1). Hint: For clique, consider two possibilities, that v is or isn’t in the optimal clique. Correspondingly, either restrict G to v and its neighbors, or remove v from G. For shortest superstring, remove two strings and replace them by a legal overlap (may even be a simple concatenation). If the length of the optimal superstring remains unchanged, work with this smaller instance. Generalize the scheduling problem a bit – assume that you are also given the number of time units already scheduled on each machine as part of the instance.

10

1

Introduction

1.15 Give a suitable deﬁnition of self-reducibility for problems in NP, i.e., decision problems and not optimization problems, which enables you to obtain a polynomial time algorithm for ﬁnding a feasible solution given an oracle for the decision version, and moreover, yields a self-reducibility tree for instances. Hint: Impose an arbitrary order among the atoms of a solution, e.g., for SAT, this was achieved by arbitrarily ordering the n variables. 1.16 Let Π1 and Π2 be two minimization problems such that there is an approximation factor preserving reduction from Π1 to Π2 . Show that if there is an α factor approximation algorithm for Π2 then there is also an α factor approximation algorithm for Π1 . Hint: First prove that if the reduction transforms instance I1 of Π1 to instance I2 of Π2 then OPTΠ1 (I1 ) = OPTΠ2 (I2 ). 1.17 Show that L ∈ ZPP iﬀ L ∈ (RP ∩ co-RP). 1.18 Show that if NP ⊆ co-RP then NP ⊆ ZPP. Hint: If SAT instance φ is satisﬁable, a satisfying truth assignment for φ can be found, with high probability, using self-reducibility and the co-RP machine for SAT. If φ is not satisﬁable, a “no” answer from the co-RP machine for SAT conﬁrms this; the machine will output such an answer with high probability.

1.4 Notes

The notion of well-characterized problems was given by Edmonds [69] and was precisely formulated by Cook [51]. In the same paper, Cook initiated the theory of NP-completeness. Independently, this discovery was also made by Levin [186]. It gained its true importance with the work of Karp [164], showing NP-completeness of a diverse collection of fundamental computational problems. Interestingly enough, approximation algorithms were designed even before the discovery of the theory of NP-completeness, by Vizing [255] for the minimum edge coloring problem, by Graham [113] for the minimum makespan problem (Problem 10.1), and by Erd¨s [73] for the MAX-CUT problem (Probo lem 2.14). However, the real signiﬁcance of designing such algorithms emerged only after belief in the P = NP conjecture grew. The notion of an approximation algorithm was formally introduced by Garey, Graham, and Ullman [91] and Johnson [150]. The ﬁrst use of linear programming in approximation

1.4

Notes

11

algorithms was due to Lov´sz [192], for analyzing the greedy set cover algoa rithm (see Chapter 13). An early work exploring the use of randomization in the design of algorithms was due to Rabin [224] – this notion is useful in the design of approximation algorithms as well. Theorem 1.7 is due to Edmonds [69] and Algorithm 1.2 is due to Gavril [93]. For basic books on algorithms, see Cormen, Leiserson, Rivest, and Stein [54], Papadimitriou and Steiglitz [217], and Tarjan [246]. For a good treatment of min–max relations, see Lov´sz and Plummer [195]. For books on approxa imation algorithms, see Hochbaum [126] and Ausiello, Crescenzi, Gambosi, Kann, Marchetti, and Protasi [17]. Books on linear programming, complexity theory, and randomized algorithms are listed in Sections 12.5, A.6, and B.4, respectively.

Part I Combinatorial Algorithms

2 Set Cover

The set cover problem plays the same role in approximation algorithms that the maximum matching problem played in exact algorithms – as a problem whose study led to the development of fundamental techniques for the entire ﬁeld. For our purpose this problem is particularly useful, since it oﬀers a very simple setting in which many of the basic algorithm design techniques can be explained with great ease. In this chapter, we will cover two combinatorial techniques: the fundamental greedy technique and the technique of layering. In Part II we will explain both the basic LP-based techniques of rounding and the primal–dual schema using this problem. Among the ﬁrst strategies one tries when designing an algorithm for an optimization problem is some form of the greedy strategy. Even if this strategy does not work for a speciﬁc problem, proving this via a counterexample can provide crucial insights into the structure of the problem. Perhaps the most natural use of this strategy in approximation algorithms is to the set cover problem. Besides the greedy set cover algorithm, we will also present the technique of layering in this chapter. Because of its generality, the set cover problem has wide applicability, sometimes even in unexpected ways. In this chapter we will illustrate such an application – to the shortest superstring problem (see Chapter 7 for an improved algorithm for the latter problem). Problem 2.1 (Set cover) Given a universe U of n elements, a collection of subsets of U , S = {S1 , . . . , Sk }, and a cost function c : S → Q+ , ﬁnd a minimum cost subcollection of S that covers all elements of U . Deﬁne the frequency of an element to be the number of sets it is in. A useful parameter is the frequency of the most frequent element. Let us denote this by f . The various approximation algorithms for set cover achieve one of two factors: O(log n) or f . Clearly, neither dominates the other in all instances. The special case of set cover with f = 2 is essentially the vertex cover problem (see Exercise 2.7), for which we gave a factor 2 approximation algorithm in Chapter 1.

16

2

Set Cover

2.1 The greedy algorithm

The greedy strategy applies naturally to the set cover problem: iteratively pick the most cost-eﬀective set and remove the covered elements, until all elements are covered. Let C be the set of elements already covered at the beginning of an iteration. During this iteration, deﬁne the cost-eﬀectiveness of a set S to be the average cost at which it covers new elements, i.e., c(S)/|S − C|. Deﬁne the price of an element to be the average cost at which it is covered. Equivalently, when a set S is picked, we can think of its cost being distributed equally among the new elements covered, to set their prices. Algorithm 2.2 (Greedy set cover algorithm) 1. C ← ∅ 2. While C = U do Find the most cost-eﬀective set in the current iteration, say S. Let α = cost(S) , i.e., the cost-eﬀectiveness of S. |S−C| Pick S, and for each e ∈ S − C, set price(e) = α. C ← C ∪ S. 3. Output the picked sets.

Number the elements of U in the order in which they were covered by the algorithm, resolving ties arbitrarily. Let e1 , . . . , en be this numbering. Lemma 2.3 For each k ∈ {1, . . . , n}, price(ek ) ≤ OPT/(n − k + 1). Proof: In any iteration, the leftover sets of the optimal solution can cover the remaining elements at a cost of at most OPT. Therefore, among these sets, there must be one having cost-eﬀectiveness of at most OPT/|C|. In the iteration in which element ek was covered, C contained at least n − k + 1 elements. Since ek was covered by the most cost-eﬀective set in this iteration, it follows that price(ek ) ≤ OPT OPT . ≤ n−k+1 |C| ✷ From Lemma 2.3 we immediately obtain: Theorem 2.4 The greedy algorithm is an Hn factor approximation algo1 rithm for the minimum set cover problem, where Hn = 1 + 1 + · · · + n . 2 Proof: Since the cost of each set picked is distributed among the new elen ments covered, the total cost of the set cover picked is equal to k=1 price(ek ). 1 1 ✷ By Lemma 2.3, this is at most 1 + 2 + · · · + n · OPT.

2.2

Layering

17

Example 2.5 The following is a tight example for Algorithm 2.2:

...

1+ε

1/n

1/(n-1)

1

When run on this instance the greedy algorithm outputs the cover consisting of the n singleton sets, since in each iteration some singleton is the most cost-eﬀective set. Thus, the algorithm outputs a cover of cost 1 1 + + · · · + 1 = Hn . n n−1 On the other hand, the optimal cover has a cost of 1 + ε. ✷

Surprisingly enough, for the minimum set cover problem the obvious algorithm given above is essentially the best one can hope for; see Sections 29.7 and 29.9. In Chapter 1 we pointed out that ﬁnding a good lower bound on OPT is a basic starting point in the design of an approximation algorithm for a minimization problem. At this point the reader may be wondering whether there is any truth to this claim. We will show in Section 13.1 that the correct way to view the greedy set cover algorithm is in the setting of the LP-duality theory – this will not only provide the lower bound on which this algorithm is based, but will also help obtain algorithms for several generalizations of this problem.

2.2 Layering

The algorithm design technique of layering is also best introduced via set cover. We note, however, that this is not a very widely applicable technique. We will give a factor 2 approximation algorithm for vertex cover, assuming arbitrary weights, and leave the problem of generalizing this to a factor f approximation algorithm for set cover, where f is the frequency of the most frequent element (see Exercise 2.13). The idea in layering is to decompose the given weight function on vertices into convenient functions, called degree-weighted, on a nested sequence of subgraphs of G. For degree-weighted functions, we will show that we will be within twice the optimal even if we pick all vertices in the cover. Let us introduce some notation. Let w : V → Q+ be the function assigning weights to the vertices of the given graph G = (V, E). We will say that a function assigning vertex weights is degree-weighted if there is a constant

18

2

Set Cover

c > 0 such that the weight of each vertex v ∈ V is c · deg(v). The signiﬁcance of such a weight function is captured in: Lemma 2.6 Let w : V → Q+ be a degree-weighted function. Then w(V ) ≤ 2 · OPT. Proof: Let c be the constant such that w(v) = c · deg(v), and let U be an optimal vertex cover in G. Since U covers all the edges, deg(v) ≥ |E|. v∈U Therefore, w(U ) ≥ c|E|. Now, since lemma follows.

v∈V

deg(v) = 2|E|, w(V ) = 2c|E|. The ✷

Let us deﬁne the largest degree-weighted function in w as follows: remove all degree zero vertices from the graph, and over the remaining vertices, compute c = min{w(v)/deg(v)}. Then, t(v) = c · deg(v) is the desired function. Deﬁne w (v) = w(v) − t(v) to be the residual weight function. The algorithm for decomposing w into degree-weighted functions is as follows. Let G0 = G. Remove degree zero vertices from G0 , say this set is D0 , and compute the largest degree-weighted function in w. Let W0 be vertices of zero residual weight; these vertices are included in the vertex cover. Let G1 be the graph induced on V −(D0 ∪W0 ). Now, the entire process is repeated on G1 w.r.t. the residual weight function. The process terminates when all vertices are of degree zero; let Gk denote this graph. The process is schematically shown in the following ﬁgure.

Gk Gk-1 Wk-1 Dk Dk-1

. . .

G1 G0 W 0 W 1 D1 D0

Let t0 , ..., tk−1 be the degree-weighted functions deﬁned on graphs G0 , ..., Gk−1 . The vertex cover chosen is C = W0 ∪. . .∪Wk−1 . Clearly, V −C = D0 ∪. . .∪Dk . Theorem 2.7 The layer algorithm achieves an approximation guarantee of factor 2 for the vertex cover problem, assuming arbitrary vertex weights. Proof: We need to show that set C is a vertex cover for G and w(C) ≤ 2 · OPT. Assume, for contradiction, that C is not a vertex cover for G. Then

2.3

Application to shortest superstring

19

there must be an edge (u, v) with u ∈ Di and v ∈ Dj , for some i, j. Assume i ≤ j. Therefore, (u, v) is present in Gi , contradicting the fact that u is a degree zero vertex. Let C ∗ be an optimal vertex cover. For proving the second part, consider a vertex v ∈ C. If v ∈ Wj , its weight can be decomposed as w(v) = i≤j ti (v).

Next, consider a vertex v ∈ V − C. If v ∈ Dj , a lower bound on its weight is given by w(v) ≥ i<j ti (v).

The important observation is that in each layer i, C ∗ ∩ Gi is a vertex cover for Gi , since Gi is a vertex-induced graph. Therefore, by Lemma 2.6, ti (C ∩ Gi ) ≤ 2 · ti (C ∗ ∩ Gi ). By the decomposition of weights given above, we get k−1 k−1

w(C) = i=0 ti (C ∩ Gi ) ≤ 2 i=0 ti (C ∗ ∩ Gi ) ≤ 2 · w(C ∗ ). ✷

Example 2.8 A tight example is provided by the family of complete bipartite graphs, Kn,n , with all vertices of unit weight. The layering algorithm will pick all 2n vertices of Kn,n in the cover, whereas the optimal cover picks only one side of the bipartition. ✷

2.3 Application to shortest superstring

The following algorithm is given primarily to demonstrate the wide applicability of set cover. A constant factor approximation algorithm for shortest superstring will be given in Chapter 7. Let us ﬁrst provide motivation for this problem. The human DNA can be viewed as a very long string over a four-letter alphabet. Scientists are attempting to decipher this string. Since it is very long, several overlapping short segments of this string are ﬁrst deciphered. Of course, the locations of these segments on the original DNA are not known. It is hypothesized that the shortest string which contains these segments as substrings is a good approximation to the original DNA string.

20

2

Set Cover

Problem 2.9 (Shortest superstring) Given a ﬁnite alphabet Σ, and a set of n strings, S = {s1 , . . . , sn } ⊆ Σ + , ﬁnd a shortest string s that contains each si as a substring. Without loss of generality, we may assume that no string si is a substring of another string sj , j = i. This problem is NP-hard. Perhaps the ﬁrst algorithm that comes to mind for ﬁnding a short superstring is the following greedy algorithm. Deﬁne the overlap of two strings s, t ∈ Σ ∗ as the maximum length of a suﬃx of s that is also a preﬁx of t. The algorithm maintains a set of strings T ; initially T = S. At each step, the algorithm selects from T two strings that have maximum overlap and replaces them with the string obtained by overlapping them as much as possible. After n − 1 steps, T will contain a single string. Clearly, this string contains each si as a substring. This algorithm is conjectured to have an approximation factor of 2. To see that the approximation factor of this algorithm is no better than 2, consider an input consisting of 3 strings: abk , bk c, and bk+1 . If the ﬁrst two strings are selected in the ﬁrst iteration, the greedy algorithm produces the string abk cbk+1 . This is almost twice as long as the shortest superstring, abk+1 c. We will obtain a 2Hn factor approximation algorithm, using the greedy set cover algorithm. The set cover instance, denoted by S, is constructed as follows. For si , sj ∈ S and k > 0, if the last k symbols of si are the same as the ﬁrst k symbols of sj , let σijk be the string obtained by overlapping these k positions of si and sj : ✛ k ✲ si sj σijk Let M be the set that consists of the strings σijk , for all valid choices of i, j, k. For a string π ∈ Σ + , deﬁne set(π) = {s ∈ S | s is a substring of π}. The universal set of the set cover instance S is S, and the speciﬁed subsets of S are set(π), for each string π ∈ S ∪ I. The cost of set(π) is |π|, i.e., the length of string π. Let OPTS and OPT denote the cost of an optimal solution to S and the length of the shortest superstring of S, respectively. As shown in Lemma 2.11, OPTS and OPT are within a factor of 2 of each other, and so an approximation algorithm for set cover can be used to obtain an approximation algorithm for shortest superstring. The complete algorithm is:

2.3

Application to shortest superstring

21

Algorithm 2.10 (Shortest superstring via set cover) 1. Use the greedy set cover algorithm to ﬁnd a cover for the instance S. Let set(π1 ), . . . , set(πk ) be the sets picked by this cover. 2. Concatenate the strings π1 , . . . , πk , in any order. 3. Output the resulting string, say s.

Lemma 2.11

OPT ≤ OPTS ≤ 2 · OPT.

Proof: Consider an optimal set cover, say {set(πi )|1 ≤ i ≤ l}, and obtain a string, say s, by concatenating the strings πi , 1 ≤ i ≤ l, in any order. Clearly, |s| = OPTS . Since each string of S is a substring of some πi , 1 ≤ i ≤ l, it is also a substring of s. Hence OPTS = |s| ≥ OPT. To prove the second inequality, let s be a shortest superstring of s1 , . . . , sn , |s| = OPT. It suﬃces to produce some set cover of cost at most 2 · OPT. Consider the leftmost occurrence of the strings s1 , . . . , sn in string s. Since no string among s1 , . . . , sn is a substring of another, these n leftmost occurrences start at distinct places in s. For the same reason, they also end at distinct places. Renumber the n strings in the order in which their leftmost occurrences start. Again, since no string is a substring of another, this is also the order in which they end. s sb1 se1 sb2 se2 sb3 se3 π1 .. π2 π3 . ...

..

.

We will partition the ordered list of strings s1 , . . . , sn in groups as described below. Each group will consist of a contiguous set of strings from this

22

2

Set Cover

list. Let bi and ei denote the index of the ﬁrst and last string in the ith group (bi = ei is allowed). Thus, b1 = 1. Let e1 be the largest index of a string that overlaps with s1 (there exists at least one such string, namely s1 itself). In general, if ei < n we set bi+1 = ei + 1 and denote by ei+1 the largest index of a string that overlaps with sbi+1 . Eventually, we will get et = n for some t ≤ n. For each pair of strings (sbi , sei ), let ki > 0 be the length of the overlap between their leftmost occurrences in s (this may be diﬀerent from their maximum overlap). Let πi = σbi ei ki . Clearly, {set(πi )|1 ≤ i ≤ t} is a solution for S, of cost i |πi |. The critical observation is that πi does not overlap πi+2 . We will prove this claim for i = 1; the same argument applies to an arbitrary i. Assume, for contradiction, that π1 overlaps π3 . Then the occurrence of sb3 in s overlaps the occurrence of se1 . However, sb3 does not overlap sb2 (otherwise, sb3 would have been put in the second group). This implies that se1 ends later than sb2 , contradicting the property of endings of strings established earlier. Because of this observation, each symbol of s is covered by at most two of the πi ’s. Hence OPTS ≤ i |πi | ≤ 2 · OPT. ✷ The size of the universal set in the set cover instance S is n, the number of strings in the given shortest superstring instance. This fact, Lemma 2.11, and Theorem 2.4 immediately give the following theorem. Theorem 2.12 Algorithm 2.10 is a 2Hn factor algorithm for the shortest superstring problem, where n is the number of strings in the given instance.

2.4 Exercises

2.1 Given an undirected graph G = (V, E), the cardinality maximum cut problem asks for a partition of V into sets S and S so that the number of edges running between these sets is maximized. Consider the following greedy algorithm for this problem. Here v1 and v2 are arbitrary vertices in G, and for A ⊂ V , d(v, A) denotes the number of edges running between vertex v and set A. Algorithm 2.13 1. Initialization: A ← {v1 } B ← {v2 } 2. For v ∈ V − {v1 , v2 } do: if d(v, A) ≥ d(v, B) then B ← B ∪ {v}, else A ← A ∪ {v}. 3. Output A and B.

2.4

Exercises

23

Show that this is a factor 1/2 approximation algorithm and give a tight example. What is the upper bound on OPT that you are using? Give examples of graphs for which this upper bound is as bad as twice OPT. Generalize the problem and the algorithm to weighted graphs. 2.2 Consider the following algorithm for the maximum cut problem, based on the technique of local search. Given a partition of V into sets, the basic step of the algorithm, called ﬂip, is that of moving a vertex from one side of the partition to the other. The following algorithm ﬁnds a locally optimal solution under the ﬂip operation, i.e., a solution which cannot be improved by a single ﬂip. The algorithm starts with an arbitrary partition of V . While there is a vertex such that ﬂipping it increases the size of the cut, the algorithm ﬂips such a vertex. (Observe that a vertex qualiﬁes for a ﬂip if it has more neighbors in its own partition than in the other side.) The algorithm terminates when no vertex qualiﬁes for a ﬂip. Show that this algorithm terminates in polynomial time, and achieves an approximation guarantee of 1/2. 2.3 Consider the following generalization of the maximum cut problem. Problem 2.14 (MAX k-CUT) Given an undirected graph G = (V, E) with nonnegative edge costs, and an integer k, ﬁnd a partition of V into sets S1 , . . . , Sk so that the total cost of edges running between these sets is maximized. 1 Give a greedy algorithm for this problem that achieves a factor of (1 − k ). Is the analysis of your algorithm tight? 2.4 Give a greedy algorithm for the following problem achieving an approximation guarantee of factor 1/4. Problem 2.15 (Maximum directed cut) Given a directed graph G = (V, E) with nonnegative edge costs, ﬁnd a subset S ⊂ V so as to maximize the total cost of edges out of S, i.e., cost({(u → v) | u ∈ S and v ∈ S}). 2.5 (N. Vishnoi) Use the algorithm in Exercise 2.2 and the fact that the vertex cover problem is polynomial time solvable for bipartite graphs to give a factor log2 ∆ algorithm for vertex cover, where ∆ is the degree of the vertex having highest degree. Hint: Let H denote the subgraph consisting of edges in the maximum cut found by Algorithm 2.13. Clearly, H is bipartite, and for any vertex v, degH (v) ≥ (1/2)degG (v). 2.6 (Wigderson [257]) Consider the following problem. Problem 2.16 (Vertex coloring) Given an undirected graph G = (V, E), color its vertices with the minimum number of colors so that the two endpoints of each edge receive distinct colors.

24

2

Set Cover

1. Give a greedy algorithm for coloring G with ∆ + 1 colors, where ∆ is the maximum degree of a vertex in G. √ 2. Give an algorithm for coloring a 3-colorable graph with O( n) colors. Hint: For any vertex v, the induced subgraph on its neighbors, N (v), √ is bipartite, and hence optimally colorable. If v has degree > n, color v ∪ N (v) using 3 distinct colors. Continue until every vertex has degree √ ≤ n. Then use the algorithm in the ﬁrst part. 2.7 Let 2SC denote the restriction of set cover to instances having f = 2. Show that 2SC is equivalent to the vertex cover problem, with arbitrary costs, under approximation factor preserving reductions. 2.8 Prove that Algorithm 2.2 achieves an approximation factor of Hk , where k is the cardinality of the largest speciﬁed subset of U . 2.9 Give a greedy algorithm that achieves an approximation guarantee of Hn for set multicover, which is a generalization of set cover in which an integral coverage requirement is also speciﬁed for each element and sets can be picked multiple numbers of times to satisfy all coverage requirements. Assume that the cost of picking α copies of set Si is α · cost(Si ). 2.10 By giving an appropriate tight example, show that the analysis of Algorithm 2.2 cannot be improved even for the cardinality set cover problem, i.e., if all speciﬁed sets have unit cost. Hint: Consider running the greedy algorithm on a vertex cover instance. 2.11 Consider the following algorithm for the weighted vertex cover problem. For each vertex v, t(v) is initialized to its weight, and when t(v) drops to 0, v is picked in the cover. c(e) is the amount charged to edge e. Algorithm 2.17 1. Initialization: C←∅ ∀v ∈ V , t(v) ← w(v) ∀e ∈ E, c(e) ← 0 2. While C is not a vertex cover do: Pick an uncovered edge, say (u, v). Let m = min(t(u), t(v)). t(u) ← t(u) − m t(v) ← t(v) − m c(u, v) ← m Include in C all vertices having t(v) = 0. 3. Output C.

2.4

Exercises

25

Show that this is a factor 2 approximation algorithm. Hint: Show that the total amount charged to edges is a lower bound on OPT and that the weight of cover C is at most twice the total amount charged to edges. 2.12 Consider the layering algorithm for vertex cover. Another weight function for which we have a factor 2 approximation algorithm is the constant function – by simply using the factor 2 algorithm for the cardinality vertex cover problem. Can layering be made to work by using this function instead of the degree-weighted function? 2.13 Use layering to get a factor f approximation algorithm for set cover, where f is the frequency of the most frequent element. Provide a tight example for this algorithm. 2.14 A tournament is a directed graph G = (V, E), such that for each pair of vertices, u, v ∈ V , exactly one of (u, v) and (v, u) is in E. A feedback vertex set for G is a subset of the vertices of G whose removal leaves an acyclic graph. Give a factor 3 algorithm for the problem of ﬁnding a minimum feedback vertex set in a directed graph. Hint: Show that it is suﬃcient to “kill” all length 3 cycles. Use the factor f set cover algorithm. 2.15 (Hochbaum [125]) Consider the following problem. Problem 2.18 (Maximum coverage) Given a universal set U of n elements, with nonnegative weights speciﬁed, a collection of subsets of U , S1 , . . . , Sl , and an integer k, pick k sets so as to maximize the weight of elements covered. Show that the obvious algorithm, of greedily picking the best set in each iteration until k sets are picked, achieves an approximation factor of 1− 1− 1 k k 1 >1− . e

2.16 Using set cover, obtain approximation algorithms for the following variants of the shortest superstring problem (here sR is the reverse of string s): 1. Find the shortest string that contains, for each string si ∈ S, both si and sR as substrings. i Hint: The universal set for the set cover instance will contain 2n elements, si and sR , for 1 ≤ i ≤ n. i 2. Find the shortest string that contains, for each string si ∈ S, either si or sR as a substring. i Hint: Deﬁne set(π) = {s ∈ S | s or sR is a substring of π}. Choose the strings π appropriately.

26

2

Set Cover

2.5 Notes

Algorithm 2.2 is due to Johnson [150], Lov´sz [192], and Chv´tal [48]. The a a hardness result for set cover, showing that this algorithm is essentially the best possible, is due to Feige [80], improving on the result of Lund and Yannakakis [199]. The application to shortest superstring is due to Li [187].

3 Steiner Tree and TSP

In this chapter, we will present constant factor algorithms for two fundamental problems, metric Steiner tree and metric TSP. The reasons for considering the metric case of these problems are quite diﬀerent. For Steiner tree, this is the core of the problem – the rest of the problem reduces to this case. For TSP, without this restriction, the problem admits no approximation factor, assuming P = NP. The algorithms, and their analyses, are similar in spirit, which is the reason for presenting these problems together.

3.1 Metric Steiner tree

The Steiner tree problem was deﬁned by Gauss in a letter he wrote to Schumacher (reproduced on the cover of this book). Today, this problem occupies a central place in the ﬁeld of approximation algorithms. The problem has a wide range of applications, all the way from ﬁnding minimum length interconnection of terminals in VLSI design to constructing phylogeny trees in computational biology. This problem and its generalizations will be studied extensively in this book, see Chapters 22 and 23. Problem 3.1 (Steiner tree) Given an undirected graph G = (V, E) with nonnegative edge costs and whose vertices are partitioned into two sets, required and Steiner, ﬁnd a minimum cost tree in G that contains all the required vertices and any subset of the Steiner vertices. We will ﬁrst show that the core of this problem lies in its restriction to instances in which the edge costs satisfy the triangle inequality, i.e., G is a complete undirected graph, and for any three vertices u, v, and w, cost(u, v) ≤ cost(u, w) + cost(v, w). Let us call this restriction the metric Steiner tree problem. Theorem 3.2 There is an approximation factor preserving reduction from the Steiner tree problem to the metric Steiner tree problem. Proof: We will transform, in polynomial time, an instance I of the Steiner tree problem, consisting of graph G = (V, E), to an instance I of the metric Steiner tree problem as follows. Let G be the complete undirected graph on

28

3

Steiner Tree and TSP

vertex set V . Deﬁne the cost of edge (u, v) in G to be the cost of a shortest u–v path in G. G is called the metric closure of G. The partition of V into required and Steiner vertices in I is the same as in I. For any edge (u, v) ∈ E, its cost in G is no more than its cost in G. Therefore, the cost of an optimal solution in I does not exceed the cost of an optimal solution in I. Next, given a Steiner tree T in I , we will show how to obtain, in polynomial time, a Steiner tree T in I of at most the same cost. The cost of an edge (u, v) in G corresponds to the cost of a path in G. Replace each edge of T by the corresponding path to obtain a subgraph of G. Clearly, in this subgraph, all the required vertices are connected. However, this subgraph may, in general, contain cycles. If so, remove edges to obtain tree T . This completes the approximation factor preserving reduction. ✷ As a consequence of Theorem 3.2, any approximation factor established for the metric Steiner tree problem carries over to the entire Steiner tree problem. 3.1.1 MST-based algorithm

Let R denote the set of required vertices. Clearly, a minimum spanning tree (MST) on R is a feasible solution for this problem. Since the problem of ﬁnding an MST is in P and the metric Steiner tree problem is NP-hard, we cannot expect the MST on R to always give an optimal Steiner tree; below is an example in which the MST is strictly costlier. t ✡❏ ❏ 5 ✡ 3 ❏5 ✡ ✡ ✟ ❜❍ ❏ 3 3❏ ❍❍ ✡✟✟ ❍t ✡ t ✟ ❏

5

Even so, an MST on R is not much more costly than an optimal Steiner tree: Theorem 3.3 The cost of an MST on R is within 2 · OPT. Proof: Consider a Steiner tree of cost OPT. By doubling its edges we obtain an Eulerian graph connecting all vertices of R and, possibly, some Steiner vertices. Find an Euler tour of this graph, for example by traversing the edges in DFS (depth ﬁrst search) order:

3.1

Metric Steiner tree

29

The cost of this Euler tour is 2 · OPT. Next obtain a Hamiltonian cycle on the vertices of R by traversing the Euler tour and “short-cutting” Steiner vertices and previously visited vertices of R:

Because of triangle inequality, the shortcuts do not increase the cost of the tour. If we delete one edge of this Hamiltonian cycle, we obtain a path that spans R and has cost at most 2 · OPT. This path is also a spanning tree on R. Hence, the MST on R has cost at most 2 · OPT. ✷ Theorem 3.3 gives a straightforward factor 2 algorithm for the metric Steiner tree problem: simply ﬁnd an MST on the set of required vertices. As in the case of set cover, the “correct” way of viewing this algorithm is in the setting of LP-duality theory. In Chapters 22 and 23 we will see that LP-duality provides the lower bound on which this algorithm is based and also helps solve generalizations of this problem. Example 3.4 For a tight example, consider a graph with n required vertices and one Steiner vertex. An edge between the Steiner vertex and a required vertex has cost 1, and an edge between two required vertices has cost 2 (not all edges of cost 2 are shown below). In this graph, any MST on R has cost 2(n − 1), while OPT = n.

30

3

Steiner Tree and TSP

s ✟ ✭✭✭ ❜ s ✭ ✟ ¤ ❜ ✟ s ✁ ❆ ❜ ✟ ¤ ✟ ❆ ✱ ✟✁ ❆ s ¤ ❇ ❍✁ ✆❉ ❍❍ ❆ ¤ ✱✱ ❇ ❍ ❆ ¤✱ ❇ ✆ ❉✁ ❍❝ s ❇ ✆✁✁ ❉✥✥✥✥ ✆s ✥ ✪❡ ❧ ❚ ❉ ❡ ❧ ✪ ❉ ✪ ❚ ❧ ❡ ... ❉❉s ✪❧ ❚❛ ❡ ✦s ❛❛ ✦ ❧s ✦ ❛✦ ✷

3.2 Metric TSP

The following is a well-studied problem in combinatorial optimization. Problem 3.5 (Traveling salesman problem (TSP)) Given a complete graph with nonnegative edge costs, ﬁnd a minimum cost cycle visiting every vertex exactly once. In its full generality, TSP cannot be approximated, assuming P = NP. Theorem 3.6 For any polynomial time computable function α(n), TSP cannot be approximated within a factor of α(n), unless P = NP. Proof: Assume, for a contradiction, that there is a factor α(n) polynomial time approximation algorithm, A, for the general TSP problem. We will show that A can be used for deciding the Hamiltonian cycle problem (which is NPhard) in polynomial time, thus implying P = NP. The central idea is a reduction from the Hamiltonian cycle problem to TSP, that transforms a graph G on n vertices to an edge-weighted complete graph G on n vertices such that • if G has a Hamiltonian cycle, then the cost of an optimal TSP tour in G is n, and • if G does not have a Hamiltonian cycle, then an optimal TSP tour in G is of cost > α(n) · n. Observe that when run on graph G , algorithm A must return a solution of cost ≤ α(n) · n in the ﬁrst case, and a solution of cost > α(n) · n in the second case. Thus, it can be used for deciding whether G contains a Hamiltonian cycle. The reduction is simple. Assign a weight of 1 to edges of G, and a weight of α(n) · n to nonedges, to obtain G . Now, if G has a Hamiltonian cycle, then the corresponding tour in G has cost n. On the other hand, if G has

3.2

Metric TSP

31

no Hamiltonian cycle, any tour in G must use an edge of cost α(n) · n, and therefore has cost > α(n) · n. ✷ Notice that in order to obtain such a strong nonapproximability result, we had to assign edge costs that violate triangle inequality. If we restrict ourselves to graphs in which edge costs satisfy triangle inequality, i.e., consider metric TSP, the problem remains NP-complete, but it is no longer hard to approximate. 3.2.1 A simple factor 2 algorithm

We will ﬁrst present a simple factor 2 algorithm. The lower bound we will use for obtaining this factor is the cost of an MST in G. This is a lower bound because deleting any edge from an optimal solution to TSP gives us a spanning tree of G. Algorithm 3.7 (Metric TSP – factor 2) 1. 2. 3. 4. Find an MST, T , of G. Double every edge of the MST to obtain an Eulerian graph. Find an Eulerian tour, T , on this graph. Output the tour that visits vertices of G in the order of their ﬁrst appearance in T . Let C be this tour.

Notice that Step 4 is similar to the “short-cutting” step in Theorem 3.3. Theorem 3.8 Algorithm 3.7 is a factor 2 approximation algorithm for metric TSP. Proof: As noted above, cost(T ) ≤ OPT. Since T contains each edge of T twice, cost(T ) = 2 · cost(T ). Because of triangle inequality, after the “shortcutting” step, cost(C) ≤ cost(T ). Combining these inequalities we get that cost(C) ≤ 2 · OPT. ✷ Example 3.9 A tight example for this algorithm is given by a complete graph on n vertices with edges of cost 1 and 2. We present the graph for n = 6 below, where thick edges have cost 1 and remaining edges have cost 2. For arbitrary n the graph has 2n − 2 edges of cost 1, with these edges forming the union of a star and an n − 1 cycle; all remaining edges have cost 2. The optimal TSP tour has cost n, as shown below for n = 6: r r ✑◗ ✑◗ ◗ ◗ ✑ ✁❆ ◗ ✑ ◗ ✑ ✁ ❆ ✏r r✑ r✑ ✑ ◗ ◗r ✁ ✏ ✏ ✏✏ ✏ ❧ ✏❆ ✂ ❇ ❇ r r ✏ ¤ ❧ ❙ ❉ ✂ ❇ ❧ ❇ ❙ ❙ ❇ ¤ ❜❜ ❉ ✂ ❇ ✟✟❙ ❉✂ ¤✟ ❜r r ❇ ❇r ❙r ❙

32

3

Steiner Tree and TSP

Suppose that the MST found by the algorithm is the spanning star created by edges of cost 1. Moreover, suppose that the Euler tour constructed in Step 3 visits vertices in order shown below for n = 6: r 3r ✂❇ 6 ✂ ❇ r r 5r ✦r ✏ ✂ ❇✦✦ ✏✏ ✏ r ✦ r1 ❇ ✂ ❙ ✂ ❇ ❙ ❙r r r 4 ❇ ✂ r2 Then the tour obtained after short-cutting contains n − 2 edges of cost 2 and has a total cost of 2n − 2. Asymptotically, this is twice the cost of the optimal TSP tour. ✷ 3.2.2 Improving the factor to 3/2

Algorithm 3.7 ﬁrst ﬁnds a low cost Euler tour spanning the vertices of G, and then short-cuts this tour to ﬁnd a traveling salesman tour. Is there a cheaper Euler tour than that found by doubling an MST? Recall that a graph has an Euler tour iﬀ all its vertices have even degrees. Thus, we only need to be concerned about the vertices of odd degree in the MST. Let V denote this set of vertices. |V | must be even since the sum of degrees of all vertices in the MST is even. Now, if we add to the MST a minimum cost perfect matching on V , every vertex will have an even degree, and we get an Eulerian graph. With this modiﬁcation, the algorithm achieves an approximation guarantee of 3/2. Algorithm 3.10 (Metric TSP – factor 3/2) 1. Find an MST of G, say T . 2. Compute a minimum cost perfect matching, M , on the set of odd-degree vertices of T . Add M to T and obtain an Eulerian graph. 3. Find an Euler tour, T , of this graph. 4. Output the tour that visits vertices of G in order of their ﬁrst appearance in T . Let C be this tour. Interestingly, the proof of this algorithm is based on a second lower bound on OPT. Lemma 3.11 Let V ⊆ V , such that |V | is even, and let M be a minimum cost perfect matching on V . Then, cost(M ) ≤ OPT/2. Proof: Consider an optimal TSP tour of G, say τ . Let τ be the tour on V obtained by short-cutting τ . By the triangle inequality, cost(τ ) ≤

3.3

Exercises

33

cost(τ ). Now, τ is the union of two perfect matchings on V , each consisting of alternate edges of τ . Thus, the cheaper of these matchings has cost ≤ cost(τ )/2 ≤ OPT/2. Hence the optimal matching also has cost at most OPT/2. ✷ Theorem 3.12 Algorithm 3.10 achieves an approximation guarantee of 3/2 for metric TSP. Proof: The cost of the Euler tour, 3 1 cost(T ) ≤ cost(T ) + cost(M ) ≤ OPT + OPT = OPT, 2 2 where the ﬁrst inequality follows by using the two lower bounds on OPT. Using the triangle inequality, cost(C) ≤ cost(T ), and the theorem follows. ✷ Example 3.13 A tight example for this algorithm is given by the following graph on n vertices, with n odd:

1

❙

1

❙1 1 ❙ ❙ 1 1 ❙

❙ ❙

...

... n/2

❙ ❙

1

❙1 ❙

1

❙ ❙

1

❙

❙1 ❙

❙ ✪

✫

Thick edges represent the MST found in step 1. This MST has only two odd vertices, and by adding the edge joining them we obtain a traveling salesman tour of cost (n − 1) + n/2 . In contrast, the optimal tour has cost n. ✷ Finding a better approximation algorithm for metric TSP is currently one of the outstanding open problems in this area. Many researchers have conjectured that an approximation factor of 4/3 may be achievable.

3.3 Exercises

3.1 The hardness of the Steiner tree problem lies in determining the optimal subset of Steiner vertices that need to be included in the tree. Show this by proving that if this set is provided, then the optimal Steiner tree can be computed in polynomial time. Hint: Find an MST on the union of this set and the set of required vertices. 3.2 Let G = (V, E) be a graph with nonnegative edge costs. S, the senders and R, the receivers, are disjoint subsets of V . The problem is to ﬁnd a minimum cost subgraph of G that has a path connecting each receiver to a

34

3

Steiner Tree and TSP

sender (any sender suﬃces). Partition the instances into two cases: S ∪R = V and S∪R = V . Show that these two cases are in P and NP-hard, respectively. For the second case, give a factor 2 approximation algorithm. Hint: Add a new vertex which is connected to each sender by a zero cost edge. Consider the new vertex and all receivers as required and the remaining vertices as Steiner, and ﬁnd a minimum cost Steiner tree. 3.3 Give an approximation factor preserving reduction from the set cover problem to the following problem, thereby showing that it is unlikely to have a better approximation guarantee than O(log n). Problem 3.14 (Directed Steiner tree) G = (V, E) is a directed graph with nonnegative edge costs. The vertex set V is partitioned into two sets, required and Steiner. One of the required vertices, r, is special. The problem is to ﬁnd a minimum cost tree in G rooted into r that contains all the required vertices and any subset of the Steiner vertices. Hint: Construct a three layer graph: layer 1 contains a required vertex corresponding to each element, layer 2 contains a Steiner vertex corresponding to each set, and layer 3 contains r. 3.4 (Hoogeveen [130]) Consider variants on the metric TSP problem in which the object is to ﬁnd a simple path containing all the vertices of the graph. Three diﬀerent problems arise, depending on the number (0, 1, or 2) of endpoints of the path that are speciﬁed. Obtain the following approximation algorithms. • If zero or one endpoints are speciﬁed, obtain a 3/2 factor algorithm. • If both endpoints are speciﬁed, obtain a 5/3 factor algorithm. Hint: Use the idea behind Algorithm 3.10. 3.5 (Papadimitriou and Yannakakis [219]) Let G be a complete undirected graph in which all edge lengths are either 1 or 2 (clearly, G satisﬁes the triangle inequality). Give a 4/3 factor algorithm for TSP in this special class of graphs. Hint: Start by ﬁnding a minimum 2-matching in G. A 2-matching is a subset S of edges so that every vertex has exactly 2 edges of S incident at it. 3.6 (Frieze, Galbiati, and Maﬃoli [89]) Give an O(log n) factor approximation algorithm for the following problem. Problem 3.15 (Asymmetric TSP) We are given a directed graph G on vertex set V , with a nonnegative cost speciﬁed for edge (u → v), for each pair u, v ∈ V . The edge costs satisfy the directed triangle inequality, i.e., for any three vertices u, v, and w, cost(u → v) ≤ cost(u → w) + cost(w → v). The problem is to ﬁnd a minimum cost cycle visiting every vertex exactly once.

3.3

Exercises

35

Hint: Use the fact that a minimum cost cycle cover (i.e., disjoint cycles covering all the vertices) can be found in polynomial time. Shrink the cycles and recurse. 3.7 Let G = (V, E) be a graph with edge costs satisfying the triangle inequality, and V ⊆ V be a set of even cardinality. Prove or disprove: The cost of a minimum cost perfect matching on V is bounded above by the cost of a minimum cost perfect matching on V . 3.8 Given n points in R2 , deﬁne the optimal Euclidean Steiner tree to be a minimum length tree containing all n points and any other subset of points from R2 . Prove that each of the additional points must have degree three, with all three angles being 120◦ . 3.9 (Rao, Sadayappan, Hwang, and Shor [230]) This exercise develops a factor 2 approximation algorithm for the following problem. Problem 3.16 (Rectilinear Steiner arborescence) Let p1 , . . . , pn be points given in R2 in the positive quadrant. A path from the origin to point pi is said to be monotone if it consists of segments traversing in the positive x direction or the positive y direction (informally, going right or up). The problem is to ﬁnd a minimum length tree containing monotone paths from the origin to each of the n points; such a tree is called rectilinear Steiner arborescence. For point p, deﬁne xp and yp to be its x and y coordinates, and |p|1 = |xp | + |yp |. Say that point p dominates point q if xp ≤ xq and yp ≤ yq . For sets of points A and B, we will say that A dominates B if for each point b ∈ B, there is a point a ∈ A such that a dominates b. For points p and q, deﬁne dom(p, q) = (x, y), where x = min(xp , xq ) and y = min(yp , yq ). If p dominates q, deﬁne segments(p, q) to be a monotone path from p to q. Consider the following algorithm. Algorithm 3.17 (Rectilinear Steiner arborescence) 1. T ← ∅. 2. P ← {p1 , . . . , pn } ∪ {(0, 0)}. 3. while |P | > 1 do: Pick p, q = arg maxp,q∈P (|dom(p, q)|1 ). P ← (P − {p, q}) ∪ {dom(p, q)}. T ← T ∪ segments(dom(p, q), p) ∪ segments(dom(p, q), q). 4. Output T .

36

3

Steiner Tree and TSP

For z ≥ 0, deﬁne z to be the line x + y = z. For a rectilinear Steiner arborescence T , let T (z) = |T ∩ z |. Prove that the length of T is

∞ z=0

T (z) dz.

Also, for every x ≥ 0 deﬁne Pz = {p ∈ P s.t. |p|1 > z}, and N (z) = min{|C| : C ⊂ Prove that

∞ z=0 z

and C dominates Pz }.

N (z) dz

is a lower bound on OPT. Use these facts to show that Algorithm 3.17 achieves an approximation guarantee of 2. 3.10 (I. M˘ndoiu) This exercise develops a factor 9 approximation algorithm a for the following problem, which ﬁnds applications in VLSI clock routing. Problem 3.18 (Rectilinear zero-skew tree) Given a set S of points in the rectilinear plane, ﬁnd a minimum length zero-skew tree (ZST) for S, i.e., a rooted tree T embedded in the rectilinear plane such that points in S are leaves of T and all root-to-leaf paths in T have equal length. By length of a path we mean the sum of the lengths of edges on it. 1. Let T length points length be an arbitrary zero-skew tree, and let R denote the common of all root-to-leaf paths. For r ≥ 0, let T (r) denote the number of of T that are at a length of R − r from the root. Prove that the of T is

R 0

T (r)dr

2. A closed 1 ball of radius r centered at point p is the set of all points whose 1 -distance from p is ≤ r. Let R denote the radius of the smallest 1 -ball that contains all points of S. For r ≥ 0, let N (r) denote the minimum number of closed 1 -balls of radius r needed to cover all points of S. Prove that

R 0

N (r)dr

is a lower bound on the length of the optimum ZST.

3.4

Notes

37

3. Consider the following algorithm. First, compute R and ﬁnd a radius R 1 -ball enclosing all points of S. The center of this ball is chosen as the root of the resulting ZST. This ball can be partitioned into 4 balls, called its quadrants, of radius R/2 each. The root can be connected to the center of any of these balls by an edge of length R/2. These balls can be further partitioned into 4 balls each of radius R/4, and so on. The ZST is constructed recursively, starting with the ball of radius R. The center of the current ball is connected to the centers of each of its quadrants that has a point of S. The algorithm then recurses on each of these quadrants. If the current ball contains exactly one point of S, then this ball is not partitioned into quadrants. Let r be the radius of this ball, c its center, and p ∈ S the point in it. Clearly, the 1 distance between c and p is ≤ r . Connect c to p by a rectilinear path of length exactly r . Show that for 0 ≤ r ≤ R, T (r) ≤ 9N (r). Hence, show that this is a factor 9 approximation algorithm.

3.4 Notes

The Steiner tree problem has its origins in a problem posed by Fermat, and was deﬁned by Gauss in a letter he wrote to his student Schumacher on March 21, 1836. Parts of the letter are reproduced on the cover of this book. Courant and Robbins [55] popularized this problem under the name of Steiner, a well known 19th century geometer. See Hwang, Richards, and Winter [133] and Schreiber [236] for the fascinating history of this problem. The factor 2 Steiner tree algorithm was discovered independently by Choukhmane [44], Iwainsky, Canuto, Taraszow, and Villa [136], Kou, Markowsky, and Berman [177], and Plesn´ [221]. The factor 3/2 metric TSP ık algorithm is due to Christoﬁdes [45], and Theorem 3.6 is due to Sahni and Gonzalez [232]. The lower bound in Exercise 3.10 is from Charikar, Kleinberg, Kumar, Rajagopalan, Sahai, and Tomkins [41]. The best factor known for the rectilinear zero-skew tree problem, due to Zelikovsky and M˘ndoiu a [263], is 3. Given n points on the Euclidean plane, the minimum spanning tree on √ these points is within a factor of 2/ 3 of the minimum Steiner tree (which is allowed to use any set of points on the plane as Steiner points). This was shown by Du and Hwang [63], thereby settling the conjecture of Gilbert and Pollak [100].

4 Multiway Cut and k-Cut

The theory of cuts occupies a central place in the study of exact algorithms. In this chapter, we will present approximation algorithms for natural generalizations of the minimum cut problem. These generalizations are NP-hard. Given a connected, undirected graph G = (V, E) with an assignment of weights to edges, w : E → R+ , a cut is deﬁned by a partition of V into two sets, say V and V − V , and consists of all edges that have one endpoint in each partition. Clearly, the removal of the cut from G disconnects G. Given terminals s, t ∈ V , consider a partition of V that separates s and t. The cut deﬁned by such a partition will be called an s–t cut. The problems of ﬁnding a minimum weight cut and a minimum weight s–t cut can be eﬃciently solved using a maximum ﬂow algorithm. Let us generalize these two notions: Problem 4.1 (Multiway cut) Given a set of terminals S = {s1 , s2 , . . . , sk } ⊆ V , a multiway cut is a set of edges whose removal disconnects the terminals from each other. The multiway cut problem asks for the minimum weight such set. Problem 4.2 (Minimum k-cut) A set of edges whose removal leaves k connected components is called a k-cut. The k-cut problem asks for a minimum weight k-cut. The problem of ﬁnding a minimum weight multiway cut is NP-hard for any ﬁxed k ≥ 3. Observe that the case k = 2 is precisely the minimum s–t cut problem. The minimum k-cut problem is polynomial time solvable for ﬁxed k; however, it is NP-hard if k is speciﬁed as part of the input. In this chapter, we will obtain factor 2 − 2/k approximation algorithms for both problems. In Chapter 19 we will improve the guarantee for the multiway cut problem to 3/2.

4.1 The multiway cut problem

Deﬁne an isolating cut for si to be a set of edges whose removal disconnects si from the rest of the terminals.

4.1

The multiway cut problem

39

Algorithm 4.3 (Multiway cut) 1. For each i = 1, . . . , k, compute a minimum weight isolating cut for si , say Ci . 2. Discard the heaviest of these cuts, and output the union of the rest, say C. Each computation in Step 1 can be accomplished by identifying the terminals in S − {si } into a single node, and ﬁnding a minimum cut separating this node from si ; this takes one max-ﬂow computation. Clearly, removing C from the graph disconnects every pair of terminals, and so is a multiway cut. Theorem 4.4 Algorithm 4.3 achieves an approximation guarantee of 2−2/k. Proof: Let A be an optimal multiway cut in G. We can view A as the union of k cuts as follows: The removal of A from G will create k connected components, each having one terminal (since A is a minimum weight multiway cut, no more than k components will be created). Let Ai be the cut separating k the component containing si from the rest of the graph. Then A = i=1 Ai . Since each edge of A is incident at two of these components, each edge will be in two of the cuts Ai . Hence, k w(Ai ) = 2w(A). i=1 Clearly, Ai is an isolating cut for si . Since Ci is a minimum weight isolating cut for si , w(Ci ) ≤ w(Ai ). Notice that this already gives a factor 2 algorithm, by taking the union of all k cuts Ci . Finally, since C is obtained by discarding the heaviest of the cuts Ci , w(C) ≤ 1− 1 k k w(Ci ) ≤ i=1 1−

1 k

k

w(Ai ) = 2 1 − i=1 1 k

w(A). ✷

Once again, Algorithm 4.3 is not based on a lower bounding scheme. Exercise 19.2 gives an algorithm with the same guarantee using an LP-relaxation as the lower bound. The use of LP-relaxations is fruitful for this problem as well. Section 19.1 gives an algorithm with an improved guarantee, using another LP-relaxation. Example 4.5 A tight example for this algorithm is given by a graph on 2k vertices consisting of a k-cycle and a distinct terminal attached to each vertex of the cycle. The edges of the cycle have weight 1 and edges attaching terminals to the cycle have weight 2 − ε for a small fraction ε > 0.

40

4

Multiway Cut and k-Cut

For example, the graph corresponding to k = 4 is: s1 r ❅ ❅ 2−ε s2 r 2−ε

❅ r ❅ 1 r

1

r 1

2 − ε r s4

1

r ❅ ❅2−ε ❅ r ❅ s3

For each terminal si , the minimum weight isolating cuts for si is given by the edge incident to si . So, the cut C returned by the algorithm has weight (k − 1)(2 − ε). On the other hand, the optimal multiway cut is given by the cycle edges, and has weight k. ✷

4.2 The minimum k-cut problem

A natural algorithm for ﬁnding a k-cut is as follows. Starting with G, compute a minimum cut in each connected component and remove the lightest one; repeat until there are k connected components. This algorithm does achieve a guarantee of 2 − 2/k, however, the proof is quite involved. Instead we will use the Gomory–Hu tree representation of minimum cuts to give a simpler algorithm achieving the same guarantee. Minimum cuts, as well as sub-optimal cuts, in undirected graphs have several interesting structural properties, as opposed to cuts in directed graphs (the algorithm of Section 28.2 is based on exploiting some of these properties). The existence of Gomory–Hu trees is one of the remarkable consequences of these properties. Let T be a tree on vertex set V ; the edges of T need not be in E. Let e be an edge in T . Its removal from T creates two connected components. Let S and S be the vertex sets of these components. The cut deﬁned in graph G by the partition (S, S) is the cut associated with e in G. Deﬁne a weight function w on the edges of T . Tree T will be said to be a Gomory–Hu tree for G if 1. for each pair of vertices u, v ∈ V , the weight of a minimum u–v cut in G is the same as that in T . 2. for each edge e ∈ T , w (e) is the weight of the cut associated with e in G, and

4.2

The minimum k-cut problem

41

A Gomory–Hu tree encodes, in a succinct manner, a minimum u–v cut in G, for each pair of vertices u, v ∈ V as follows. A minimum u–v cut in T is given by a minimum weight edge on the unique path from u to v in T , say e. By the properties stated above, the cut associated with e in G is a minimum u–v cut, and has weight w (e). So, for the n pairs of vertices u, v ∈ V , we 2 need only n − 1 cuts, those encoded by the edges of a Gomory–Hu tree, to give minimum u–v cuts in G. The following ﬁgure shows a weighted graph and its associated Gomory– Hu tree. Exercise 4.6 shows how to construct a Gomory–Hu tree for an undirected graph, using only n − 1 max-ﬂow computations. b 10 a 8 f 3 e 3 2 2 4 4 c 5 2 7 d

a 18

b

17 f 13 e 14 d

15

c

We will need the following lemma. Lemma 4.6 Let S be the union of cuts in G associated with l edges of T . Then, the removal of S from G leaves a graph with at least l + 1 components. Proof: Removing the corresponding l edges from T leaves exactly l + 1 connected components, say with vertex sets V1 , V2 , . . . , Vl+1 . Clearly, removing S from G will disconnect each pair Vi and Vj . Hence we must get at least l + 1 connected components. ✷ As a consequence of Lemma 4.6, the union of k − 1 cuts picked from T will form a k-cut in G. The complete algorithm is given below. Algorithm 4.7 (Minimum k-cut) 1. Compute a Gomory–Hu tree T for G. 2. Output the union of the lightest k − 1 cuts of the n − 1 cuts associated with edges of T in G; let C be this union.

42

4

Multiway Cut and k-Cut

By Lemma 4.6, the removal of C from G will leave at least k components. If more than k components are created, throw back some of the removed edges until there are exactly k components. Theorem 4.8 Algorithm 4.7 achieves an approximation factor of 2 − 2/k. Proof: Let A be an optimal k-cut in G. As in Theorem 4.4, we can view A as the union of k cuts: Let V1 , V2 , . . . , Vk be the k components formed by removing A from G, and let Ai denote the cut separating Vi from the rest of the graph. Then A = A1 ∪ · · · ∪ Ak , and, since each edge of A lies in two of these cuts, k w(Ai ) = 2w(A). i=1 Without loss of generality assume that Ak is the heaviest of these cuts. The idea behind the rest of the proof is to show that there are k − 1 cuts deﬁned by the edges of T whose weights are dominated by the weight of the cuts A1 , A2 , . . . , Ak−1 . Since the algorithm picks the lightest k − 1 cuts deﬁned by T , the theorem follows. The k −1 cuts are identiﬁed as follows. Let B be the set of edges of T that connect across two of the sets V1 , V2 , . . . , Vk . Consider the graph on vertex set V and edge set B, and shrink each of the sets V1 , V2 , . . . , Vk to a single vertex. This shrunk graph must be connected (since T was connected). Throw edges away until a tree remains. Let B ⊆ B be the left over edges, |B | = k − 1. The edges of B deﬁne the required k − 1 cuts. Next, root this tree at Vk (recall that Ak was assumed to be the heaviest cut among the cuts Ai ). This helps in deﬁning a correspondence between the edges in B and the sets V1 , V2 , . . . , Vk−1 : each edge corresponds to the set it comes out of in the rooted tree.

Vk

v c’(u,v)<c(A i ) edge of B’ u edge of B-B’

Vi

4.2

The minimum k-cut problem

43

Suppose edge (u, v) ∈ B corresponds to set Vi in this manner. The weight of a minimum u–v cut in G is w (u, v). Since Ai is a u–v cut in G, w(Ai ) ≥ w (u, v). Thus each cut among A1 , A2 , . . . , Ak−1 is at least as heavy as the cut deﬁned in G by the corresponding edge of B . This, together with the fact that C is the union of the lightest k − 1 cuts deﬁned by T , gives: k−1 w(C) ≤ e∈B w (e) ≤ i=1 w(Ai ) ≤

1−

1 k

k

w(Ai ) = 2 1 − i=1 1 k

w(A). ✷

Example 4.9 The tight example given above for multiway cuts on 2k vertices also serves as a tight example for the k-cut algorithm (of course, there is no need to mark vertices as terminals). Below we give the example for k = 4, together with its Gomory–Hu tree. r ❅ ❅ 2−ε r 2 − ε

❅ r ❅ 1 r

1

r 1

2 − ε r r ❅ ❅ 2−ε

1

r ❅

❅2 − ε ❅ r ❅

❅ r ❅ 2 r

2

r 2 r ❅

r 2 − ε

2 − ε r

❅2 − ε ❅ r ❅

The lightest k − 1 cuts in the Gomory–Hu tree have weight 2 − ε each, corresponding to picking edges of weight 2 − ε of G. So, the k-cut returned

44

4

Multiway Cut and k-Cut

by the algorithm has weight (k − 1)(2 − ε). On the other hand, the optimal k-cut picks all edges of weight 1, and has weight k. ✷

4.3 Exercises

4.1 Show that Algorithm 4.3 can be used as a subroutine for ﬁnding a k-cut within a factor of 2 − 2/k of the minimum k-cut. How many subroutine calls are needed? 4.2 A natural greedy algorithm for computing a multiway cut is the following. Starting with G, compute minimum si –sj cuts for all pairs si , sj that are still connected and remove the lightest of these cuts; repeat this until all pairs si , sj are disconnected. Prove that this algorithm also achieves a guarantee of 2 − 2/k. The next 4 exercises provide background and an algorithm for ﬁnding Gomory–Hu trees. 4.3 Let G = (V, E) be a graph and w : E → R+ be an assignment of nonnegative weights to its edges. For u, v ∈ V let f (u, v) denote the weight of a minimum u–v cut in G. 1. Let u, v, w ∈ V , and suppose f (u, v) ≤ f (u, w) ≤ f (v, w). Show that f (u, v) = f (u, w), i.e., the two smaller numbers are equal. 2. Show that among the n values f (u, v), for all pairs u, v ∈ V , there are 2 at most n − 1 distinct values. 3. Show that for u, v, w ∈ V , f (u, v) ≥ min{f (u, w), f (w, v)}. 4. Show that for u, v, w1 , . . . , wr ∈ V f (u, w) ≥ min{f (u, w1 ), f (w1 , w2 ), . . . , f (wr , v)} (4.1)

4.4 Let T be a tree on vertex set V with weight function w on its edges. We will say that T is a ﬂow equivalent tree if it satisﬁes the ﬁrst of the two Gomory–Hu conditions. i.e., for each pair of vertices u, v ∈ V , the weight of a minimum u–v cut in G is the same as that in T . Let K be the complete graph on V . Deﬁne the weight of each edge (u, v) in K to be f (u, v). Show that any maximum weight spanning tree in K is a ﬂow equivalent tree for G. Hint: For u, v ∈ V , let u, w1 , . . . , wr , v be the unique path from u to v in T . Use (4.1) and the fact that since T is a maximum weight spanning tree, f (u, v) ≤ min{f (u, w1 ), . . . , f (wr , v)}.

4.3

Exercises

45

¯ 4.5 Let (A, A) be a minimum s–t cut such that s ∈ A. Let x and y be any two vertices in A. Consider the graph G obtained by collapsing all vertices ¯ of A to a single vertex vA . The weight of any edge (a, vA ) in G is deﬁned to ¯ ¯ ¯ be the sum of the weights of edges (a, b) where b ∈ A. Clearly, any cut in G deﬁnes a cut in G. Show that a minimum x–y cut in G deﬁnes a minimum x–y cut in G. 4.6 Now we are ready to state the Gomory–Hu algorithm. The algorithm maintains a partition of V , (S1 , S2 , . . . St ), and a spanning tree T on the vertex set {S1 , . . . , St }. Let w be the function assigning weights to the edges of T . Tree T satisﬁes the following invariant. Invariant: For any edge (Si , Sj ) in T there are vertices a and b in Si and Sj respectively, such that w (Si , Sj ) = f (a, b), and the cut deﬁned by edge (Si , Sj ) is a minimum a–b cut in G. The algorithm starts with the trivial partition V , and proceeds in n − 1 iterations. In each iteration, it selects a set Si in the partition such that |Si | ≥ 2 and reﬁnes the partition by splitting Si , and ﬁnding a tree on the reﬁned partition satisfying the invariant. This is accomplished as follows. Let x and y be two distinct vertices in Si . Root the current tree T at Si , and consider the subtrees rooted at the children of Si . Each of these subtrees is collapsed into a single vertex, to obtain graph G (besides these collapsed vertices, G contains all vertices of Si ). A minimum x–y cut is found in G . Let (A, B) be the partition of the vertices of G deﬁning this cut, with x ∈ A x and y ∈ B, and let wxy be the weight of this cut. Compute Si = S ∩ A and y Si = S ∩ B, the two sets into which Si splits. The algorithm updates the partition and the tree as follows. It reﬁnes the y x partition by replacing Si with two sets Si and Si . The new tree has the edge y x (Si , Si ), with weight wxy . Consider a subtree T that was incident at Si in T . Assume w.l.o.g. that the node corresponding to T lies in A. Then, T is x connected by an edge to Si . The weight of this connecting edge is the same as the weight of the edge connecting T to Si . All edges in T retain their weights. Show that the new tree satisﬁes the invariant. Hence show that the algorithm terminates (when the partition consists of singleton vertices) with a Gomory–Hu tree for G. Consider the graph: b 10 a 8 f 3 e 3 2 2 4 4 c 5 2 7 d

46

4

Multiway Cut and k-Cut

The execution of the Gomory–Hu algorithm is demonstrated below:

Initial partition: a, b, c, d, e, f

Select b and f:

a, b

17

c, d, e, f

Select a and b:

a

18

b

17

c, d, e, f

Select c and f:

a

18

b

17

f

13

c, d, e

Select d and e:

a

18

b

17

f

13

c, e 14 d

Select c and e:

a

18

b

17

f

13

e 14 d

15

c

4.7 Prove that if the Gomory–Hu tree for an edge-weighted undirected graph G contains all n − 1 distinct weights, then G can have only one minimum weight cut.

4.4 Notes

Algorithm 4.3 is due to Dahlhaus, Johnson, Seymour, Papadimitriou and Yannakakis [57]. Algorithm 4.7 is due to Saran and Vazirani [233]; the proof given here is due to R. Ravi. For Gomory–Hu trees see Gomory and Hu [110].

5 k-Center

Consider the following application. Given a set of cities, with intercity distances speciﬁed, pick k cities for locating warehouses in so as to minimize the maximum distance of a city from its closest warehouse. We will study this problem, called the k-center problem, and its weighted version, under the restriction that the edge costs satisfy the triangle inequality. Without this restriction, the k-center problem cannot be approximated within factor α(n), for any computable function α(n), assuming P = NP (see Exercise 5.1). We will introduce the algorithmic technique of parametric pruning for solving this problem. In Chapter 17 we will use this technique in a linear programming setting. Problem 5.1 (Metric k-center) Let G = (V, E) be a complete undirected graph with edge costs satisfying the triangle inequality, and k be a positive integer. For any set S ⊆ V and vertex v ∈ V , deﬁne connect(v, S) to be the cost of the cheapest edge from v to a vertex in S. The problem is to ﬁnd a set S ⊆ V , with |S| = k, so as to minimize maxv {connect(v, S)}.

5.1 Parametric pruning applied to metric k-center

If we know the cost of an optimal solution, we may be able to prune away irrelevant parts of the input and thereby simplify the search for a good solution. However, as stated in Chapter 1, computing the cost of an optimal solution is precisely the diﬃcult core of NP-hard NP-optimization problems. The technique of parametric pruning gets around this diﬃculty as follows. A parameter t is chosen, which can be viewed as a “guess” on the cost of an optimal solution. For each value of t, the given instance I is pruned by removing parts that will not be used in any solution of cost > t. Denote the pruned instance by I(t). The algorithm consists of two steps. In the ﬁrst step, the family of instances I(t) is used for computing a lower bound on OPT, say t∗ . In the second step, a solution is found in instance I(α · t∗ ), for a suitable choice of α. A restatement of the k-center problem shows how parametric pruning applies naturally to it. Sort the edges of G in nondecreasing order of cost, i.e., cost(e1 ) ≤ cost(e2 ) ≤ . . . ≤ cost(em ), and let Gi = (V, Ei ), where Ei =

48

5

k-Center

{e1 , e2 , . . . , ei }. A dominating set in an undirected graph H = (U, F ) is a subset S ⊆ U such that every vertex in U − S is adjacent to a vertex in S. Let dom(H) denote the size of a minimum cardinality dominating set in H. Computing dom(H) is NP-hard. The k-center problem is equivalent to ﬁnding the smallest index i such that Gi has a dominating set of size at most k, i.e., Gi contains k stars spanning all vertices, where a star is the graph K1,p , with p ≥ 1. If i∗ is the smallest such index, then cost(ei∗ ) is the cost of an optimal k-center. We will denoted this by OPT. We will work with the family of graphs G1 , . . . , Gm . Deﬁne the square of graph H to be the graph containing an edge (u, v) whenever H has a path of length at most two between u and v, u = v. We will denote it by H 2 . The following structural result gives a method for lower bounding OPT. Lemma 5.2 Given a graph H, let I be an independent set in H 2 . Then, |I| ≤ dom(H). Proof: Let D be a minimum dominating set in H. Then, H contains |D| stars spanning all vertices. Since each of these stars will be a clique in H 2 , H 2 contains |D| cliques spanning all vertices. Clearly, I can pick at most one vertex from each clique, and the lemma follows. ✷ The k-center algorithm is: Algorithm 5.3 (Metric k-center) 1. 2. 3. 4. Construct G2 , G2 , . . . , G2 . m 1 2 Compute a maximal independent set, Mi , in each graph G2 . i Find the smallest index i such that |Mi | ≤ k, say j. Return Mj .

The lower bound on which this algorithm is based is: Lemma 5.4 For j as deﬁned in the algorithm, cost(ej ) ≤ OPT. Proof: For every i < j we have that |Mi | > k. Now, by Lemma 5.2, ✷ dom(Gi ) > k, and so i∗ > i. Hence, j ≤ i∗ . Theorem 5.5 Algorithm 5.3 achieves an approximation factor of 2 for the metric k-center problem. Proof: The key observation is that a maximal independent set, I, in a graph is also a dominating set (for, if some vertex v is not dominated by I, then I ∪{v} must also be an independent set, contradicting I’s maximality). Thus,

5.1

Parametric pruning applied to metric k-center

49

there exist stars in G2 , centered on the vertices of Mj , covering all vertices. j By the triangle inequality, each edge used in constructing these stars has cost at most 2 · cost(ej ). The theorem follows from Lemma 5.4. ✷ Example 5.6 A tight example for the previous algorithm is given by a wheel graph on n + 1 vertices, where all edges incident to the center vertex have cost 1, and the rest of the edges have cost 2: ✭r ✭✭✭ ✟¤❜ r ✟ ¤ ❜ ✁✟ ❆ ✟ r ❜ ❜ ¤ ✟ ❆ ✟✁ ❆ ✱❇ ¤ ✟✁ r ❍ ❆ ¤ ✱✱ ❇ ✆❉ ❍❍ ✁ ❇ ❍❍❆ ¤ ✱ ✆ ❉✁ ❇ r ❆r ✆ ✁❉ ✥ ✥❍ ¤✱ ✥✥✥ ✪❡ ✆r ✥❉ ✥ ✁ ✪ ❧ ❡ ❚ ❉ ❡ ❉ ❚❧❧ ✪ . ❡ .. ❚❉ ✪ ❧ ✪ ❉r❛ ❚❛ ❧ ❡r ❛❛ ✦✦ ❧ ✦✦ ❧r ❛ (Here, thin edges have cost 1 and thick edges have cost 2; not all edges of cost 2 are shown.) For k = 1, the optimal solution is the center of the wheel, and OPT = 1. The algorithm will compute index j = n. Now, G2 is a clique and, if a n peripheral vertex is chosen as the maximal independent set, then the cost of the solution found is 2. ✷ Next, we will show that 2 is essentially the best approximation factor achievable for the metric k-center problem. Theorem 5.7 Assuming P = NP, there is no polynomial time algorithm achieving a factor of 2 − ε, ε > 0, for the metric k-center problem. Proof: We will show that such an algorithm can solve the dominating set problem in polynomial time. The idea is similar to that of Theorem 3.6 and involves giving a reduction from the dominating set problem to metric kcenter. Let G = (V, E), k be an instance of the dominating set problem. Construct a complete graph G = (V, E ) with edge costs given by cost(u, v) = 1, if (u, v) ∈ E, 2, if (u, v) ∈ E.

Clearly, G satisﬁes the triangle inequality. This reduction satisﬁes the conditions: • if dom(G) ≤ k, then G has a k-center of cost 1, and

50

5

k-Center

• if dom(G) > k, then the optimum cost of a k-center in G is 2. In the ﬁrst case, when run on G , the (2 − ε)-approximation algorithm must give a solution of cost 1, since it cannot use an edge of cost 2. Hence, using this algorithm, we can distinguish between the two possibilities, thus solving the dominating set problem. ✷

5.2 The weighted version

We will use the technique of parametric pruning to obtain a factor 3 approximation algorithm for the following generalization of the metric k-center problem. Problem 5.8 (Metric weighted k-center) In addition to a cost function on edges, we are given a weight function on vertices, w : V → R+ , and a bound W ∈ R+ . The problem is to pick S ⊆ V of total weight at most W , minimizing the same objective function as before, i.e., max{min{cost(u, v)}}. v∈V u∈S

Let wdom(G) denote the weight of a minimum weight dominating set in G. Then, with respect to the graphs Gi deﬁned above, we need to ﬁnd the smallest index i such that wdom(Gi ) ≤ W . If i∗ is this index, then the cost of the optimal solution is OPT = cost(ei∗ ). Given a vertex weighted graph H, let I be an independent set in H 2 . For each u ∈ I, let s(u) denote a lightest neighbor of u in H, where u is also considered a neighbor of itself. (Notice that the neighbor is picked in H and not in H 2 .) Let S = {s(u)| u ∈ I}. The following fact, analogous to Lemma 5.2, will be used to derive a lower bound on OPT: Lemma 5.9 w(S) ≤ wdom(H).

Proof: Let D be a minimum weight dominating set of H. Then there exists a set of disjoint stars in H, centered on the vertices of D and covering all the vertices. Since each of these stars becomes a clique in H 2 , I can pick at most one vertex from each of them. Thus, each vertex in I has the center of the corresponding star available as a neighbor in H. Hence, w(S) ≤ w(D). ✷ The algorithm is given below. In it, si (u) will denote a lightest neighbor of u in Gi ; for this deﬁnition, u will also be considered a neighbor of itself.

5.2

The weighted version

51

Algorithm 5.10 (Metric weighted k-center) 1. 2. 3. 4. 5. Construct G2 , G2 , . . . , G2 . m 1 2 Compute a maximal independent set, Mi , in each graph G2 . i Compute Si = {si (u)| u ∈ Mi }. Find the minimum index i such that w(Si ) ≤ W , say j. Return Sj .

Theorem 5.11 Algorithm 5.10 achieves an approximation factor of 3 for the weighted k-center problem. Proof: By Lemma 5.9, cost(ej ) is a lower bound on OPT; the argument is identical to that in Lemma 5.4 and is omitted here. Since Mj is a dominating set in G2 , we can cover V with stars of G2 centered in vertices of Mj . By the j j triangle inequality these stars use edges of cost at most 2 · cost(ej ). .r .. .. ❏ .. ❏ .. r. . .. . ..❏ ✦r .. ≤ 2c(ei ) ✦✦... .... .. ❏ .... .. .... ✦ ..... .... .. ❏ . ✦✦ ..... .... .. .. .. .... u ✦✦ .. ✭r .... .... .. ❏✦ ✭✭✭ .... .. .... ✭ .. ≤ c(ei ) ...... ✭✭.✭. . .. ✭. r.....✭✭ . . .. ... .. .... ......... .... ≤ 3c(ei ) .... ... .. ......... .... .. ......... ..... . ......... .... .. ....... ............... ... .. . ......r . si (u) Each star center is adjacent to a vertex in Sj , using an edge of cost at most cost(ej ). Move each of the centers to the adjacent vertex in Sj and redeﬁne the stars. Again, by the triangle inequality, the largest edge cost used in constructing the ﬁnal stars is at most 3 · cost(ej ). ✷ Example 5.12 A tight example is provided by the following graph on n + 4 vertices. Vertex weights and edge costs are as marked; all missing edges have a cost given by the shortest path. ∞ r ❅ ❅ 1+ε ❅ ∞ r ❅ b c d a 1 1 1 ❅ r r r 1 + ε ❅r . 1 2 2 2 . . + ε 1 ∞ r

52

5

k-Center

It is not diﬃcult to see that for W = 3 the optimum cost of a k-center is 1 + ε: a k-center achieving this cost is {a, c}. For any i < n + 3, the set Si computed by the algorithm will contain a vertex of inﬁnite weight. Suppose that, for i = n + 3, the algorithm chooses Mn+3 = {b} as a maximal independent set. Then Sn+3 = {a}, and this is the output of the algorithm. The cost of this solution is 3. ✷

5.3 Exercises

5.1 Show that if the edge costs do not satisfy the triangle inequality, then the k-center problem cannot be approximated within factor α(n) for any computable function α(n). Hint: Put together ideas from Theorems 3.6 and 5.7. 5.2 Consider Step 2 of Algorithm 5.3, in which a maximal independent set is found in G2 . Perhaps a more natural choice would have been to ﬁnd a i minimal dominating set. Modify Algorithm 5.3 so that Mi is picked to be a minimal dominating set in G2 . Show that this modiﬁed algorithm does i not achieve an approximation guarantee of 2 for the k-center problem. What approximation factor can you establish for this algorithm? Hint: With this modiﬁcation, the lower bounding method does not work, since Lemma 5.2 does not hold if I is picked to be a minimal dominating set in H 2 . 5.3 (Gonzalez [111]) Consider the following problem. Problem 5.13 (Metric k-cluster) Let G = (V, E) be a complete undirected graph with edge costs satisfying the triangle inequality, and let k be a positive integer. The problem is to partition V into sets V1 , . . . , Vk so as to minimize the costliest edge between two vertices in the same set, i.e., minimize

1≤i≤k, u,v∈Vi

max

cost(u, v).

1. Give a factor 2 approximation algorithm for this problem, together with a tight example. 2. Show that this problem cannot be approximated within a factor of 2 − ε, for any ε > 0, unless P = NP. 5.4 (Khuller, Pless, and Sussmann [169]) The fault-tolerant version of the metric k-center problem has an additional input, α ≤ k, which speciﬁes the

5.4

Notes

53

number of centers that each city should be connected to. The problem again is to pick k centers so that the length of the longest edge used is minimized. A set S ⊆ V in an undirected graph H = (V, E) is an α-dominating set if each vertex v ∈ V is adjacent to at least α vertices in S (assuming that a vertex is adjacent to itself). Let domα (H) denote the size of a minimum cardinality α-dominating set in H. 1. Let I be an independent set in H 2 . Show that α|I| ≤ domα (H). 2. Give a factor 3 approximation algorithm for the fault-tolerant k-center problem. Hint: Compute a maximal independent set Mi in G2 , for 1 ≤ i ≤ m. i k Find the smallest index i such that |Mi | ≤ α , and moreover, the degree of each vertex of Mi in Gi is ≥ α − 1. 5.5 (Khuller, Pless, and Sussmann [169]) Consider a modiﬁcation of the problem of Exercise 5.4 in which vertices of S have no connectivity requirements and only vertices of V −S have connectivity requirements. Each vertex of V − S needs to be connected to α vertices in S. The object again is to pick S, |S| = k, so that the length of the longest edge used is minimized. The algorithm for this problem works on each graph Gi . It starts with Si = ∅. Vertex v ∈ V − Si is said to be j-connected if it is adjacent to j vertices in Si , using edges of G2 . While there is a vertex v ∈ V − Si that is i not k-connected, pick the vertex with minimum connectivity, and include it in Si . Finally, ﬁnd the minimum index i such that |Si | ≤ k, say l. Output Sl . Prove that this is a factor 2 approximation algorithm.

5.4 Notes

Both k-center algorithms presented in this chapter are due to Hochbaum and Shmoys [127], and Theorem 5.7 is due to Hsu and Nemhauser [132].

6 Feedback Vertex Set

In this chapter we will use the technique of layering, introduced in Chapter 2, to obtain a factor 2 approximation algorithm for: Problem 6.1 (Feedback vertex set) Given an undirected graph G = (V, E) and a function w assigning nonnegative weights to its vertices, ﬁnd a minimum weight subset of V whose removal leaves an acyclic graph.

6.1 Cyclomatic weighted graphs

Order the edges of G in an arbitrary order. The characteristic vector of a simm ple cycle C in G is a vector in GF[2] , m = |E|, which has 1’s in components corresponding to edges of C and 0’s in the remaining components. The cycle m space of G is the subspace of GF[2] that is spanned by the characteristic vectors of all simple cycles of G, and the cyclomatic number of G, denoted cyc(G), is the dimension of this space. Theorem 6.2 cyc(G) = |E| − |V | + κ(G), where κ(G) denotes the number of connected components of G. Proof: The cycle space of a graph is the direct sum of the cycle spaces of its connected components, and so its cyclomatic number is the sum of the cyclomatic numbers of its connected components. Therefore, it is suﬃcient to prove the theorem for a connected graph G. Let T be a spanning tree in G. For each nontree edge e, deﬁne its fundamental cycle to be the unique cycle formed in T ∪{e}. The set of characteristic vectors of all such cycles is linearly independent (each cycle includes an edge that is in no other fundamental cycle). Thus, cyc(G) ≥ |E| − |V | + 1. Each edge e of T deﬁnes a fundamental cut (S, S) in G, S ⊂ V (S and S are the vertex sets of two connected components formed by removing e from m T ). Deﬁne the characteristic vector of a cut to be a vector in GF[2] that has 1’s in components corresponding to the edges of G in the cut and 0’s in the remaining components. Consider the |V | − 1 vectors deﬁned by edges of T . Since each cycle must cross each cut an even number of times, these vectors are orthogonal to the cycle space of G. Furthermore, these |V | − 1

6.1

Cyclomatic weighted graphs

55

vectors are linearly independent, since each cut has an edge (the tree edge deﬁning this cut) that is not in any of the other |V | − 2 cuts. Therefore the dimension of the orthogonal complement to the cycle space is at least |V | − 1. Hence, cyc(G) ≤ |E| − |V | + 1. Combining with the previous inequality we get cyc(G) = |E| − |V | + 1. ✷ Denote by δG (v) the decrease in the cyclomatic number of the graph on removing vertex v. Since the removal of a feedback vertex set F = {v1 , . . . , vf } decreases the cyclomatic number of G down to 0, f cyc(G) = i=1 δGi−1 (vi ),

where G0 = G and, for i > 0, Gi = G − {v1 , . . . , vi }. By Lemma 6.4 below, we get: cyc(G) ≤ v∈F δG (v).

(6.1)

Let us say that a function assigning vertex weights is cyclomatic if there is a constant c > 0 such that the weight of each vertex v is c · δG (v). By inequality (6.1), for such a weight function, c · cyc(G) is a lower bound on OPT. The importance of cyclomatic weight functions is established in Lemma 6.5 below, which shows that for such a weight function, any minimal feedback vertex set has a weight within twice the optimal. Let degG (v) denote the degree of v in G, and comps(G − v) denote the number of connected components formed by removing v from G. The claim below follows in a straightforward way by applying Theorem 6.2 to G and G − v. Claim 6.3 For a connected graph G, δG (v) = degG (v) − comps(G − v). Lemma 6.4 Let H be a subgraph of G (not necessarily vertex induced). Then, δH (v) ≤ δG (v). Proof: It is suﬃcient to prove the lemma for the connected components of G and H containing v. We may thus assume w.l.o.g. that G and H are connected (H may be on a smaller set of vertices). By Claim 6.3, proving the following inequality is suﬃcient: degH (v) − comps(H − v) ≤ degG (v) − comps(G − v). We will show that edges in G − H can only help this inequality. Let c1 , c2 , . . . , ck be components formed by removing v from H. Edges of G − H not incident at v can only help merge some of these components (and of

56

6

Feedback Vertex Set

course, they don’t change the degree of v). An edge of G − H that is incident at v can lead to an additional component, but this is compensated by the contribution the edge has to the degree of v. ✷ Lemma 6.5 If F is a minimal feedback vertex set of G, then δG (v) ≤ 2 · cyc(G). v∈F Proof: Since the cycle space of G is the direct sum of the cycle spaces of its connected components, it suﬃces to prove the lemma for a connected graph G. Let F = {v1 , . . . , vf }, and let k be the number of connected components obtained by deleting F from G. Partition these components into two types: those that have edges incident to only one of the vertices of F , and those that have edges incident to two or more vertices of F . Let t and k − t be the number of components of the ﬁrst and second type, respectively. We will prove that f f

δG (vi ) = i=1 i=1

(degG (vi ) − comps(G − vi )) ≤ 2(|E| − |V |), f i=1

thereby proving the lemma. Clearly, we are left to prove f comps(G − vi ) = f + t. Therefore,

degG (vi ) ≤ 2(|E| − |V |) + f + t. i=1 v1

v2

...

vk

... comp. 1 comp. 2 comp. k

Since F is a feedback vertex set, each of the k components is acyclic and is therefore a tree. Thus, the number of edges in these components is |V |−f −k. Next, we put a lower bound on the number of edges in the cut (F, V − F ).

6.2

Layering applied to feedback vertex set

57

Since F is minimal, each vi ∈ F must be in a cycle that contains no other vertices of F . Therefore, each vi must have at least two edges incident at one of the components. For each vi , arbitrarily remove one of these edges from G, thus removing a total of f edges. Now, each of the t components must still have at least one edge and each of the k − t components must still have at least two edges incident at F . Therefore, the number of edges in the cut (F, V − F ) is at least f + t + 2(k − t) = f + 2k − t. These two facts imply that f degG (vi ) ≤ 2|E| − 2(|V | − f − k) − (f + 2k − t). i=1 The lemma follows.

✷

Corollary 6.6 Let w be a cyclomatic weight function on the vertices of G, and let F be a minimal feedback vertex set in it. Then w(F ) ≤ 2 · OPT.

6.2 Layering applied to feedback vertex set

Let us now deal with arbitrary weighted graphs. Consider the following basic operation: Given graph G = (V, E) and a weight function w, let c = min v∈V w(v) δG (v)

.

The weight function t(v) = cδG (v) is the largest cyclomatic weight function in w. Deﬁne w (v) = w(v) − t(v) to be the residual weight function. Finally, let V be the set of vertices having positive residual weight (clearly, V ⊂ V ), and let G be the subgraph of G induced on V . Using this basic operation, decompose G into a nested sequence of induced subgraphs, until an acyclic graph is obtained, each time ﬁnding the largest cyclomatic weight function in the current residual weight function. Let these graphs be G = G0 ⊃ G1 ⊃ · · · ⊃ Gk , where Gk is acyclic; Gi is the induced subgraph of G on vertex set Vi , where V = V0 ⊃ V1 ⊃ · · · ⊃ Vk . Let ti , i = 0, . . . , k − 1 be the cyclomatic weight function deﬁned on graph Gi . Thus, w0 = w is the residual weight function for G0 , t0 is the largest cyclomatic weight function in w0 , w1 = w0 − t0 is the residual weight function for G1 , and so on. Finally, wk is the residual weight function for Gk . For convenience, deﬁne tk = wk . Since the weight of a vertex v has been decomposed into the weights t0 , t1 , . . . , tk , we have ti (v) = w(v). i: v∈Vi

58

6

Feedback Vertex Set

The next fact suggests an algorithm for constructing a feedback vertex set on which Lemma 6.5 can be applied. Lemma 6.7 Let H be a subgraph of G = (V, E), induced on vertex set V ⊂ V . Let F be a minimal feedback vertex set in H, and let F ⊆ V − V be a minimal set such that F ∪ F is a feedback vertex set for G. Then F ∪ F is a minimal feedback vertex set for G. Proof: Since F is minimal for H, for each v ∈ F , there is a cycle, say C, in H that does not use any other vertex of F . Since F ∩ V = ∅, C uses only ✷ one vertex, v, from F ∪ F as well, and so v is not redundant. After the entire decomposition, Fk = ∅ is a minimal feedback vertex set of Gk . For i = k, k − 1, . . . , 1, the minimal feedback vertex set Fi found in Gi is extended in a minimal way using vertices of Vi−1 − Vi to yield a minimal feedback vertex set, say Fi−1 , for Gi−1 . The last set, F0 , is a feedback vertex set for G.

Gk Gk-1

...

Fk-1

G1 G0

F - F2 1 F0 - F1

Algorithm 6.8 (Feedback vertex set) 1. Decomposition phase H ← G, w ← w, i ← 0 While H is not acyclic, w c ← minu∈H δH(u) (u)

Gi ← H, ti ← c · δGi , w ← w − ti H ← the subgraph of Gi induced by vertices u with w (u) > 0 i ← i + 1, k ← i, Gk ← H 2. Extension phase Fk ← ∅ For i = k, . . . , 1, extend Fi to a feedback vertex set Fi−1 of Gi−1 by adding a minimal set of vertices from Vi−1 − Vi . Output F0 .

6.2

Layering applied to feedback vertex set

59

Theorem 6.9 Algorithm 6.8 achieves an approximation guarantee of factor 2 for the feedback vertex set problem. Proof: Let F ∗ be an optimal feedback vertex set for G. Since Gi is an induced subgraph of G, F ∗ ∪ Vi must be a feedback vertex set for Gi (not necessarily optimal). Since the weights of vertices have been decomposed into the functions ti , we have OPT = w(F ∗ ) = k i=0

ti (F ∗ ∩ Vi ) ≥

k

OPTi , i=0 where OPTi is the weight of an optimal feedback vertex set of Gi with weight function ti . By decomposing the weight of F0 , we get k k

w(F0 ) = i=0 ti (F0 ∩ Vi ) = i=0 ti (Fi ).

By Lemma 6.7, Fi is a minimal feedback vertex set in Gi . Since for 0 ≤ i ≤ k − 1, ti is a cyclomatic weight function, by Lemma 6.5, ti (Fi ) ≤ 2OPTi ; recall that Fk = ∅. Therefore, k w(F0 ) ≤ 2 i=0 OPTi ≤ 2 · OPT. ✷

Example 6.10 A tight example for the algorithm is given by the graph obtained by removing a perfect matching from a complete bipartite graph and duplicating every edge. (Note that the algorithm works for parallel edges as well. If a tight example without parallel edges is desired, then a vertex with very high weight can be placed on every edge.) s ✏s ◗ ✑ ❙ ◗ ✏✏✏ ✏✑✑ ❙ ◗✏✑ ✏ s ✏ ❙ s ✏ ◗✑ ✏ ◗ ❅ ✑ ◗✏✏ ❙✏✏ ✑✏◗ ❅ ✑✏❙ ◗ ◗ ✏ ✑ s ✏❅ ❙ s ◗ ✑ ❙ ✑ ◗ ❅ ✑ ◗ ❅✑ ❙ ◗ . . . . . ✑✑❅❙ . ◗◗❅ ❙ ✑ ◗❅ ✑ ❙ ◗❙s ❅ ◗ ✑ s

..................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....................................................................................................

60

6

Feedback Vertex Set

Assuming that the graph is cyclomatic weighted, each vertex receives the same weight. The decomposition obtained by the algorithm consists of only one nontrivial graph, G itself, on which the algorithm computes a minimal feedback vertex set. A possible output of the algorithm is the set shown above; this set contains 2n − 2 vertices as compared with the optimum of n given by one side of the bipartition. ✷

6.3 Exercises

6.1 A natural greedy algorithm for ﬁnding a minimum feedback vertex set is to repeatedly pick and remove the most cost-eﬀective vertex, i.e., a vertex minimizing w(v)/δH (v), where H is the current graph, until there are no more cycles left. Give examples to show that this is not a constant factor algorithm. What is the approximation guarantee of this algorithm? 6.2 Give an approximation factor preserving reduction from the vertex cover problem to the feedback vertex set problem (thereby showing that improving the factor for the latter problem will also improve it for the former; also see Section 30.1).

6.4 Notes

Algorithm 6.8 is due to Bafna, Berman, and Fujito [19] (see also Becker and Geiger [23] and Chudak, Goemans, Hochbaum, and Williamson [46] for other factor 2 algorithms for the feedback vertex set problem).

7 Shortest Superstring

In Chapter 2 we deﬁned the shortest superstring problem (Problem 2.9) and gave a preliminary approximation algorithm using set cover. In this chapter, we will ﬁrst give a factor 4 algorithm, and then we will improve this to factor 3.

7.1 A factor 4 algorithm

We begin by developing a good lower bound on OPT. Let us assume that s1 , s2 , . . . , sn are numbered in order of leftmost occurrence in the shortest superstring, s. pref(s1 , s2 ) ✛ ✲ pref(sn−1 , sn )pref(sn , s✲✛ 1 ) over(sn , s1 ) ✛ ✲✛ ✲

s s1

s2

..

. sn−1 sn s1

Let overlap(si , sj ) denote the maximum overlap between si and sj , i.e., the longest suﬃx of si that is a preﬁx of sj . Also, let preﬁx(si , sj ) be the preﬁx of si obtained by removing its overlap with sj . The overlap in s between two consecutive si ’s is maximum possible, because otherwise a shorter superstring can be obtained. Hence, assuming that no si is a substring of another, we get OPT = |preﬁx(s1 , s2 )| + |preﬁx(s2 , s3 )| + . . . + |preﬁx(sn , s1 )| + |overlap(sn , s1 )|. (7.1)

Notice that we have repeated s1 at the end in order to obtain the last two terms of (7.1). This equality shows the close relation between the shortest

62

7

Shortest Superstring

superstring of S and the minimum traveling salesman tour on the preﬁx graph of S, deﬁned as the directed graph on vertex set {1, . . . , n} that contains an edge i → j of weight |preﬁx(si , sj )| for each i, j (self loops included). Clearly, |preﬁx(s1 , s2 )| + |preﬁx(s2 , s3 )| + . . . + |preﬁx(sn , s1 )| represents the weight of the tour 1 → 2 → . . . → n → 1. Hence, by (7.1), the minimum weight of a traveling salesman tour of the preﬁx graph gives a lower bound on OPT. As such, this lower bound is not very useful, since we cannot eﬃciently compute a minimum traveling salesman tour. The key idea is to lower-bound OPT using the minimum weight of a cycle cover of the preﬁx graph (a cycle cover is a collection of disjoint cycles covering all vertices). Since the tour 1 → 2 → . . . → n → 1 is a cycle cover, from (7.1) we get that the minimum weight of a cycle cover lower-bounds OPT. Unlike minimum TSP, a minimum weight cycle cover can be computed in polynomial time. Corresponding to the preﬁx graph, construct the following bipartite graph, H. U = {u1 , . . . , un } and V = {v1 , . . . , vn } are the vertex sets of the two sides of the bipartition. For each i, j ∈ {1, . . . , n} add edge (ui , vj ) of weight |preﬁx(si , sj )|. It is easy to see that each cycle cover of the preﬁx graph corresponds to a perfect matching of the same weight in H and vice versa. Hence, ﬁnding a minimum weight cycle cover reduces to ﬁnding a minimum weight perfect matching in H. If c = (i1 → i2 → . . . il → i1 ) is a cycle in the preﬁx graph, let α(c) = preﬁx(si1 , si2 ) ◦ . . . ◦ preﬁx(sil−1 , sil ) ◦ preﬁx(sil , si1 ). Notice that each string si1 , si2 , . . . , sil is a substring of (α(c))∞ . Next, let σ(c) = α(c) ◦ si1 . Then σ(c) is a superstring of si1 , . . . , sil .1 In the above construction, we “opened” cycle c at an arbitrary string si1 . For the rest of the algorithm, we will call si1 the representative string for c. We can now state the complete algorithm: Algorithm 7.1 (Shortest superstring – factor 4) 1. Construct the preﬁx graph corresponding to strings in S. 2. Find a minimum weight cycle cover of the preﬁx graph, C = {c1 , . . . , ck }. 3. Output σ(c1 ) ◦ . . . ◦ σ(ck ).

1

This remains true even for the shorter string α(c) ◦ overlap(sl , s1 ). We will work with σ(c), since it will be needed for the factor 3 algorithm presented in the next section, where we use the property that σ(c) begins and ends with a copy of si1 .

7.1

A factor 4 algorithm

63

Clearly, the output is a superstring of the strings in S. Notice that if in each of the cycles we can ﬁnd a representative string of length at most the weight of the cycle, then the string output is within 2 · OPT. Thus, the hard case is when all strings of some cycle c are long. But since they must all be substrings of (α(c))∞ , they must be periodic. This will be used to prove Lemma 7.3, which establishes another lower bound on OPT. Lemma 7.2 If each string in S ⊆ S is a substring of t∞ for a string t, then there is a cycle of weight at most |t| in the preﬁx graph covering all the vertices corresponding to strings in S . Proof: For each string in S , locate the starting point of its ﬁrst occurrence in t∞ . Clearly, all these starting points will be distinct (since no string in S is a substring of another) and will lie in the ﬁrst copy of t. Consider the cycle in the preﬁx graph visiting the corresponding vertices in this order. Clearly, the weight of this cycle is at most |t|. ✷ Lemma 7.3 Let c and c be two cycles in C, and let r, r be representative strings from these cycles. Then |overlap(r, r )| < wt(c) + wt(c ). Proof: Suppose, for contradiction, that |overlap(r, r )| ≥ wt(c) + wt(c ). Denote by α (α ) the preﬁx of length wt(c) (wt(c ), respectively) of overlap(r, r ). ✛ r ✛ r α overlap(r, r ) α α◦α =α ◦α α α ✲ α ✲

Clearly, overlap(r, r ) is a preﬁx of both α∞ and (α )∞ . In addition, α is a preﬁx of (α )∞ and α is a preﬁx of α∞ . Since overlap(r, r ) ≥ |α| + |α |, it follows that α and α commute, i.e., α ◦ α = α ◦ α. But then, α∞ = (α )∞ . This is so because for any k > 0, αk ◦ (α )k = (α )k ◦ αk . Hence, for any N > 0, the preﬁx of length N of α∞ is the same as that of (α )∞ . Now, by Lemma 7.2, there is a cycle of weight at most wt(c) in the preﬁx graph covering all strings in c and c , contradicting the fact that C is a minimum weight cycle cover. ✷

64

7

Shortest Superstring

Theorem 7.4 Algorithm 7.1 achieves an approximation factor of 4 for the shortest superstring problem. Proof: Let wt(C) = k k i=1

wt(ci ). The output of the algorithm has length k |σ(ci )| = wt(C) + i=1 i=1

|ri |,

where ri denotes the representative string from cycle ci . We have shown that wt(C) ≤ OPT. Next, we show that the sum of the lengths of representative strings is at most 3 · OPT. Assume that r1 , . . . , rk are numbered in order of their leftmost occurrence in the shortest superstring of S. Using Lemma 7.3, we get the following lower bound on OPT: k k−1 k k

OPT ≥ i=1 |ri | − i=1 |overlap(ri , ri+1 )| ≥ i=1 |ri | − 2 i=1 wt(ci ).

Hence, k k

|ri | ≤ OPT + 2 i=1 i=1

wt(ci ) ≤ 3 · OPT. ✷

7.2 Improving to factor 3

Notice that any superstring of the strings σ(ci ), i = 1, . . . , k, is also a superstring of all strings in S. Instead of simply concatenating these strings, let us make them overlap as much as possible (this may sound circular, but it is not!). Let X be a set of strings. We will denote by ||X|| the sum of the lengths of the strings in X. Let us deﬁne the compression achieved by a superstring s as the diﬀerence between the sum of the lengths of the input strings and |s|, i.e., ||S|| − |s|. Clearly, maximum compression is achieved by the shortest superstring. Several algorithms are known to achieve at least half the optimal compression. For instance, the greedy superstring algorithm, described in Section 2.3, does so; however, its proof is based on a complicated case analysis. For a less eﬃcient algorithm, see Section 7.2.1. Either of these algorithms can be used in Step 3 of Algorithm 7.5.

7.2

Improving to factor 3

65

Algorithm 7.5 (Shortest superstring – factor 3) 1. Construct the preﬁx graph corresponding to strings in S. 2. Find a minimum cycle cover of the preﬁx graph, C = {c1 , . . . , ck }. 3. Run the greedy superstring algorithm on {σ(c1 ), . . . , σ(ck )} and output the resulting string, say τ . Let OPTσ denote the length of the shortest superstring of the strings in Sσ = {σ(c1 ) . . . σ(ck )}, and let ri be the representative string of ci . Lemma 7.6 |τ | ≤ OPTσ + wt(C).

Proof: Assume w.l.o.g. that σ(c1 ), . . . , σ(ck ) appear in this order in a shortest superstring of Sσ . The maximum compression that can be achieved on Sσ is given by k−1 |overlap(σ(ci ), σ(ci+1 ))|. i=1 Since each string σ(ci ) has ri as a preﬁx as well as suﬃx, by Lemma 7.3, |overlap(σ(ci ), σ(ci+1 ))| ≤ wt(ci ) + wt(ci+1 ). Hence, the maximum compression achievable on Sσ is at most 2 · wt(C), i.e., ||Sσ || − OPTσ ≤ 2 · wt(C). The compression achieved by the greedy superstring algorithm on Sσ is at least half the maximum compression. Therefore, ||Sσ || − |τ | ≥ Therefore, 2(|τ | − OPTσ ) ≤ ||Sσ || − OPTσ ≤ 2 · wt(C). The lemma follows. Finally, we relate OPTσ to OPT. Lemma 7.7 OPTσ ≤ OPT + wt(C). ✷ 1 (||Sσ || − OPTσ ). 2

Proof: Let OPTr denote the length of the shortest superstring of the strings in Sr = {r1 , . . . , rk }. The key observation is that each σ(ci ) begins and ends with ri . Therefore, the maximum compression achievable on Sσ is at least as large as that achievable on Sr , i.e.,

66

7

Shortest Superstring

||Sσ || − OPTσ ≥ ||Sr || − OPTr . Clearly, ||Sσ || = ||Sr || + wt(C). This gives OPTσ ≤ OPTr + wt(C). The lemma follows by noticing that OPTr ≤ OPT. Combining the previous two lemmas we get: Theorem 7.8 Algorithm 7.5 achieves an approximation factor of 3 for the shortest superstring problem. 7.2.1 Achieving half the optimal compression ✷

We give a superstring algorithm that achieves at least half the optimal compression. Suppose that the strings to be compressed, s1 , · · · , sn , are numbered in the order in which they appear in a shortest superstring. Then, the optimal compression is given by n−1 |overlap(σi , σi+1 )|. i=1 This is the weight of the traveling salesman path 1 → 2 → . . . → n in the overlap graph, H, of the strings s1 , · · · , sn . H is a directed graph that has a vertex vi corresponding to each string si , and contains an edge (vi → vj ) of weight |overlap(si , sj )| for each i = j, 1 ≤ i, j ≤ n (H has no self loops). The optimal compression is upper bounded by the cost of a maximum traveling salesman tour in H, which in turn is upper bounded by the cost of a maximum cycle cover. The latter can be computed in polynomial time using matching, similar to the way we computed a minimum weight cycle cover. Since H has no self loops, each cycle has length at least 2. Remove the lightest edge from each cycle of the maximum cycle cover to obtain a set of disjoint paths. The sum of weights of edges on these paths is at least half the optimal compression. Overlap strings s1 , · · · , sn according to the edges of these paths and concatenate the resulting strings. This gives a superstring achieving at least half the optimal compression.

7.3 Exercises

7.1 Show that Lemma 7.3 cannot be strengthened to |overlap(r, r )| < max {wt(c), wt(c )}.

7.4

Notes

67

7.2 (Jiang, Li, and Du [148]) Obtain constant factor approximation algorithms for the variants of the shortest superstring problem given in Exercise 2.16.

7.4 Notes

The algorithms given in this chapter are due to Blum, Jiang, Li, Tromp, and Yannakakis [27].

8 Knapsack

In Chapter 1 we mentioned that some NP-hard optimization problems allow approximability to any required degree. In this chapter, we will formalize this notion and will show that the knapsack problem admits such an approximability. Let Π be an NP-hard optimization problem with objective function fΠ . We will say that algorithm A is an approximation scheme for Π if on input (I, ε), where I is an instance of Π and ε > 0 is an error parameter, it outputs a solution s such that: • fΠ (I, s) ≤ (1 + ε) · OPT if Π is a minimization problem. • fΠ (I, s) ≥ (1 − ε) · OPT if Π is a maximization problem. A will be said to be a polynomial time approximation scheme, abbreviated PTAS, if for each ﬁxed ε > 0, its running time is bounded by a polynomial in the size of instance I. The deﬁnition given above allows the running time of A to depend arbitrarily on ε. This is rectiﬁed in the following more stringent notion of approximability. If the previous deﬁnition is modiﬁed to require that the running time of A be bounded by a polynomial in the size of instance I and 1/ε, then A will be said to be a fully polynomial approximation scheme, abbreviated FPTAS. In a very technical sense, an FPTAS is the best one can hope for an NPhard optimization problem, assuming P = NP; see Section 8.3.1 for a short discussion on this issue. The knapsack problem admits an FPTAS. Problem 8.1 (Knapsack) Given a set S = {a1 , . . . , an } of objects, with speciﬁed sizes and proﬁts, size(ai ) ∈ Z+ and proﬁt(ai ) ∈ Z+ , and a “knapsack capacity” B ∈ Z+ , ﬁnd a subset of objects whose total size is bounded by B and total proﬁt is maximized. An obvious algorithm for this problem is to sort the objects by decreasing ratio of proﬁt to size, and then greedily pick objects in this order. It is easy to see that as such this algorithm can be made to perform arbitrarily badly (Exercise 8.1).

8.2

An FPTAS for knapsack

69

8.1 A pseudo-polynomial time algorithm for knapsack

Before presenting an FPTAS for knapsack, we need one more concept. For any optimization problem Π, an instance consists of objects, such as sets or graphs, and numbers, such as cost, proﬁt, size, etc. So far, we have assumed that all numbers occurring in a problem instance I are written in binary. The size of the instance, denoted |I|, was deﬁned as the number of bits needed to write I under this assumption. Let us say that Iu will denote instance I with all numbers occurring in it written in unary. The unary size of instance I, denoted |Iu |, is deﬁned as the number of bits needed to write Iu . An algorithm for problem Π is said to be eﬃcient if its running time on instance I is bounded by a polynomial in |I|. Let us consider the following weaker deﬁnition. An algorithm for problem Π whose running time on instance I is bounded by a polynomial in |Iu | will be called a pseudo-polynomial time algorithm. The knapsack problem, being NP-hard, does not admit a polynomial time algorithm; however, it does admit a pseudo-polynomial time algorithm. This fact is used critically in obtaining an FPTAS for it. All known pseudopolynomial time algorithms for NP-hard problems are based on dynamic programming. Let P be the proﬁt of the most proﬁtable object, i.e., P = maxa∈S proﬁt(a). Then nP is a trivial upperbound on the proﬁt that can be achieved by any solution. For each i ∈ {1, . . . , n} and p ∈ {1, . . . , nP }, let Si,p denote a subset of {a1 , . . . , ai } whose total proﬁt is exactly p and whose total size is minimized. Let A(i, p) denote the size of the set Si,p (A(i, p) = ∞ if no such set exists). Clearly A(1, p) is known for every p ∈ {1, . . . , nP }. The following recurrence helps compute all values A(i, p) in O(n2 P ) time: A(i + 1, p) = min {A(i, p), size(ai+1 ) + A(i, p − proﬁt(ai+1 ))} if proﬁt(ai+1 ) < p A(i + 1, p) = A(i, p) otherwise The maximum proﬁt achievable by objects of total size bounded by B is max {p| A(n, p) ≤ B}. We thus get a pseudo-polynomial algorithm for knapsack.

8.2 An FPTAS for knapsack

Notice that if the proﬁts of objects were small numbers, i.e., they were bounded by a polynomial in n, then this would be a regular polynomial time algorithm, since its running time would be bounded by a polynomial in |I|. The key idea behind obtaining an FPTAS is to exploit precisely this fact: we will ignore a certain number of least signiﬁcant bits of proﬁts of objects

70

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Knapsack

(depending on the error parameter ε), so that the modiﬁed proﬁts can be viewed as numbers bounded by a polynomial in n and 1/ε. This will enable us to ﬁnd a solution whose proﬁt is at least (1 − ε) · OPT in time bounded by a polynomial in n and 1/ε. Algorithm 8.2 (FPTAS for knapsack) 1. Given ε > 0, let K = proﬁt(ai ) 2. For each object ai , deﬁne proﬁt (ai ) = . K 3. With these as proﬁts of objects, using the dynamic programming algorithm, ﬁnd the most proﬁtable set, say S . 4. Output S . εP n

.

Lemma 8.3 Let A denote the set output by the algorithm. Then, proﬁt(A) ≥ (1 − ε) · OPT. Proof: Let O denote the optimal set. For any object a, because of rounding down, K · proﬁt (a) can be smaller than proﬁt(a), but by not more than K. Therefore, proﬁt(O) − K · proﬁt (O) ≤ nK. The dynamic programming step must return a set at least as good as O under the new proﬁts. Therefore, proﬁt(S ) ≥ K · proﬁt (O) ≥ proﬁt(O) − nK = OPT − εP ≥ (1 − ε) · OPT , where the last inequality follows from the observation that OPT ≥ P . ✷

Theorem 8.4 Algorithm 8.2 is a fully polynomial approximation scheme for knapsack. Proof: By Lemma 8.3, the solution found is within (1 − ε) factor of OPT. P Since the running time of the algorithm is O n2 K = O n2 n , which ε is polynomial in n and 1/ε, the theorem follows. ✷

8.3

Strong NP-hardness and the existence of FPTAS’s

71

8.3 Strong NP-hardness and the existence of FPTAS’s

In this section, we will prove in a formal sense that very few of the known NPhard problems admit an FPTAS. First, here is a strengthening of the notion of NP-hardness in a similar sense in which a pseudo-polynomial algorithm is a weakening of the notion of an eﬃcient algorithm. A problem Π is strongly NP-hard if every problem in NP can be polynomially reduced to Π in such a way that numbers in the reduced instance are always written in unary. The restriction automatically forces the transducer to use polynomially bounded numbers only. Most known NP-hard problems are in fact strongly NP-hard; this includes all the problems in previous chapters for which approximation algorithms were obtained. A strongly NP-hard problem cannot have a pseudo-polynomial time algorithm, assuming P = NP (Exercise 8.4). Thus, knapsack is not strongly NP-hard, assuming P = NP. We will show below that under some very weak restrictions, any NPhard problem admitting an FPTAS must admit a pseudo-polynomial time algorithm. Theorem 8.5 is proven for a minimization problem; a similar proof holds for a maximization problem. Theorem 8.5 Let p be a polynomial and Π be an NP-hard minimization problem such that the objective function fΠ is integer valued and on any instance I, OPT(I) < p(|Iu |). If Π admits an FPTAS, then it also admits a pseudo-polynomial time algorithm. Proof: Suppose there is an FPTAS for Π whose running time on instance I and error parameter ε is q(|I|, 1/ε), where q is a polynomial. On instance I, set the error parameter to ε = 1/p(|Iu |), and run the FPTAS. Now, the solution produced will have objective function value less than or equal to: (1 + ε)OPT(I) < OPT(I) + εp(|Iu |) = OPT(I) + 1. In fact, with this error parameter, the FPTAS will be forced to produce an optimal solution. The running time will be q(|I|, p(|Iu |)), i.e., polynomial in |Iu |. Therefore, we have obtained a pseudo-polynomial time algorithm for Π. ✷ The following corollary applies to most known NP-hard problems. Corollary 8.6 Let Π be an NP-hard optimization problem satisfying the restrictions of Theorem 8.5. If Π is strongly NP-hard, then Π does not admit an FPTAS, assuming P = NP. Proof: If Π admits an FPTAS, then it admits a pseudo-polynomial time algorithm by Theorem 8.5. But then it is not strongly NP-hard, assuming P = NP, leading to a contradiction. ✷

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Knapsack

The stronger assumption that OPT < p(|I|) in Theorem 8.5 would have enabled us to prove that there is a polynomial time algorithm for Π. However, this stronger assumption is less widely applicable. For instance, it is not satisﬁed by the minimum makespan problem, which we will study in Chapter 10. 8.3.1 Is an FPTAS the most desirable approximation algorithm?

The design of almost all known FPTAS’s and PTAS’s is based on the idea of trading accuracy for running time – the given problem instance is mapped to a coarser instance, depending on the error parameter ε, which is solved optimally by a dynamic programming approach. The latter ends up being an exhaustive search of polynomially many diﬀerent possibilities (for instance, for knapsack, this involves computing A(i, p) for all i and p). In most such algorithms, the running time is prohibitive even for reasonable n and ε. Further, if the algorithm had to resort to exhaustive search, does the problem really oﬀer “footholds” to home in on a solution eﬃciently? Is an FPTAS or PTAS the best one can hope for for an NP-hard problem? Clearly, the issue is complex and there is no straightforward answer.

8.4 Exercises

8.1 Consider the greedy algorithm for the knapsack problem. Sort the objects by decreasing ratio of proﬁt to size, and then greedily pick objects in this order. Show that this algorithm can be made to perform arbitrarily badly. 8.2 Consider the following modiﬁcation to the algorithm given in Exercise 8.1. Let the sorted order of objects be a1 , . . . , an . Find the lowest k such that the size of the ﬁrst k objects exceeds B. Now, pick the more proﬁtable of {a1 , . . . , ak−1 } and {ak } (we have assumed that the size of each object is at most B). Show that this algorithm achieves an approximation factor of 2. 8.3 (Bazgan, Santha, and Tuza [22]) Obtain an FPTAS for the following problem. Problem 8.7 (Subset-sum ratio problem) Given n positive integers, a1 < . . . < an , ﬁnd two disjoint nonempty subsets S1 , S2 ⊆ {1, . . . , n} with i∈S1 ai ≥ i∈S2 ai , such that the ratio ai i∈S2 ai i∈S1 is minimized.

8.5

Notes

73

Hint: First, obtain a pseudo-polynomial time algorithm for this problem. Then, scale and round appropriately. 8.4 Show that a strongly NP-hard problem cannot have a pseudo-polynomial time algorithm, assuming P = NP.

8.5 Notes

Algorithm 8.2 is due to Ibarra and Kim [134]. Theorem 8.5 is due to Garey and Johnson [92].

9 Bin Packing

Consider the following problem. Problem 9.1 (Bin packing) Given n items with sizes a1 , . . . , an ∈ (0, 1], ﬁnd a packing in unit-sized bins that minimizes the number of bins used. This problem ﬁnds many industrial applications. For instance, in the stock-cutting problem, bins correspond to a standard length of paper and items correspond to speciﬁed lengths that need to be cut. It is easy to obtain a factor 2 approximation algorithm for this problem. For instance, let us consider the algorithm called First-Fit. This algorithm considers items in an arbitrary order. In the ith step, it has a list of partially packed bins, say B1 , . . . , Bk . It attempts to put the next item, ai , in one of these bins, in this order. If ai does not ﬁt into any of these bins, it opens a new bin Bk+1 , and puts ai in it. If the algorithm uses m bins, then at least m − 1 bins are more than half full. Therefore, n ai > i=1 m−1 . 2

Since the sum of the item sizes is a lower bound on OPT, m − 1 < 2OPT, i.e., m ≤ 2OPT (see Notes for a better analysis). On the negative side: Theorem 9.2 For any ε > 0, there is no approximation algorithm having a guarantee of 3/2 − ε for the bin packing problem, assuming P = NP. Proof: If there were such an algorithm, then we show how to solve the NPhard problem of deciding if there is a way to partition n nonnegative numbers a1 , . . . , an into two sets, each adding up to 1 i ai . Clearly, the answer to 2 this question is ‘yes’ iﬀ the n items can be packed in 2 bins of size 1 i ai . If 2 the answer is ‘yes’ the 3/2 − ε factor algorithm will have to give an optimal packing, and thereby solve the partitioning problem. ✷

9.1 An asymptotic PTAS

Notice that the argument in Theorem 9.2 uses very special instances: those for which OPT is a small number, such as 2 or 3, even though the number

9.1

An asymptotic PTAS

75

of items is unbounded. What can we say about “typical” instances, those for which OPT increases with n? Theorem 9.3 For any ε, 0 < ε ≤ 1/2, there is an algorithm Aε that runs in time polynomial in n and ﬁnds a packing using at most (1 + 2ε)OPT + 1 bins. The sequence of algorithms, Aε , form an asymptotic polynomial time approximation scheme for bin packing, since for each ε > 0 ∃N > 0, and a polynomial time algorithm in this sequence, say B, such that B has an approximation guarantee of 1 + ε for all instances having OPT ≥ N . However, Theorem 9.3 should not be considered a practical solution to the bin packing problem, since the running times of the algorithms Aε are very high. We will prove Theorem 9.3 in three steps. Lemma 9.4 Let ε > 0 be ﬁxed, and let K be a ﬁxed nonnegative integer. Consider the restriction of the bin packing problem to instances in which each item is of size at least ε and the number of distinct item sizes is K. There is a polynomial time algorithm that optimally solves this restricted problem. Proof: The number of items in a bin is bounded by 1/ε . Denote this by +K M . Therefore, the number of diﬀerent bin types is bounded by R = MM (see Exercise 9.4), which is a (large!) constant. Clearly, the total number of bins used is at most n. Therefore, the number of possible feasible packings is bounded by P = n+R , which is polynomial in n (see Exercise 9.4). R Enumerating them and picking the best packing gives the optimal answer. ✷ Lemma 9.5 Let ε > 0 be ﬁxed. Consider the restriction of the bin packing problem to instances in which each item is of size at least ε. There is a polynomial time approximation algorithm that solves this restricted problem within a factor of (1 + ε). Proof: Let I denote the given instance. Sort the n items by increasing size, and partition them into K = 1/ε2 groups each having at most Q = nε2 items. Notice that two groups may contain items of the same size.

J

J’

Construct instance J by rounding up the size of each item to the size of the largest item in its group. Instance J has at most K diﬀerent item sizes.

76

9

Bin Packing

Therefore, by Lemma 9.4, we can ﬁnd an optimal packing for J. Clearly, this will also be a valid packing for the original item sizes. We show below that OPT(J) ≤ (1 + ε)OPT(I), thereby proving the lemma. The following clever argument accomplishes this. Let us construct another instance, say J , by rounding down the size of each item to that of the smallest item in its group. Clearly OPT(J ) ≤ OPT(I). The crucial observation is that a packing for instance J yields a packing for all but the largest Q items of instance J (Exercise 9.6 asks for a formal proof). Therefore, OPT(J) ≤ OPT(J ) + Q ≤ OPT(I) + Q. Since each item in I has size at least ε, OPT(I) ≥ nε. Therefore, Q = nε2 ≤ εOPT. Hence, OPT(J) ≤ (1 + ε)OPT(I). ✷ Proof of Theorem 9.3: Let I denote the given instance, and I denote the instance obtained by discarding items of size < ε from I. By Lemma 9.5, we can ﬁnd a packing for I using at most (1 + ε)OPT(I ) bins. Next, we start packing the small items (of size < ε) in a First-Fit manner in the bins opened for packing I . Additional bins are opened if an item does not ﬁt into any of the already open bins. If no additional bins are needed, then we have a packing in (1+ε)OPT(I ) ≤ (1 + ε)OPT(I) bins. In the second case, let M be the total number of bins used. Clearly, all but the last bin must be full to the extent of at least 1 − ε. Therefore, the sum of the item sizes in I is at least (M − 1)(1 − ε). Since this is a lower bound on OPT, we get M≤ OPT + 1 ≤ (1 + 2ε)OPT + 1, (1 − ε)

where we have used the assumption that ε ≤ 1/2. Hence, for each value of ε, 0 < ε ≤ 1/2, we have a polynomial time algorithm achieving a guarantee of (1 + 2ε)OPT + 1. ✷ Algorithm Aε is summarized below. Algorithm 9.6 (Algorithm Aε for bin packing) 1. 2. 3. 4. 5. Remove items of size < ε. Round to obtain constant number of item sizes (Lemma 9.5). Find optimal packing (Lemma 9.4). Use this packing for original item sizes. Pack items of size < ε using First-Fit.

9.2

Exercises

77

9.2 Exercises

9.1 Give an example on which First-Fit does at least as bad as 5/3 · OPT. 9.2 (Johnson [149]) Consider a more restricted algorithm than First-Fit, called Next-Fit, which tries to pack the next item only in the most recently started bin. If it does not ﬁt, it is packed in a new bin. Show that this algorithm also achieves factor 2. Give a factor 2 tight example. 9.3 (C. Kenyon) Say that a bin packing algorithm is monotonic if the number of bins it uses for packing a subset of the items is at most the number of bins it uses for packing all n items. Show that whereas Next-Fit is monotonic, First-Fit is not. 9.4 Prove the bounds on R and P stated in Lemma 9.4. Hint: Use the fact that the number of ways of throwing n identical balls into k distinct bins is n+k−1 . n 9.5 Consider an alternative way of establishing Lemma 9.5. All items having sizes in the interval (ε(1 + ε)r , ε(1 + ε)r+1 ] are rounded up to min(ε(1 + ε)r+1 , 1), for r ≥ 0. Clearly, this yields a constant number of item sizes. Does the rest of the proof go through? Hint: Consider the situation that there are lots of items of size 1/2, and 1/2 = ε(1 + ε)r for any r ≥ 0. 9.6 Prove the following statement made in Lemma 9.5, “A packing for instance J yields a packing for all but the largest Q items of instance J.” Hint: Throw away the Q largest items of J and the Q smallest items of J , and establish a domination. 9.7 Use the fact that integer programming with a ﬁxed number of variables is in P to give an alternative proof of Lemma 9.4. (Because of the exorbitant running time of the integer programming algorithm, this variant is also impractical.) 9.8 Show that if there is an algorithm for bin packing having a guarantee of OPT(I) + log2 (OPT(I)), then there is a fully polynomial approximation scheme for this problem. 9.9 (C. Kenyon) Consider the following problem. Problem 9.7 (Bin covering) Given n items with sizes a1 , . . . , an ∈ (0, 1], maximize the number of bins opened so that each bin has items summing to at least 1. Give an asymptotic PTAS for this problem when restricted to instances in which item sizes are bounded below by c, for a ﬁxed constant c > 0. Hint: The main idea of Algorithm 9.6 applies to this problem as well.

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Bin Packing

9.3 Notes

The ﬁrst nontrivial bin packing result, showing that First-Fit requires at most (17/10)OPT + 3 bins, was due to Ullman [248]. The asymptotic PTAS is due to Fernandez de la Vega and Lueker [86]. An improved algorithm, having a guarantee of OPT(I) + log2 (OPT(I)) was given by Karmarkar and Karp [163]. For further results, see the survey of Coﬀman, Garey, and Johnson [50]. The result cited in Exercise 9.7, showing that integer programming with a ﬁxed number of variables is in P, is due to Lenstra [185]. Bin packing has also been extensively studied in the on-line model. For these and other on-line algorithms see Borodin and El-Yaniv [31].

10 Minimum Makespan Scheduling

A central problem in scheduling theory is the following. Problem 10.1 (Minimum makespan scheduling) Given processing times for n jobs, p1 , p2 , . . . , pn , and an integer m, ﬁnd an assignment of the jobs to m identical machines so that the completion time, also called the makespan, is minimized. We will give a simple factor 2 algorithm for this problem before presenting a PTAS for it.

10.1 Factor 2 algorithm

The algorithm is very simple: schedule the jobs one by one, in an arbitrary order, each job being assigned to a machine with least amount of work so far. This algorithm is based on the following two lower bounds on the optimal makespan, OPT: 1. The average time for which a machine has to run, ( 2. The largest processing time. Let LB denote the combined lower bound, i.e., LB = max 1 m pi , max{pi } . i i i

pi ) /m; and

Algorithm 10.2 (Minimum makespan scheduling) 1. Order the jobs arbitrarily. 2. Schedule jobs on machines in this order, scheduling the next job on the machine that has been assigned the least amount of work so far.

Theorem 10.3 Algorithm 10.2 achieves an approximation guarantee of 2 for the minimum makespan problem.

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10

Minimum Makespan Scheduling

Proof: Let Mi be the machine that completes its jobs last in the schedule produced by the algorithm, and let j be the index of the last job scheduled on this machine. M1 . . . Mi . . . Mm Let startj be the time at which job j starts execution on Mi . Since the algorithm assigns a job to the least loaded machine, it follows that all machines are busy until startj . This implies that startj ≤ 1 m pi ≤ OPT. i ✛

≤

1 m

i

pi

✲✛

pj

✲

Further, pj ≤ OPT. Thus, the makespan of the schedule is startj + pj ≤ 2 · OPT. ✷ Example 10.4 A tight example for this algorithm is provided by a sequence of m2 jobs with unit processing time, followed by a single job of length m. The schedule obtained by the algorithm has a makespan of 2m, while OPT = m + 1. ✷

10.2 A PTAS for minimum makespan

The minimum makespan problem is strongly NP-hard; thus, by Corollary 8.6, it does not admit an FPTAS, assuming P = NP. We will obtain a PTAS for it. The minimum makespan problem is closely related to the bin packing problem by the following observation. There exists a schedule with makespan t iﬀ n objects of sizes p1 , p2 , . . . , pn can be packed into m bins of capacity t each. This suggests a reduction from minimum makespan to bin packing as follows. Denoting the sizes of the n objects, p1 , . . . , pn , by I, let bins(I, t) represent the minimum number of bins of size t required to pack these n objects. Then, the minimum makespan is given by min{t : bins(I, t) ≤ m}.

10.2

A PTAS for minimum makespan

81

As shown above, LB and 2 · LB are lower and upper bounds on the minimum makespan. Thus, we can determine the minimum makespan by a binary search in this interval. At ﬁrst sight, this reduction may not seem very useful since the bin packing problem is also NP-hard. However, it turns out that this problem is polynomial time solvable if the object sizes are drawn from a set of ﬁxed cardinality. We will use this fact critically for solving the minimum makespan problem. 10.2.1 Bin packing with ﬁxed number of object sizes

We ﬁrst present a dynamic programming algorithm for the restricted bin packing problem, thereby improving on the result of Lemma 9.4 in two ways. We will not require a lower bound on item sizes and will improve on the running time. Let k be the ﬁxed number of object sizes, and assume that bins have capacity 1. Fix an ordering on the object sizes. Now, an instance of the bin packing problem can be described by a k-tuple, (i1 , i2 , . . . , ik ), specifying the number of objects of each size. Let BINS(i1 , i2 , . . . , ik ) denote the minimum number of bins needed to pack these objects. k For a given instance, (n1 , n2 , . . . , nk ), i=1 ni = n, we ﬁrst compute Q, the set of all k-tuples (q1 , q2 , . . . , qk ) such that BINS(q1 , q2 , . . . , qk ) = 1 and 0 ≤ qi ≤ ni , 1 ≤ i ≤ k. Clearly, Q contains at most O(nk ) elements. Next, we compute all entries of the k-dimensional table BINS(i1 , i2 , . . . , ik ) for every (i1 , i2 , . . . , ik ) ∈ {0, . . . , n1 } × {0, . . . , n2 } × . . . × {0, . . . , nk }. The table is initialized by setting BINS(q) = 1 for every q ∈ Q. Then, we use the following recurrence to compute the remaining entries: BINS(i1 , i2 , . . . , ik ) = 1 + min BINS(i1 − q1 , . . . , ik − qk ). q∈Q (10.1)

Computing each entry takes O(nk ) time. Thus, the entire table can be computed in O(n2k ) time, thereby determining BINS(n1 , n2 , . . . , nk ). 10.2.2 Reducing makespan to restricted bin packing

The basic idea is that if we can tolerate some error in computing the minimum makespan, then we can reduce this problem to the restricted version of bin packing in polynomial time. There will be two sources of error: • rounding object sizes so that there are a bounded number of diﬀerent sizes, and • terminating the binary search to ensure polynomial running time. Each error can be made as small as needed, at the expense of running time. Moreover, for any ﬁxed error bound, the running time is polynomial in n, and thus we obtain a polynomial approximation scheme.

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10

Minimum Makespan Scheduling

Let ε be an error parameter and t be in the interval [LB, 2 · LB]. We say that an object is small if its size is less than tε; small objects are discarded for now. The rest of the objects are rounded down as follows: each pj in the interval tε(1 + ε)i , tε(1 + ε)i+1 is replaced by pj = tε(1+ε)i , for i ≥ 0. The resulting pj ’s can assume at most k = log1+ε 1 distinct values. Determine ε an optimal packing for the rounded objects in bins of size t using the dynamic programming algorithm. Since rounding reduces the size of each object by a factor of at most 1 + ε, if we consider the original sizes of the objects, then the packing determined is valid for a bin size of t(1 + ε). Keeping this as the bin size, pack the small objects greedily in leftover spaces in the bins; open new bins only if needed. Clearly, any time a new bin is opened, all previous bins must be full to the extent of at least t. Denote with α(I, t, ε) the number of bins used by this algorithm; recall that these bins are of size t(1 + ε). Let us call the algorithm presented above the core algorithm since it will form the core of the PTAS for computing makespan. As shown in Lemma 10.5 and its corollary, the core algorithm also helps establish a lower bound on the optimal makespan. Lemma 10.5 α(I, t, ε) ≤ bins(I, t).

Proof: If the algorithm does not open any new bins for the small objects, then the assertion clearly holds since the rounded down pieces have been packed optimally in bins of size t. In the other case, all but the last bin are packed at least to the extent of t. Hence, the optimal packing of I in bins of size t must also use at least α(I, t, ε) bins. ✷ Since OPT = min{t : bins(I, t) ≤ m}, Lemma 10.5 gives: Corollary 10.6 min{t : α(I, t, ε) ≤ m} ≤ OPT. If min{t : α(I, t, ε) ≤ m} could be determined with no additional error during the binary search, then clearly we could use the core algorithm to obtain a schedule with a makespan of (1 + ε)OPT. Next, we will specify the details of the binary search and show how to control the error it introduces. The binary search is performed on the interval [LB, 2 · LB]. Thus, the length of the available interval is LB at the start of the search, and it reduces by a factor of 2 in each iteration. We continue the search until the available interval drops to a length of ε · LB. This will require log2 1 iterations. Let ε T be the right endpoint of the interval we terminate with. Lemma 10.7 T ≤ (1 + ε) · OPT.

Proof: Clearly, min{t : α(I, t, ε) ≤ m} must be in the interval [T − ε · LB, T ]. Hence, T ≤ min{t : α(I, t, ε) ≤ m} + ε · LB.

10.4

Notes

83

Now, using Corollary 10.6 and the fact that LB ≤ OPT, the lemma follows. ✷ Finally, the output of the core algorithm for t = T gives a schedule whose makespan is at most T · (1 + ε). We get: Theorem 10.8 The algorithm produces a valid schedule having makespan at most (1 + ε)2 · OPT ≤ (1 + 3ε) · OPT. The running time of the entire algorithm is O n2k log2 log1+ε 1 . ε

1 ε

, where k =

10.3 Exercises

10.1 (Graham [114]) The tight example for the factor 2 algorithm, Example 10.4, involves scheduling a very long job last. This suggests sorting the jobs by decreasing processing times before scheduling them. Show that this leads to a 4/3 factor algorithm. Provide a tight example for this algorithm. 10.2 (Horowitz and Sahni [131]) Give an FPTAS for the variant of the minimum makespan scheduling problem in which the number of machines, m, is a ﬁxed constant.

10.4 Notes

Algorithm 10.2 is due to Graham [113]. The PTAS is due to Hochbaum and Shmoys [128].

11 Euclidean TSP

In this chapter, we will give a PTAS for the special case of the traveling salesman problem in which the points are given in a d-dimensional Euclidean space. As before, the central idea of the PTAS is to deﬁne a “coarse solution”, depending on the error parameter ε, and to ﬁnd it using dynamic programming. A feature this time is that we do not know a deterministic way of specifying the coarse solution – it is speciﬁed probabilistically. Problem 11.1 (Euclidean TSP) For ﬁxed d, given n points in Rd , the problem is to ﬁnd the minimum length tour of the n points. The distance between any two points x and y is deﬁned to be the Euclidean distance between them, i.e., d i=1

(xi − yi )2

1/2

.

11.1 The algorithm

We will give the algorithm for points on the plane, i.e., d = 2. The extension to arbitrary d is straightforward. The algorithm involves numerous details. In the interest of highlighting the main ideas, some of these details will be left as exercises. Deﬁne the bounding box of the instance to be the smallest axis-parallel square that contains all n points. Via a simple perturbation of the instance, we may assume that the length of this square, L, is 4n2 and that there is a unit grid deﬁned on the square such that each point lies on a gridpoint (see Exercise 11.1). Further, assume w.l.o.g. that n is a power of 2, and let L = 2k , k = 2 + log2 n. The basic dissection of the bounding box is a recursive partitioning into smaller squares. Thus, the L×L square is divided into four L/2×L/2 squares, and so on. It will be convenient to view this dissection as a 4-ary tree, T , whose root is the bounding box. The four children of the root are the four L/2 × L/2 squares, and so on. The nodes of T are assigned levels. The root is at level 0, its children at level 1, and so on. The squares represented by nodes get levels accordingly. Thus, squares at level i have dimensions L/2i × L/2i . The dissection is continued until we obtain unit squares. Clearly, T has depth k = O(log n). By a useful square we mean a square represented by a node in T.

11.1

The algorithm

85

Next, let us deﬁne levels for the horizontal and vertical lines that accomplish the basic dissection (these are all the lines of the grid deﬁned on the bounding box). The two lines that divide the bounding box into four squares have level 1. In general, the 2i lines that divide the level i − 1 squares into level i squares each have level i. Therefore, a line of level i forms the edge of useful squares at levels i, i + 1, . . . , i.e., the largest useful square on it has dimensions L/2i × L/2i :

Level 1 line

Level 2 line Level 3 line

Each line will have a special set of points called portals. The coarse solution we will be seeking is allowed to cross a line only at a portal. The portals on each line are equidistant points. On a line of level i, these points are L/(2i m) apart, where the parameter m is ﬁxed to be a power of 2 in the range [k/ε, 2k/ε]. Clearly, m = O(log n/ε). Since the largest useful square on a level i line has dimensions L/2i × L/2i , each useful square has a total of at most 4m portals on its four sides and corners. We have chosen m to be a power of 2 so that a portal in a lower level square is a portal for all higher level squares it lies in. We will say that a tour τ is well behaved w.r.t. the basic dissection if it is a tour on the n points and any subset of the portals. In addition, this tour is allowed to visit portals multiple times, but other than that it must be non-self-intersecting. The key structural fact to be established is that there is such a tour of length at most (1 + ε) · OPT. This requires a probabilistic argument, and we will return to it. First let us show why a PTAS follows from this fact. We will say that tour τ is well behaved w.r.t. the basic dissection and has limited crossings if it is well behaved w.r.t. the basic dissection, and furthermore, it visits each portal at most twice. Lemma 11.2 Let tour τ be well behaved w.r.t. the basic dissection. Then there must be a tour that is well behaved with limited crossings, whose length is at most that of τ . Proof: The basic reason is that removing self-intersections by “shortcutting” can only result in a shorter tour, since Euclidean distance satisﬁes

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the triangle inequality. If τ uses a portal on line l more than twice, we can keep “short-cutting” on the two sides of l until the portal is used at most twice. If this introduces additional self-intersections, they can also be removed. ✷ Lemma 11.3 The optimal well behaved tour w.r.t. the basic dissection, having limited crossings, can be computed in time 2O(m) = nO(1/ε) . Proof: We will build a table, using dynamic programming, that contains, for each useful square, the cost of each valid visit. We will sketch the main ideas, leaving details as Exercise 11.2. Let τ be the optimal tour we wish to ﬁnd. Clearly, the total number of times τ can enter and exit a useful square, S, is at most 8m. The part of τ inside S is simply a set of at most 4m paths, each entering and exiting S at portals, and together covering all the points inside the square. Furthermore, the paths must be internally non-self-intersecting, i.e., two paths can intersect only at their entrance or exit points. This means that the pairing of entrance and exit points of the paths must form a balanced arrangement of parentheses.

Invalid pairing

Valid pairing

Let us call such a listing of portals, together with their pairing as entrance and exit points, a valid visit. The number of useful squares is clearly poly(n). Let us ﬁrst show that the number of valid visits in a useful square is at most nO(1/ε) , thereby showing that the number of entries in the table is bounded by nO(1/ε) . Consider a useful square S. Each of its portals is used 0, 1, or 2 times, a total of 34m = nO(1/ε) possibilities. Of these, retain only those possibilities that involve an even number of portal usages. Consider one such possibility, and suppose that it uses 2r portals. Next, we need to consider all possible pairings of these portals that form a balanced arrangement of parentheses. The number of such arrangements is the rth Catalan number, and is bounded by 22r = nO(1/ε) . Hence, the total number of valid visits in S is bounded by nO(1/ε) . For each entry in the table, we need to compute the optimal length of this valid visit. The table is built up the decomposition tree, starting at its leaves. Consider a valid visit V in a square S. Let S be a level i square. We have already ﬁxed the entrances and exits on the boundary of S. Square S has four children at level i + 1, which have four sides internal to S, with a total of at most 4m more portals. Each of these portals is used 0, 1, or 2 times, giving rise again to nO(1/ε) possibilities. Consider one such possibility, and consider all its portal usages together with portal usages of a valid visit V . Obtain all possible valid pairings of these portals that are consistent with those of visit V . Again, using Catalan numbers, their number is bounded by nO(1/ε) . Each such pairing will give rise to valid visits in the four squares.

11.2

Proof of correctness

87

The cost of the optimal way of executing these valid visits in the four squares has already been computed. Compute their sum. The smallest of these sums is the optimal way of executing visit V in square S. ✷

11.2 Proof of correctness

For the proof of correctness, it suﬃces to show that there is a well behaved tour w.r.t. the basic dissection whose length is bounded by (1 + ε)OPT. It turns out that this is not always the case (see Exercise 11.3). Instead, we will construct a larger family of dissections and will show that, for any placement of the n points, at least half these dissections have short well behaved tours with limited crossings. So, picking a random dissection from this set suﬃces. Let us deﬁne L2 diﬀerent dissections of the bounding box, which are shifts of the basic dissection. Given integers a, b with 0 ≤ a, b < L, the (a, b)-shifted dissection is obtained by moving each vertical line from its original location x to (a + x) mod L, and moving each horizontal line from its original location y to (b + y) mod L. Thus, the middle lines of the shifted dissection are located at (a + L/2) mod L and (b + L/2) mod L, respectively.

b

11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111 00000000 1111 0000 1111 0000 11111 00000 11111111111111 00000000000000 11111111 00000000 1111 0000 11111 00000 1111 0000 11111 00000 11111111 00000000 1111 0000 11111 00000 1111 0000 11111 00000 11111111 00000000 1111 0000 11111 00000 1111 0000 11111 00000 11111111 00000000 1111 0000 11111 00000 1111 0000 11111 00000 11111111 00000000 1111 0000 11111 00000 1111 0000 11111 00000 00000000 1111 11111111 0000 11111 00000 11111111111111 00000000000000 0000 1111 00000 11111 00000000 11111111 11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 a The entire bounding box is thought of as being “wrapped around”. Useful squares that extend beyond L in their x or y coordinates will thus be thought of as “wrapped around”, and will still be thought of as a single square. Of course, the positions of the given n points remains unchanged; only the dissection is shifted. Let π be the optimal tour, and N (π) be the total number of times π crosses horizontal and vertical grid lines. If π uses a point at the intersection of two grid lines, then we will count it as two crossings. The following fact is left as Exercise 11.4. Lemma 11.4 N (π) ≤ 2 · OPT.

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Euclidean TSP

Following is the central fact leading to the PTAS. Theorem 11.5 Pick a and b uniformly at random from [0, L). Then, the expected increase in cost in making π well behaved w.r.t. the (a, b)-shifted dissection is bounded by 2ε · OPT. Proof: Given any dissection, consider the process of making π well behaved w.r.t. it. This involves replacing a segment of π that does not cross a line l at a portal by two segments so that the crossing is at the closest portal on l. The corresponding increase in the length of the tour is bounded by the interportal distance on line l. Consider the expected increase in length due to one of the crossings of tour π with a line. Let l be this line. l will be a level i line in the randomly picked dissection with probability 2i /L. If l is a level i line, then the interportal distance on it is L/(2i m). Thus, the expected increase in the length of the tour due to this crossing is at most L 2i k = ≤ ε, im L 2 m

i

where we have used the fact that m lies in [k/ε, 2k/ε]. The theorem follows by summing over all N (π) crossings and using Lemma 11.4. ✷ Remark 11.6 The ideas leading up to Theorem 11.5 can be summarized as follows. Since lower level lines have bigger useful squares incident at them, we had to place portals on them further apart to ensure that any useful square had at most 4m portals on it (thereby ensuring that dynamic programming could be carried out in polynomial time). But this enabled us to construct instances for which there was no short well behaved tour w.r.t. the basic dissection (Exercise 11.3). On the other hand, there are fewer lines having lower levels – Theorem 11.5 exploits this fact. Now, using Markov’s inequality we get: Corollary 11.7 Pick a and b uniformly at random from [0, L). Then, the probability that there is a well behaved tour of length at most 4ε · OPT w.r.t. the (a, b)-shifted dissection is greater or equal to 1/2. Notice that Lemma 11.2 holds in the setting of an (a, b)-shifted dissection as well. The PTAS is now straightforward. Simply pick a random dissection, and ﬁnd an optimal well behaved tour with limited crossings w.r.t. this dissection using the dynamic programming procedure of Lemma 11.3. Notice that the same procedure holds even for a shifted dissection. The algorithm can be derandomized by trying all possible shifts and outputting the shortest tour obtained. Thus, we get: Theorem 11.8 There is a PTAS for the Euclidean TSP problem in R2 .

11.4

Notes

89

11.3 Exercises

11.1 Show that we may assume that the length of the bounding square can be taken to be L = 4n2 and that there is a unit grid deﬁned on the square such that each point lies on a gridpoint. Hint: Since we started with the smallest axis-parallel bounding square, its length is a lower bound on OPT. Therefore, moving each point to a grid point can increase the length of the tour by at most OPT/n2 . 11.2 Provide the missing details in the proof of Lemma 11.3. 11.3 Give an instance of the Euclidean TSP problem for which, w.r.t. the basic dissection, the process of making the optimal tour well behaved increases its length by a ﬁxed constant factor. Hint: Make the optimal tour cross the middle line of the dissection that has the largest interportal distance numerous times. 11.4 Prove Lemma 11.4. Hint: Notice that√ left-hand side simply measures the 1 length of tour the π. The bound of 2 2 · OPT is easier to prove, since this applies to single edges as well. This bound suﬃces for the PTAS. 11.5 Extend the arguments given to obtain a PTAS for the Euclidean TSP problem in Rd . 11.6 Generalize the algorithm to norms other than the Euclidean norm. 11.7 (Arora [10]) Obtain a PTAS for the Euclidean Steiner tree problem. Given n points in Rd , ﬁnd the minimum length tree containing all n points and any other subset of points. The latter points are called Steiner. The distance between two points is assumed to be their Euclidean distance. 11.8 Consider the Euclidean Steiner tree problem in R2 . Show that in any optimal Steiner tree each Steiner point has degree 3 and the three angles so formed are of 120◦ each. (See Gauss’ ﬁgures on cover for an illustration of this fact.)

11.4 Notes

The ﬁrst PTAS for Euclidean TSP was given by Arora [9], following a PTAS for the planar graph TSP problem due to Grigni, Koutsoupias, and Papadimitriou [115]. Subsequently, Mitchell [207] independently obtained the same result. Later, Arora [10] went on to give an n(log n)O(1/ε) algorithm for the problem for any ﬁxed d. For a PTAS with an improved running time see Rao and Smith [229]. This chapter is based on Arora [10] and Arora, Raghavan, and Rao [13].

Part II LP-Based Algorithms

12 Introduction to LP-Duality

A large fraction of the theory of approximation algorithms, as we know it today, is built around linear programming (LP). In Section 12.1 we will review some key concepts from this theory. In Section 12.2 we will show how the LP-duality theorem gives rise to min-max relations which have far-reaching algorithmic signiﬁcance. Finally, in Section 12.3 we introduce the two fundamental algorithm design techniques of rounding and the primal–dual schema, as well as the method of dual ﬁtting, which yield all the algorithms of Part II of this book.

12.1 The LP-duality theorem

Linear programming is the problem of optimizing (i.e., minimizing or maximizing) a linear function subject to linear inequality constraints. The function being optimized is called the objective function. Perhaps the most interesting fact about this problem from our perspective is that it is well-characterized (see deﬁnition in Section 1.2). Let us illustrate this through a simple example. minimize subject to 7x1 + x2 + 5x3 x1 − x2 + 3x3 ≥ 10 x3 ≥ 6 5x1 + 2x2 − x1 , x2 , x3 ≥ 0 Notice that in this example all constraints are of the kind “≥” and all variables are constrained to be nonnegative. This is the standard form of a minimization linear program; a simple transformation enables one to write any minimization linear program in this manner. The reason for choosing this form will become clear shortly. Any solution, i.e., a setting for the variables in this linear program, that satisﬁes all the constraints is said to be a feasible solution. Let z ∗ denote the optimum value of this linear program. Let us consider the question, “Is z ∗ at most α?” where α is a given rational number. For instance, let us ask whether z ∗ ≤ 30. A Yes certiﬁcate for this question is simply a feasible solution whose

94

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objective function value is at most 30. For example, x = (2, 1, 3) constitutes such a certiﬁcate since it satisﬁes the two constraints of the problem, and the objective function value for this solution is 7 · 2 + 1 + 5 · 3 = 30. Thus, any Yes certiﬁcate to this question provides an upper bound on z ∗ . How do we provide a No certiﬁcate for such a question? In other words, how do we place a good lower bound on z ∗ ? In our example, one such bound is given by the ﬁrst constraint: since the xi ’s are restricted to be nonnegative, term-by-term comparison of coeﬃcients shows that 7x1 + x2 + 5x3 ≥ x1 − x2 + 3x3 . Since the right-hand side of the ﬁrst constraint is 10, the objective function is at least 10 for any feasible solution. A better lower bound can be obtained by taking the sum of the two constraints: for any feasible solution x, 7x1 + x2 + 5x3 ≥ (x1 − x2 + 3x3 ) + (5x1 + 2x2 − x3 ) ≥ 16. The idea behind this process of placing a lower bound is that we are ﬁnding suitable nonnegative multipliers for the constraints so that when we take their sum, the coeﬃcient of each xi in the sum is dominated by the coeﬃcient in the objective function. Now, the right-hand side of this sum is a lower bound on z ∗ since any feasible solution has a nonnegative setting for each xi . Notice the importance of ensuring that the multipliers are nonnegative: they do not reverse the direction of the constraint inequality. Clearly, the rest of the game lies in choosing the multipliers in such a way that the right-hand side of the sum is as large as possible. Interestingly enough, the problem of ﬁnding the best such lower bound can be formulated as a linear program: maximize subject to 10y1 + 6y2 y1 + 5y2 3y1 − y 1 , y2 ≥ 0 Here y1 and y2 were chosen to be the nonnegative multipliers for the ﬁrst and the second constraint, respectively. Let us call the ﬁrst linear program the primal program and the second the dual program. There is a systematic way of obtaining the dual of any linear program; one is a minimization problem and the other is a maximization problem. Further, the dual of the dual is the primal program itself (Exercise 12.1). By construction, every feasible solution to the dual program gives a lower bound on the optimum value of the primal. Observe that the reverse also holds. Every feasible solution to the primal program gives an upper bound on the optimal value of the dual. Therefore, if we can ﬁnd feasible solutions for the dual and the primal with ≤ 7 −y1 + 2y2 ≤ 1 y2 ≤ 5

12.1

The LP-duality theorem

95

matching objective function values, then both solutions must be optimal. In our example, x = (7/4, 0, 11/4) and y = (2, 1) both achieve objective function values of 26, and thus both are optimal solutions (see ﬁgure below). The reader may wonder whether our example was ingeniously constructed to make this happen. Surprisingly enough, this is not an exception, but the rule! This is the central theorem of linear programming: the LP-duality theorem.

dual opt = primal opt

0

26

∞ ✲

✛ dual solutions

✲✛ primal solutions

✲

In order to state this theorem formally, let us consider the following minimization problem, written in standard form, as the primal program; equivalently, we could have started with a maximization problem as the primal program. n minimize j=1 n

cj xj

(12.1)

subject to j=1 aij xj ≥ bi , xj ≥ 0,

i = 1, . . . , m j = 1, . . . , n

where aij , bi , and cj are given rational numbers. Introducing variables yi for the ith inequality, we get the dual program: m maximize i=1 m

bi y i aij yi ≤ cj , yi ≥ 0, i=1 (12.2) j = 1, . . . , n i = 1, . . . , m

subject to

Theorem 12.1 (LP-duality theorem) The primal program has ﬁnite optimum iﬀ its dual has ﬁnite optimum. Moreover, if x∗ = (x∗ , . . . , x∗ ) and n 1

96

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Introduction to LP-Duality

∗ ∗ y ∗ = (y1 , . . . , ym ) are optimal solutions for the primal and dual programs, respectively, then n j=1

cj x∗ j

m

= i=1 ∗ bi y i .

Notice that the LP-duality theorem is really a min–max relation, since one program is a minimization problem and the other is a maximization problem. A corollary of this theorem is that the linear programming problem is well-characterized. Feasible solutions to the primal (dual) provide Yes (No) certiﬁcates to the question, “Is the optimum value less than or equal to α?” Thus, as a corollary of this theorem we get that linear programming is in NP ∩ co-NP. Going back to our example, by construction, any feasible solution to the dual program gives a lower bound on the optimal value of the primal. In fact, it also gives a lower bound on the objective function value achieved by any feasible solution to the primal. This is the easy half of the LP-duality theorem, sometimes called the weak duality theorem. We give a formal proof of this theorem, since some steps in the proof will lead to the next important fact. The design of several exact algorithms have their basis in the LP-duality theorem. In contrast, in approximation algorithms, typically the weak duality theorem suﬃces. Theorem 12.2 (Weak duality theorem) If x = (x1 , . . . , xn ) and y = (y1 , . . . , ym ) are feasible solutions for the primal and dual program, respectively, then n m

cj xj ≥ j=1 i=1

bi y i .

(12.3)

Proof: Since y is dual feasible and xj ’s are nonnegative, n n m

cj xj ≥ j=1 j=1 i=1

aij yi

xj .

(12.4)

Similarly, since x is primal feasible and yi ’s are nonnegative, m i=1

n j=1

aij xj yi ≥

m

bi y i . i=1 (12.5)

The theorem follows by observing that

12.2 n j=1 m m

Min–max relations and LP-duality

n j=1

97

aij yi i=1 xj = i=1 aij xj yi . ✷

By the LP-duality theorem, x and y are both optimal solutions iﬀ (12.3) holds with equality. Clearly, this happens iﬀ both (12.4) and (12.5) hold with equality. Hence, we get the following result about the structure of optimal solutions: Theorem 12.3 (Complementary slackness conditions) Let x and y be primal and dual feasible solutions, respectively. Then, x and y are both optimal iﬀ all of the following conditions are satisﬁed: Primal complementary slackness conditions m For each 1 ≤ j ≤ n: either xj = 0 or i=1 aij yi = cj ; and Dual complementary slackness conditions n For each 1 ≤ i ≤ m: either yi = 0 or j=1 aij xj = bi . The complementary slackness conditions play a vital role in the design of eﬃcient algorithms, both exact and approximation; see Chapter 15 for details. (For a better appreciation of their importance, we recommend that the reader study algorithms for the weighted matching problem, see Section 12.5.)

12.2 Min–max relations and LP-duality

In order to appreciate the role of LP-duality theory in approximation algorithms, it is important to ﬁrst understand its role in exact algorithms. To do so, we will review some of these ideas in the context of the max-ﬂow min-cut theorem. In particular, we will show how this and other min–max relations follow from the LP-duality theorem. Some of the ideas on cuts and ﬂows developed here will also be used in the study of multicommodity ﬂow in Chapters 18, 20, and 21. The problem of computing a maximum ﬂow in a network is: given a directed1 graph, G = (V, E) with two distinguished nodes, source s and sink t, and positive arc capacities, c : E → R+ , ﬁnd the maximum amount of ﬂow that can be sent from s to t subject to 1. capacity constraint: for each arc e, the ﬂow sent through e is bounded by its capacity, and

1

The maximum ﬂow problem in undirected graphs reduces to that in directed graphs: replace each edge (u, v) by two directed edges, (u → v) and (v → u), each of the same capacity as (u, v).

98

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Introduction to LP-Duality

2. ﬂow conservation: at each node v, other than s and t, the total ﬂow into v should equal the total ﬂow out of v. An s–t cut is deﬁned by a partition of the nodes into two sets X and X so that s ∈ X and t ∈ X, and consists of the set of arcs going from X to X. The capacity of this cut, c(X, X), is deﬁned to be the sum of capacities of these arcs. Because of the capacity constraints on ﬂow, the capacity of any s–t cut is an upper bound on any feasible ﬂow. Thus, if the capacity of an s–t cut, say (X, X), equals the value of a feasible ﬂow, then (X, X) must be a minimum s–t cut and the ﬂow must be a maximum ﬂow in G. The max-ﬂow min-cut theorem proves that it is always possible to ﬁnd a ﬂow and an s–t cut so that equality holds. Let us formulate the maximum ﬂow problem as a linear program. First, introduce a ﬁctitious arc of inﬁnite capacity from t to s, thus converting the ﬂow to a circulation; the objective now is to maximize the ﬂow on this arc, denoted by fts . The advantage of making this modiﬁcation is that we can now require ﬂow conservation at s and t as well. If fij denotes the amount of ﬂow sent through arc (i, j) ∈ E, we can formulate the maximum ﬂow problem as follows: maximize subject to fts fij ≤ cij , fji − j: (j,i)∈E j: (i,j)∈E

(i, j) ∈ E fij ≤ 0, i∈V (i, j) ∈ E

fij ≥ 0,

The second set of inequalities say that for each node i, the total ﬂow into i is at most the total ﬂow out of i. Notice that if this inequality holds at each node, then in fact it must be satisﬁed with equality at each node, thereby implying ﬂow conservation at each node (this is so because a deﬁcit in ﬂow balance at one node implies a surplus at some other node). With this trick, we get a linear program in standard form. To obtain the dual program we introduce variables dij and pi corresponding to the two types of inequalities in the primal. We will view these variables as distance labels on arcs and potentials on nodes, respectively. The dual program is: minimize

(i,j)∈E

cij dij dij − pi + pj ≥ 0, ps − pt ≥ 1 dij ≥ 0, (i, j) ∈ E (i, j) ∈ E

(12.6)

subject to

12.2

Min–max relations and LP-duality

99

pi ≥ 0,

i∈V

(12.7)

For developing an intuitive understanding of the dual program, it will be best to ﬁrst transform it into an integer program that seeks 0/1 solutions to the variables: minimize

(i,j)∈E

cij dij dij − pi + pj ≥ 0, ps − pt ≥ 1 dij ∈ {0, 1}, pi ∈ {0, 1}, (i, j) ∈ E i∈V (i, j) ∈ E

subject to

Let (d∗ , p∗ ) be an optimal solution to this integer program. The only way to satisfy the inequality p∗ − p∗ ≥ 1 with a 0/1 substitution is to set p∗ = 1 s t s and p∗ = 0. This solution naturally deﬁnes an s–t cut (X, X), where X is the t set of potential 1 nodes, and X the set of potential 0 nodes. Consider an arc (i, j) with i ∈ X and j ∈ X. Since p∗ = 1 and p∗ = 0, by the ﬁrst constraint, i j d∗ ≥ 1. But since we have a 0/1 solution, d∗ = 1. The distance label for ij ij each of the remaining arcs can be set to either 0 or 1 without violating the ﬁrst constraint; however, in order to minimize the objective function value, it must be set to 0. The objective function value must thus be equal to the capacity of the cut (X, X), and (X, X) must be a minimum s–t cut. Thus, the previous integer program is a formulation of the minimum s–t cut problem! What about the dual program? The dual program can be viewed as a relaxation of the integer program where the integrality constraint on the variables is dropped. This leads to the constraints 1 ≥ dij ≥ 0 for (i, j) ∈ E and 1 ≥ pi ≥ 0 for i ∈ V . Next, we notice that the upper bound constraints on the variables are redundant; their omission cannot give a better solution. Dropping these constraints gives the dual program in the form given above. We will say that this program is the LP-relaxation of the integer program. Consider an s–t cut C. Set C has the property that any path from s to t in G contains at least one edge of C. Using this observation, we can interpret any feasible solution to the dual program as a fractional s–t cut: the distance labels it assigns to arcs satisfy the property that on any path from s to t the distance labels add up to at least 1. To see this, consider an s–t path (s = v0 , v1 , . . . , vk = t). Now, the sum of the potential diﬀerences on the endpoints of arcs on this path is k−1 (pi − pi+1 ) = ps − pt . i=0 100

12

Introduction to LP-Duality

By the ﬁrst constraint, the sum of the distance labels on the arcs must add up to at least ps −pt , which is ≥ 1. Let us deﬁne the capacity of this fractional s–t cut to be the dual objective function value achieved by it. In principle, the best fractional s–t cut could have lower capacity than the best integral cut. Surprisingly enough, this does not happen. Consider the polyhedron deﬁning the set of feasible solutions to the dual program. Let us call a feasible solution an extreme point solution if it is a vertex of this polyhedron, i.e., it cannot be expressed as a convex combination of two feasible solutions. From linear programming theory we know that for any objective function, i.e., assignment of capacities to the arcs of G, there is an extreme point solution that is optimal (for this discussion let us assume that for the given objective function, an optimal solution exists). Now, it can be proven that each extreme point solution of the polyhedron is integral, with each coordinate being 0 or 1 (see Exercise 12.6). Thus, the dual program always has an integral optimal solution. By the LP-duality theorem maximum ﬂow in G must equal capacity of a minimum fractional s–t cut. But since the latter equals the capacity of a minimum s–t cut, we get the max-ﬂow min-cut theorem. The max-ﬂow min-cut theorem is therefore a special case of the LP-duality theorem; it holds because the dual polyhedron has integral vertices. In fact, most min–max relations in combinatorial optimization hold for a similar reason. Finally, let us illustrate the usefulness of complementary slackness conditions by utilizing them to derive additional properties of optimal solutions to the ﬂow and cut programs. Let f ∗ be an optimum solution to the primal LP (i.e., a maximum s–t ﬂow). Also, let (d∗ , p∗ ) be an integral optimum solution to the dual LP, and let (X, X) be the cut deﬁned by (d∗ , p∗ ). Consider an arc (i, j) such that i ∈ X and j ∈ X. We have proven above that d∗ = 1. ij ∗ Since d∗ = 0, by the dual complementary slackness condition, fij = cij . ij Next, consider an arc (k, l) such that k ∈ X and l ∈ X. Since p∗ − p∗ = −1, k l and d∗ ∈ {0, 1}, the constraint d∗ − p∗ + p∗ ≥ 0 must be satisﬁed as a strict kl kl k l ∗ inequality. By the primal complementary slackness condition, fkl = 0. Thus, we have proven that arcs going from X to X are saturated by f ∗ and the reverse arcs carry no ﬂow. (Observe that it was not essential to invoke complementary slackness conditions to prove these facts; they also follow from the fact that ﬂow across cut (X, X) equals its capacity.)

12.3 Two fundamental algorithm design techniques

We can now explain why linear programming is so useful in approximation algorithms. Many combinatorial optimization problems can be stated as integer programs. Once this is done, the linear relaxation of this program provides a natural way of lower bounding the cost of the optimal solution. As stated in Chapter 1, this is typically a key step in the design of an approximation

12.3

Two fundamental algorithm design techniques

101

algorithm. As in the case of the minimum s–t cut problem, a feasible solution to the LP-relaxation can be thought of as a fractional solution to the original problem. However, in the case of an NP-hard problem, we cannot expect the polyhedron deﬁning the set of feasible solutions to have integer vertices. Thus, our task is not to look for an optimal solution to the LP-relaxation, but rather a near-optimal integral solution. There are two basic techniques for obtaining approximation algorithms using linear programming. The ﬁrst, and more obvious, method is to solve the linear program and then convert the fractional solution obtained into an integral solution, trying to ensure that in the process the cost does not increase much. The approximation guarantee is established by comparing the cost of the integral and fractional solutions. This technique is called LProunding or simply rounding. The second, less obvious and perhaps more sophisticated, method is to use the dual of the LP-relaxation in the design of the algorithm. This technique is called the primal–dual schema. Let us call the LP-relaxation the primal program. Under this schema, an integral solution to the primal program and a feasible solution to the dual program are constructed iteratively. Notice that any feasible solution to the dual also provides a lower bound on OPT. The approximation guarantee is established by comparing the two solutions. Both these techniques have been used extensively to obtain algorithms for many fundamental problems. Fortunately, once again, these techniques can be illustrated in the simple setting of the set cover problem. This is done in Chapters 14 and 15. Later chapters will present ever more sophisticated use of these techniques for solving a variety of problems. LP-duality theory has also been useful in analyzing combinatorially obtained approximation algorithms, using the method of dual ﬁtting. In Chapter 13 we will give an alternative analysis of the greedy set cover algorithm, Algorithm 2.2, using this method. This method has also been used to analyze greedy algorithms for the metric uncapacitated facility location problem (see Exercise 24.12). The method seems quite basic and should ﬁnd other applications as well. 12.3.1 A comparison of the techniques and the notion of integrality gap The reader may suspect that from the viewpoint of approximation guarantee, the primal–dual schema is inferior to rounding, since an optimal solution to the primal gives a tighter lower bound than a feasible solution to the dual. It turns out that this is not so. In order to give a formal explanation, we need to introduce the crucial notion of integrality gap of an LP-relaxation. Given an LP-relaxation for a minimization problem Π, let OPTf (I) denote the cost of an optimal fractional solution to instance I, i.e., the objective function value of an optimal solution to the LP-relaxation. Deﬁne the inte-

102

12

Introduction to LP-Duality

grality gap, sometimes also called the integrality ratio, of the relaxation to be sup

I

OPT(I) , OPTf (I)

i.e., the supremum of the ratio of the optimal integral and fractional solutions. In the case of a maximization problem, the integrality gap will be deﬁned to be the inﬁmum of this ratio. As stated in Section 12.2, most min–max relations arise from LP-relaxations that always have integral optimal solutions. Clearly, the integrality gap of such an LP is 1. We will call such an LP-relaxation an exact relaxation. If the cost of the solution found by the algorithm is compared directly with the cost of an optimal fractional solution (or a feasible dual solution), as is done in most algorithms, the best approximation factor we can hope to prove is the integrality gap of the relaxation (see Exercise 12.4). Interestingly enough, for many problems, both techniques have been successful in yielding algorithms having guarantees essentially equal to the integrality gap of the relaxation. The main diﬀerence in performance between the two techniques lies in the running times of the algorithms produced. An LP-rounding algorithm needs to ﬁnd an optimal solution to the linear programming relaxation. Since linear programming is in P, this can be done in polynomial time if the relaxation has polynomially many constraints. Even if the relaxation has exponentially many constraints, this may still be achievable, if a polynomial time separation oracle can be constructed, i.e., a polynomial time algorithm that given a point in Rn , where n is the number of variables in the relaxation, either conﬁrms that this point is a feasible solution (i.e., satisﬁes all constraints), or produces a violated constraint (see the notes in Section 12.5 for references). The running time for both possibilities is high; for the second it may be exorbitant. Let us remark that for certain problems, extreme point solutions have additional structural properties and some LP-rounding algorithms require such a solution to the linear programming relaxation. Such solutions can also be found in polynomial time. On the other hand, the primal–dual schema leaves enough room to exploit the special combinatorial structure of individual problems and is thereby able to yield algorithms having good running times. It provides only a broad outline of the algorithm – the details have to be designed individually for speciﬁc problems. In fact, for many problems, once the algorithm has been designed using the primal–dual schema, the scaﬀolding of linear programming can be completely dispensed with to get a purely combinatorial algorithm. This brings us to another advantage of the primal–dual schema – this time not objectively quantiﬁable. A combinatorial algorithm is more malleable than an algorithm that requires an LP-solver. Once a basic problem is solved using the primal–dual schema, one can also solve variants and generalizations

12.4

Exercises

103

of the basic problem. Exercises in Chapters 22 and 24 illustrate this point. From a practical standpoint, a combinatorial algorithm is more useful, since it is easier to adapt it to speciﬁc applications and ﬁne tune its performance for speciﬁc types of inputs.

12.4 Exercises

12.1 Show that the dual of the dual of a linear program is the original program itself. 12.2 Show that any minimization program can be transformed into an equivalent program in standard form, i.e., the form of LP (12.1). 12.3 Change some of the constraints of the primal program (12.1) into equalities, i.e., so they are of the form n aij xj = bi , i ∈ I. j=1 Show that the dual of this program involves modifying program (12.2) so that the corresponding dual variables yi , i ∈ I are unconstrained, i.e., they are not constrained to be nonnegative. Additionally, if some of the variables xj , j ∈ J in program (12.1) are unconstrained, then the corresponding constraints in the dual become equalities. 12.4 Is the following a theorem: An approximation algorithm designed using an LP-relaxation cannot achieve a better approximation guarantee than the integrality gap of the relaxation. Hint: In principle it may be possible to show, using additional structural properties, that whenever an instance has a bad gap, the cost of the solution found by the algorithm is much less that αOPT, where α is the integrality gap of the relaxation. (Observe that if the instance has a bad gap, the cost of the solution found cannot be much less than αOPTf .) 12.5 Use the max-ﬂow min-cut theorem to derive Menger’s Theorem:

Theorem 12.4 Let G = (V, E) be a directed graph with s, t ∈ V . Then, the maximum number of edge-disjoint (vertex-disjoint) s–t paths is equal to the minimum number of edges (vertices) whose removal disconnects s from t. 12.6 Show that each extreme point solution for LP (12.6) is 0/1, and hence represents a valid cut.

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Introduction to LP-Duality

Hint: An n × m matrix A is said to be totally unimodular if the determinant of every square submatrix of A is 1, −1, or 0. Show, by induction, that the constraint matrix of this LP is totally unimodular. Also, use the fact that a feasible solution for a set of linear inequalities in Rn is an extreme point solution iﬀ it satisﬁes n linearly independent inequalities with equality. 12.7 This exercise develops a proof of the K¨nig-Egerv´ry Theorem (Theoo a rem 1.6). Let G = (V, E) be a bipartite graph. 1. Show that the following is an exact LP-relaxation (i.e., always has an integral optimal solution) for the maximum matching problem in G. maximize e xe xe ≤ 1, e: e

(12.8) v∈V e∈E

subject to xe ≥ 0, incident at v Hint: Using the technique of Exercise 12.6 show that each extreme point solution for LP (12.8) is 0/1, and hence represents a valid matching. 2. Obtain the dual of this LP and show that it is an exact LP-relaxation for the problem of ﬁnding a minimum vertex cover in bipartite graph G. 3. Use the previous result to derive the K¨nig-Egerv´ry Theorem. o a 12.8 (Edmonds [68]) 1. Let G = (V, E) be an undirected graph, with weights we on edges. The following is an exact LP-relaxation for the problem of ﬁnding a maximum weight matching in G. (By e : e ∈ S we mean edges e that have both endpoints in S.) maximize e we xe xe ≤ 1, e: e

(12.9) v∈V S ⊂ V, |S| odd e∈E

subject to incident at v e: e∈S

|S| − 1 xe ≤ , 2

xe ≥ 0,

Obtain the dual of this LP. If the weight function is integral, the dual is also exact. Observe that Theorem 1.7 follows from these facts.

12.4

Exercises

105

2. Assume that |V | is even. The following is an exact LP-relaxation for the minimum weight perfect matching problem in G (a matching is perfect if it matches all vertices). Obtain the dual of this LP. Use complementary slackness conditions to give conditions satisﬁed by a pair of optimal primal (integral) and dual solutions for both formulations. minimize e we xe xe = 1, e: e

(12.10) v∈V S ⊂ V, |S| odd e∈E

subject to

e: e∈S

incident at v |S| − 1 , xe ≤ 2

xe ≥ 0,

12.9 (Edmonds [71]) Show that the following is an integer programming formulation for the minimum spanning tree (MST) problem. Assume we are given graph G = (V, E), |V | = n, with cost function c : E → Q+ . For A ⊆ E, we will denote by κ(A) the number of connected components in graph GA = (V, A). minimize e ce xe xe = n − κ(A), e∈A (12.11) A⊂E

subject to

xe = n − 1 e∈E xe ∈ {0, 1},

e∈E

The rest of this exercise develops a proof that the LP-relaxation of this integer program is exact for the MST problem. 1. First, it will be convenient to change the objective function of IP (12.11) to max e −ce xe . Obtain the LP-relaxation and dual of this modiﬁed formulation. 2. Consider the primal solution produced by Kruskal’s algorithm. Let e1 , . . . , em be the edges sorted by increasing cost, |E| = m. This algorithm greedily picks a maximal acyclic subgraph from this sorted list. Obtain a suitable dual feasible solution so that all complementary slackness conditions are satisﬁed. Hint: Let At = {e1 , . . . , et }. Set yAt = et+1 − et , for 1 ≤ t < m, and yE = −cm , where y is the dual variable.

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12

Introduction to LP-Duality

3. Show that x is a feasible solution to the above-stated primal program iﬀ it is a feasible solution to the following LP. That is, prove that this is also an exact relaxation for the MST problem. minimize e ce xe xe = |S| − 1, e∈S (12.12) S⊂V

subject to

xe = n − 1 e∈E xe ≥ 0,

e∈E

12.10 In this exercise, you will derive von Neumann’s minimax theorem in game theory from the LP-duality theorem. A ﬁnite two-person zero-sum game is speciﬁed by an m × n matrix A with real entries. In each round, the row player, R, selects a row, say i; simultaneously, the column player, C, selects a column, say j. The payoﬀ to R at the end of this round is aij . Thus, |aij | is the amount that C pays R (R pays C) if aij is positive (aij is negative); no money is exchanged if aij is zero. Zero-sum game refers to the fact that the total amount of money possessed by R and C together is conserved. The strategy of each player is speciﬁed by a vector whose entries are nonnegative and add up to one, giving the probabilities with which the player picks each row or column. Let R’s strategy be given by m-dimensional vector x, and C’s strategy be given by n-dimensional vector y. Then, the expected payoﬀ to R in a round is xT Ay. The job of each player is to pick a strategy that guarantees maximum possible expected winnings (equivalently, minimum possible expected losses), regardless of the strategy chosen by the other player. If R chooses strategy x, he can be sure of winning only miny xT Ay, where the minimum is taken over all possible strategies of C. Thus, the optimal choice for R is given by maxx miny xT Ay. Similarly, C will minimize her losses by choosing the strategy given by miny maxx xT Ay. The minimax theorem states that for every matrix A, maxx miny xT Ay = miny maxx xT Ay. Let us say that a strategy is pure if it picks a single row or column, i.e., the vector corresponding to it consists of one 1 and the rest 0’s. A key observation is that for any strategy x of R, miny xT Ay is attained for a pure strategy of C: Suppose the minimum is attained for strategy y. Consider the pure strategy corresponding to any nonzero component of y. The fact that the components of y are nonnegative and add up to one leads to an easy proof that this pure strategy attains the same minimum. Thus, R’s optimum m strategy is given by maxx minj i=1 aij xi . The second critical observation

12.5

Notes

107

is that the problem of computing R’s optimal strategy can be expressed as a linear program: maximize subject to z m z− m i=1 i=1

aij xi ≤ 0, xi = 1

j = 1, . . . , n

xi ≥ 0,

i = 1, . . . , m

Find the dual of this LP and show that it computes the optimal strategy for C. (Use the fact that for any strategy y of C, maxx xT Ay is attained for a pure strategy of R.) Hence, prove the minimax theorem using the LP-duality theorem.

12.5 Notes

For a good introduction to theory of linear programming, see Chv´tal [49]. a There are numerous other books on the topic, e.g., Dantzig [58], Karloﬀ [160], Nemhauser and Wolsey [212], and Schrijver [237]. Linear programming has been extensively used in combinatorial optimization, see Cook, Cunningham, Pulleyblank, and Schrijver [52], Gr¨tschel, Lov´sz, and Schrijver [117], Lov´sz o a a [194], Lov´sz and Plummer [195], and Papadimitriou and Steiglitz [217]. For a a good explanation of Edmonds’ weighted matching algorithm, see Lov´sz a and Plummer [195]. For algorithms for ﬁnding a solution to an LP, given a separation oracle, see Gr¨tschel, Lov´sz, and Schrijver [116, 117] and Schrijver o a [237].

13 Set Cover via Dual Fitting

In this chapter we will introduce the method of dual ﬁtting, which helps analyze combinatorial algorithms using LP-duality theory. Using this method, we will present an alternative analysis of the natural greedy algorithm (Algorithm 2.2) for the set cover problem (Problem 2.1). Recall that in Section 2.1 we deferred giving the lower bounding method on which this algorithm was based. We will provide the answer below. The power of this approach will become apparent when we show the ease with which it extends to solving several generalizations of the set cover problem (see Section 13.2). The method of dual ﬁtting can be described as follows, assuming a minimization problem: The basic algorithm is combinatorial – in the case of set cover it is in fact the simple greedy algorithm. Using the linear programming relaxation of the problem and its dual, one shows that the primal integral solution found by the algorithm is fully paid for by the dual computed; however, the dual is infeasible. By fully paid for we mean that the objective function value of the primal solution found is at most the objective function value of the dual computed. The main step in the analysis consists of dividing the dual by a suitable factor and showing that the shrunk dual is feasible, i.e., it ﬁts into the given instance. The shrunk dual is then a lower bound on OPT, and the factor is the approximation guarantee of the algorithm.

13.1 Dual-ﬁtting-based analysis for the greedy set cover algorithm

To formulate the set cover problem as an integer program, let us assign a variable xS for each set S ∈ S, which is allowed 0/1 values. This variable will be set to 1 iﬀ set S is picked in the set cover. Clearly, the constraint is that for each element e ∈ U we want that at least one of the sets containing it be picked. minimize

S∈S

c(S)xS xS ≥ 1,

S: e∈S

(13.1) e∈U

subject to

13.1

Dual-ﬁtting-based analysis for the greedy set cover algorithm

109

xS ∈ {0, 1},

S∈S

The LP-relaxation of this integer program is obtained by letting the domain of variables xS be 1 ≥ xS ≥ 0. Since the upper bound on xS is redundant, we get the following LP. A solution to this LP can be viewed as a fractional set cover. minimize

S∈S

c(S)xS xS ≥ 1,

S: e∈S

(13.2) e∈U S∈S

subject to

xS ≥ 0,

Example 13.1 Let us give a simple example to show that a fractional set cover may be cheaper than the optimal integral set cover. Let U = {e, f, g} and the speciﬁed sets be S1 = {e, f }, S2 = {f, g}, S3 = {e, g}, each of unit cost. An integral cover must pick two of the sets for a cost of 2. On the other hand, picking each set to the extent of 1/2 gives a fractional cover of cost 3/2. ✷ Introducing a variable ye corresponding to each element e ∈ U , we obtain the dual program. maximize e∈U ye ye ≤ c(S), e: e∈S

(13.3) S∈S e∈U

subject to

ye ≥ 0,

Intuitively, why is LP (13.3) the dual of LP (13.2)? In our experience, this is not the right question to be asked. As stated in Section 12.1, there is a purely mechanical procedure for obtaining the dual of a linear program. Once the dual is obtained, one can devise intuitive, and possibly physically meaningful, ways of thinking about it. Using this mechanical procedure, one can obtain the dual of a complex linear program in a fairly straightforward manner. Indeed, the LP-duality-based approach derives its wide applicability from this fact. An intuitive way of thinking about LP (13.3) is that it is packing “stuﬀ” into elements, trying to maximize the total amount packed, subject to the constraint that no set is overpacked. A set is said to be overpacked if the total amount packed into its elements exceeds the cost of the set. Whenever the coeﬃcients in the constraint matrix, objective function, and right-hand side are all nonnegative, the minimization LP is called a covering LP and

110

13

Set Cover via Dual Fitting

the maximization LP is called a packing LP. Thus, (13.2) and (13.3) are a covering-packing pair of linear programs. Such pairs of programs will arise frequently in subsequent chapters. 0 OPTf OPT ✛ ∞ ✲ ✲ ✲

✛

✲✛

primal integral solutions primal fractional solutions

dual fractional solutions

At this point, we can state the lower bounding scheme being used by Algorithm 2.2. Denote by OPTf the cost of an optimal fractional set cover, i.e., an optimal solution to LP (13.2). Clearly OPTf ≤ OPT, the cost of an optimal (integral) set cover. The cost of any feasible solution to the dual program, LP (13.3), is a lower bound on OPTf , and hence also on OPT. Algorithm 2.2 uses this as the lower bound. Algorithm 2.2 deﬁnes dual variables price(e), for each element, e. Observe that the cover picked by the algorithm is fully payed for by this dual solution. However, in general, this dual solution is not feasible (see Exercise 13.2). We will show below that if this dual is shrunk by a factor of Hn , it ﬁts into the given set cover instance, i.e., no set is overpacked. For each element e deﬁne, ye = price(e) . Hn

Algorithm 2.2 uses the dual feasible solution, y, as the lower bound on OPT. Lemma 13.2 The vector y deﬁned above is a feasible solution for the dual program (13.3). Proof: We need to show that no set is overpacked by the solution y. Consider a set S ∈ S consisting of k elements. Number the elements in the order in which they are covered by the algorithm, breaking ties arbitrarily, say e1 , . . . , ek . Consider the iteration in which the algorithm covers element ei . At this point, S contains at least k −i+1 uncovered elements. Thus, in this iteration, S itself can cover ei at an average cost of at most c(S)/(k − i + 1). Since the algorithm chose the most cost-eﬀective set in this iteration, price(ei ) ≤ c(S)/(k − i + 1). Thus, y ei ≤ 1 c(S) . · Hn k − i + 1

Summing over all elements in S,

13.1 k Dual-ﬁtting-based analysis for the greedy set cover algorithm

111

yei ≤ i=1 c(S) · Hn

1 1 1 + + ··· + k k−1 1

=

Hk · c(S) ≤ c(S). Hn ✷

Therefore, S is not overpacked.

Theorem 13.3 The approximation guarantee of the greedy set cover algorithm is Hn . Proof: The cost of the set cover picked is price(e) = Hn e∈U e∈U

ye

≤ Hn · OPT,

where OPT denotes the cost of the optimal fractional set cover. The last inequality follows from the fact that y is dual feasible. ✷ 13.1.1 Can the approximation guarantee be improved?

Consider the three questions raised in Section 1.1.2 regarding improving the approximation guarantee for vertex cover. Let us ask analogous questions for set cover. The ﬁrst and third questions are already answered in Section 2.1. As a corollary of Theorem 13.3 we get an upper bound of Hn on the integrality gap of relaxation (13.2). Example 13.4 shows that this bound is essentially tight. Since the integrality gap of the LP-relaxation used bounds the best approximation factor one can hope to achieve using this relaxation, the answer to the second question is also essentially “no”. Example 13.4 Consider the following set cover instance. Let n = 2k − 1, where k is a positive integer, and let U = {e1 , e2 , . . . , en }. For 1 ≤ i ≤ n, consider i written as a k-bit number. We can view this as a k-dimensional vector over GF [2]. Let i denote this vector. For 1 ≤ i ≤ n deﬁne set Si = {ej | i · j = 1}, where i · j denotes the inner product of these two vectors. Finally, let S = {S1 , . . . , Sn }, and deﬁne the cost of each set to be 1. It is easy to check that each set contains 2k−1 = (n + 1)/2 elements, and each element is contained in (n + 1)/2 sets. Thus, xi = 2/(n + 1), 1 ≤ i ≤ n, is a fractional set cover. Its cost is 2n/(n + 1). Next, we will show that any integral set cover must pick at least k of the sets. Consider the union of any p sets, where p < k. Let i1 , . . . , ip be the indices of these p sets, and let A be a p × k matrix over GF [2] whose rows consist of vectors i1 , . . . , ip , respectively. Since the rank of A is < k, the dimension of its null space is ≥ 1, and so the null space contains a nonzero vector, say j. Since Aj = 0, the element ej is not in any of the p sets. Hence the p sets do not form a cover.

112

13

Set Cover via Dual Fitting

Therefore, any integral set cover has cost at least k = log2 (n + 1). Hence, the lower bound on the integrality gap established by this example is n+1 2n · log2 (n + 1) > log2 n . 2 ✷

13.2 Generalizations of set cover

The greedy algorithm and its analysis using dual ﬁtting extend naturally to several generalizations of the set cover problem (see Exercise 13.4). • Set multicover: Each element, e, needs to be covered a speciﬁed integer number, re , of times. The objective again is to cover all elements up to their coverage requirements at minimum cost. We will assume that the cost of picking a set S k times is kcost(S). • Multiset multicover: We are given a collection of multisets, rather than sets, of U . A multiset contains a speciﬁed number of copies of each element. Let M (S, e) denote the multiplicity of element e in set S. The instance satisﬁes the condition that the multiplicity of an element in a set is at most its coverage requirement, i.e., ∀S, e M (S, e) ≤ re . The objective is the same as before. • Covering integer programs: These are integer programs of the form minimize subject to c·x Ax ≥ b,

where all entries in A, b, c are nonnegative and x is required to be nonnegative and integral. 13.2.1 Dual ﬁtting applied to constrained set multicover

In this section, we will present an Hn factor approximation algorithm for set multicover with the additional constraint that each set can be picked at most once. Let us call this the constrained set multicover problem. One interesting feature of this problem is that its linear relaxation and dual contain negative coeﬃcients and thus do not form a covering-packing pair of LP’s. Let re ∈ Z+ be the coverage requirement for each element e ∈ U . The integer programming formulation of constrained set multicover is not very diﬀerent from that of set cover.

13.2

Generalizations of set cover

113

minimize

S∈S

c(S)xS xS ≥ re ,

S: e∈S

(13.4) e∈U S∈S

subject to

xS ∈ {0, 1},

Notice, however, that in the LP-relaxation, the constraints xS ≤ 1 are no longer redundant. If we drop them, then a set may be picked multiple times to satisfy the coverage requirement of the elements. Thus, the LP-relaxation looks diﬀerent from that for set cover. In particular, because of the negative numbers in the constraint matrix and the right-hand side, it is not even a covering linear program. The analysis given below deals with this added complexity. minimize

S∈S

c(S)xS xS ≥ re ,

S: e∈S

(13.5) e∈U S∈S S∈S

subject to

−xS ≥ −1, xS ≥ 0,

The additional constraints in the primal lead to new variables, zS , in the dual. The dual also has negative numbers in the constraint matrix and is not a packing program. Now, a set S can be overpacked with the ye ’s. However, this can be done only if we raise zS to ensure feasibility, which in turn decreases the objective function value. Overall, overpacking may still be advantageous, since the ye ’s appear with coeﬃcients of re in the objective function. maximize e∈U re y e −

S∈S

zS

(13.6)

subject to e: e∈S

ye ye ≥ 0, zS ≥ 0,

− zS ≤ c(S),

S∈S e∈U S∈S

The algorithm is again greedy. Let us say that element e is alive if it occurs in fewer than re of the picked sets. In each iteration, the algorithm picks, from amongst the currently unpicked sets, the most cost-eﬀective set, where the cost-eﬀectiveness of a set is deﬁned to be the average cost at which it covers alive elements. The algorithm halts when there are no more alive elements, i.e., each element has been covered to the extent of its requirement.

114

13

Set Cover via Dual Fitting

When a set S is picked, its cost is distributed equally among the alive elements it covers as follows: if S covers e for the jth time, we set price(e, j) to the current cost-eﬀectiveness of S. Clearly, the cost-eﬀectiveness of sets picked is nondecreasing. Hence, for each element e, price(e, 1) ≤ price(e, 2) ≤ . . . ≤ price(e, re ). At the end of the algorithm, the dual variables are set as follows: For each e ∈ U , let αe = (1/Hn ) · price(e, re ). For each S ∈ S that is picked by the algorithm, let βS = 1 · (price(e, re ) − price(e, je )) , Hn e covered by S

where je is the copy of e that is covered by S. Notice that since price(e, je ) ≤ price(e, re ), βS is nonnegative. If S is not picked by the algorithm, βS is deﬁned to be 0. Lemma 13.5 The multicover picked by the algorithm is fully paid for by the dual solution (α, β). Proof: Since the cost of the sets picked by the algorithm is distributed among the covered elements, it follows that the total cost of the multicover produced by the algorithm is re price(e, j). e∈U j=1

The objective function value of the dual solution (α, β) is re re αe − e∈U S∈S

βS = e∈U j=1

price(e, j). ✷

The lemma follows.

The dual solution deﬁned above is, in general, infeasible. We will show that when scaled by a factor of Hn , a feasible solution results. Deﬁne for each element e ∈ U and each set S ∈ S, ye = αe βS and zS = . Hn Hn

Lemma 13.6 The pair (y, z) is a feasible solution for the dual program (13.6).

13.2

Generalizations of set cover

115

Proof: Consider a set S ∈ S consisting of k elements. Number its elements in the order in which their requirements are fulﬁlled, i.e., the order in which they stopped being alive. Let the ordered elements be e1 , . . . , ek . First, assume that S is not picked by the algorithm. When the algorithm is about to cover the last copy of ei , S contains at least k−i+1 alive elements, so price(ei , rei ) ≤ c(S) . k−i+1

Since zS is zero, we get k y ei i=1 − zS = ≤

1 Hn

k

price(ei , rei ) i=1 c(S) · Hn

1 1 1 + + ··· + k k−1 1

≤ c(S) .

Next, assume that S is picked by the algorithm, and before this happens, k ≥ 0 elements of S are already completely covered. Then k yei i=1 − zS k k

= =

1 · Hn

price(ei , rei ) − i=1 k i=k +1 k i=k +1

(price(ei , rei ) − price(ei , ji )) price(ei , ji ) ,

1 · price(ei , rei ) + Hn i=1

where S covers the ji th copy of ei , for each i ∈ {k + 1, . . . , k}. k But i=k +1 price(ei , ji ) = cost(S), since the cost of S is equally distributed among the copies it covers. Finally consider elements ei , i ∈ {1, . . . , k }. When the last copy of ei is being covered, S is not yet picked and covers at least k − i + 1 alive elements. Thus, price(ei , rei ) ≤ c(S)/(k − i + 1). Therefore, k y ei i=1 − zS ≤

c(S) · Hn

1 1 + ··· + +1 k k−k +1

≤ c(S). ✷

Hence, (y, z) is feasible for the dual program.

Theorem 13.7 The greedy algorithm achieves an approximation guarantee of Hn for the constrained set multicover problem.

116

13

Set Cover via Dual Fitting

Proof: By Lemmas 13.5 and 13.6, the total cost of the multicover produced by the algorithm is re αe − e∈U S∈S

βS = Hn · e∈U re y e −

S∈S

zS ≤ Hn · OPT. ✷

Observe that as a corollary of Theorem 13.7 we get that the integrality gap of LP (13.5) is bounded by Hn . In contrast, the integrality gap of the corresponding LP for multiset multicover, with the restriction that each set be picked at most once, is not bounded by any function of n (see Exercise 13.5).

13.3 Exercises

13.1 Show that the dual-ﬁtting-based analysis for the greedy set cover and constrained set multicover algorithms actually establishes an approximation guarantee of Hk , where k is size of the largest set in the given instance. (Notice the ease with which this can be established using the LP-duality approach; compare with Exercise 2.8.) 13.2 Give an example in which the dual solution, price(e), for each element e, computed by Algorithm 2.2 overpacks some sets, S, by a factor of essentially H|S| . 13.3 Give examples to show that the lower bound used by Algorithm 2.2, y, can be smaller than OPT by a factor of O(log n). 13.4 Give the following approximation algorithms. 1. Hn factor for set multicover. 2. Hm factor for multiset multicover, where m is the size of the largest multiset in the given instance (the size of a multiset counts elements with multiplicity). 3. O(log n) factor for covering integer programs. Hint: For Hm factor algorithm for multiset multicover, set the dual variables according to the average price for covering elements, i.e., ye = 1 Hm re price(e, i)/re . i=1 13.3

Exercises

117

Use scaling and rounding to reduce covering integer programs to multiset multicover, with m polynomially bounded in n, at the expense of a small error (which goes into the approximation factor). 13.5 Show that the integrality gap of the relaxation for the following two variants of multiset multicover, based on LP (13.2), is not bounded by any function of n. 1. Remove the restriction that M (S, e) ≤ re . 2. Impose the constraint that each set can be picked at most once. What is the best approximation guarantee you can establish for the greedy algorithm for the second variant. Why does the proof of factor Hn given in Section 13.2 not extend to this case? 13.6 (Mihail [206]) Consider the following variant on the set multicover problem. Let U be the universal set, |U | = n, and S a collection of subsets of U . For each S ∈ S, its cost is given as a function of time, t ∈ {1, . . . , T }. Each of these cost functions is nonincreasing with time. In addition, for each element in U , a coverage requirement is speciﬁed, again as a function of time; these functions are nondecreasing with time. The problem is to pick sets at a minimum total cost so that the coverage requirements are satisﬁed for each element at each time. A set can be picked any number of times; the cost of picking a set depends on the time at which it is picked. Once picked, the set remains in the cover for all future times at no additional cost. Give an Hn factor algorithm for this problem. (An H(n·T ) factor algorithm is straightforward.) 13.7 In many realistic situations, the cost of picking an item a multiple number of times does not grow linearly. Instead it is given by a concave function. The following variant of the set multicover problem models this situation. For each set Si we are given a concave function fi specifying the cost of picking this set multiple times. The problem again is to satisfy all coverage requirements of elements at minimum cost. Give a factor Hn algorithm for this problem. Hint: Reduce the problem to a multiset multicover problem. For each set Si , j j construct sets Si , j ≥ 1. Set Si contains each element of Si with multiplicity j and has a cost of fi (j). The greedy algorithm run on this instance achieves the required factor. Next show that there is no need to explicitly construct all j the sets Si . In each iteration of the greedy algorithm, the most cost-eﬀective set can be computed directly in polynomial time, even if the requirements are exponentially large.

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13.4 Notes

The dual-ﬁtting-based analysis of set cover is due to Lov´sz [192] and Chv´tal a a [48]. The analysis of constrained set multicover is due to Rajagopalan and Vazirani [227]. For algorithms for covering integer programs, see Dobson [61] and Rajagopalan and Vazirani [227].

14 Rounding Applied to Set Cover

We will introduce the technique of LP-rounding by using it to design two approximation algorithms for the set cover problem, Problem 2.1. The ﬁrst is a simple rounding algorithm achieving a guarantee of f , where f is the frequency of the most frequent element. The second algorithm, achieving an approximation guarantee of O(log n), illustrates the use of randomization in rounding. Consider the polyhedron deﬁned by feasible solutions to an LP-relaxation. For some problems, one can ﬁnd special properties of extreme point solutions of this polyhedron, which can yield rounding-based algorithms. One such property is half-integrality, i.e., in each extreme point solution, every coordinate is 0, 1, or 1/2. In Section 14.3 we will show that the vertex cover problem possesses this remarkable property. This directly gives a factor 2 algorithm for weighted vertex cover; namely, ﬁnd an optimal extreme point solution and round all the halves to 1. A more general property, together with an enhanced rounding algorithm, called iterated rounding, is introduced in Chapter 23.

14.1 A simple rounding algorithm

A linear programming relaxation for the set cover problem is given in LP(13.2). One way of converting a solution to this linear program into an integral solution is to round up all nonzero variables to 1. It is easy to construct examples showing that this could increase the cost by a factor of Ω(n) (see Example 14.3). However, this simple algorithm does achieve the desired approximation guarantee of f (see Exercise 14.1). Let us consider a slight modiﬁcation of this algorithm that is easier to prove and picks fewer sets in general: Algorithm 14.1 (Set cover via LP-rounding) 1. Find an optimal solution to the LP-relaxation. 2. Pick all sets S for which xS ≥ 1/f in this solution.

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Theorem 14.2 Algorithm 14.1 achieves an approximation factor of f for the set cover problem. Proof: Let C be the collection of picked sets. Consider an arbitrary element e. Since e is in at most f sets, one of these sets must be picked to the extent of at least 1/f in the fractional cover. Thus, e is covered by C, and hence C is a valid set cover. The rounding process increases xS , for each set S ∈ C, by a factor of at most f . Therefore, the cost of C is at most f times the cost of the fractional cover, thereby proving the desired approximation guarantee. ✷ The set cover instance arising from a vertex cover problem has f = 2. Therefore, Algorithm 14.1 gives a factor 2 approximation algorithm for the weighted vertex cover problem, thus matching the approximation guarantee established in Theorem 2.7. Example 14.3 Let us give a tight example for Algorithm 14.1. For simplicity, we will view a set cover instance as a hypergraph: sets correspond to vertices and elements correspond to hyperedges (this is a generalization of the transformation that helped us view a set cover instance with each element having frequency 2 as a vertex cover instance). Let V1 , . . . , Vk be k disjoint sets of cardinality n each. The hypergraph has vertex set V = V1 ∪ . . . ∪ Vk , and nk hyperedges; each hyperedge picks one vertex from each Vi . In the set cover instance, elements correspond to hyperedges and sets correspond to vertices. Once again, inclusion corresponds to incidence. Each set has cost 1. Picking each set to the extent of 1/k gives an optimal fractional cover of cost n. Given this fractional solution, the rounding algorithm will pick all nk sets. On the other hand, picking all sets corresponding to vertices in V1 gives a set cover of cost n. ✷

14.2 Randomized rounding

A natural idea for rounding an optimal fractional solution is to view the fractions as probabilities, ﬂip coins with these biases and round accordingly. Let us show how this idea leads to an O(log n) factor randomized approximation algorithm for the set cover problem. First, we will show that each element is covered with constant probability by the sets picked by this process. Repeating this process O(log n) times, and picking a set if it is chosen in any of the iterations, we get a set cover with high probability, by a standard coupon collector argument. The expected cost of cover picked in this manner is O(log n)·OPTf ≤ O(log n)·OPT, where OPTf is the cost of an optimal solution to the LP-relaxation. Applying Markov’s Inequality, we convert this into a high probability statement. We provide details below.

14.2

Randomized rounding

121

Let x = p be an optimal solution to the linear program. For each set S ∈ S, pick S with probability pS , the entry corresponding to S in p. Let C be the collection of sets picked. The expected cost of C, E[cost(C)] =

S∈S

Pr[S is picked] · cS =

S∈S

pS · cS = OPTf .

Next, let us compute the probability that an element a ∈ U is covered by C. Suppose that a occurs in k sets of S. Let the probabilities associated with these sets be p1 , . . . , pk . Since a is fractionally covered in the optimal solution, p1 + p2 + · · · + pk ≥ 1. Using elementary calculus, it is easy to show that under this condition, the probability that a is covered by C is minimized when each of the pi ’s is 1/k. Thus, Pr[a is covered by C] ≥ 1 − 1 − 1 k k 1 ≥1− , e

where e is the base of natural logarithms. Hence each element is covered with constant probability by C. To get a complete set cover, independently pick c log n such subcollections, and compute their union, say C , where c is a constant such that 1 e Now, Pr[a is not covered by C ] ≤ 1 e c log n c log n

≤

1 . 4n

≤

1 . 4n

Summing over all elements a ∈ U , we get Pr[C is not a valid set cover] ≤ n · 1 1 ≤ . 4n 4

Clearly, E[C ] ≤ OPTf ·c log n. Applying Markov’s Inequality (see Section B.2) with t = OPTf · 4c log n, we get Pr[cost(C ) ≥ OPTf · 4c log n] ≤ 1 . 4

The probability of the union of the two undesirable events is ≤ 1/2. Hence, Pr[C is a valid set cover and has cost ≤ OPTf · 4c log n] ≥ 1 . 2

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Observe that we can verify in polynomial time whether C satisﬁes both these conditions. If not, we repeat the entire algorithm. The expected number of repetitions needed at most 2.

14.3 Half-integrality of vertex cover

Consider the vertex cover problem with arbitrary weights. Let c : V → Q+ be the function assigning nonnegative weights to the vertices. The integer program for this problem is: minimize v∈V c(v)xv xu + xv ≥ 1, xv ∈ {0, 1}, (u, v) ∈ E v∈V

(14.1)

subject to

The LP-relaxation of this integer program is: minimize v∈V c(v)xv xu + xv ≥ 1, xv ≥ 0, (u, v) ∈ E v∈V

(14.2)

subject to

Recall that an extreme point solution of a set of linear inequalities is a feasible solution that cannot be expressed as convex combination of two other feasible solutions. A half-integral solution to LP (14.2) is a feasible solution in which each variable is 0, 1, or 1/2. Lemma 14.4 Let x be a feasible solution to LP (14.2) that is not halfintegral. Then, x is the convex combination of two feasible solutions and is therefore not an extreme point solution for the set of inequalities in LP (14.2). Proof: Consider the set of vertices for which solution x does not assign half-integral values. Partition this set as follows. V+ = v 1 < xv < 1 2 , V− = v 0 < xv < 1 2 .

For ε > 0, deﬁne the following two solutions. xv + ε, xv ∈ V+ xv − ε, xv ∈ V+ yv = xv − ε, xv ∈ V− , zv = xv + ε, xv ∈ V− xv , otherwise xv , otherwise.

14.4

Exercises

123

By assumption, V+ ∪ V− = ∅, and so x is distinct from y and z. Furthermore, x is a convex combination of y and z, since x = 1 (y + z). We 2 will show, by choosing ε > 0 small enough, that y and z are both feasible solutions for LP (14.2), thereby establishing the lemma. Ensuring that all coordinates of y and z are nonnegative is easy. Next, consider the edge constraints. Suppose xu + xv > 1. Clearly, by choosing ε small enough, we can ensure that y and z do not violate the constraint for such an edge. Finally, consider an edge such that xu + xv = 1. There are essentially three possibilities for xu and xv . xu = xv = 1 ; xu = 0, xv = 1; 2 and u ∈ V+ , v ∈ V− . In all three cases, for any choice of ε, xu + xv = yu + yv = zu + zv = 1. The lemma follows. This leads to: Theorem 14.5 Any extreme point solution for the set of inequalities in LP (14.2) is half-integral. Theorem 14.5 directly leads to a factor 2 approximation algorithm for weighted vertex cover: ﬁnd an extreme point solution, and pick all vertices that are set to half or one in this solution. ✷

14.4 Exercises

14.1 Modify Algorithm 14.1 so that it picks all sets that are nonzero in the fractional solution. Show that the algorithm also achieves a factor of f . Hint: Use the primal complementary slackness conditions to prove this. 14.2 Consider the collection of sets, C, picked by the randomized rounding algorithm. Show that with some constant probability, C covers at least half the elements at a cost of at most O(OPT). 14.3 Give O(log n) factor randomized rounding algorithms for the set multicover and multiset multicover problems (see Section 13.2). 14.4 Give a (non-bipartite) tight example for the half-integrality-based algorithm for weighted vertex cover. 14.5 (J. Cheriyan) Give a polynomial time algorithm for the following problem. Given a graph G with nonnegative vertex weights and a valid, though not 2 necessarily optimal, coloring of G, ﬁnd a vertex cover of weight ≤ (2− k )OPT, where k is the number of colors used.

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14.6 Give a counterexample to the following claim. A set cover instance in which each element is in exactly f sets has a (1/f )-integral optimal fractional solution (i.e., in which each set is picked an integral multiple of 1/f ). 14.7 This exercise develops a combinatorial algorithm for ﬁnding an optimal half integral vertex cover. Given undirected graph G = (V, E) and a nonnegative cost function c on vertices, obtain bipartite graph H(V , V , E ) as follows. Corresponding to each vertex v ∈ V , there is a vertex v ∈ V and v ∈ V each of cost c(v)/2. Corresponding to each edge (u, v) ∈ E, there are two edges (u , v ), (u , v ) ∈ E . Show that a vertex cover in H can be mapped to a half-integral vertex cover in G preserving total cost and vice versa. Use the fact that an optimal vertex cover in a bipartite graph can be found in polynomial time to obtain an optimal half-integral vertex cover in G. 14.8 Consider LP (12.8), introduced in Exercise 12.7, for a non-bipartite graph G = (V, E). 1. Show that it is not an exact relaxation for the maximum matching problem in G. 2. Show that this LP always has a half-integral optimal solution. 14.9 In an attempt to improve the running time of the algorithm obtained in Exercise 9.7 for bin packing, consider going to the LP-relaxation of the integer programming and using LP-rounding. What guarantee can you establish for bin packing through this method?

14.5 Notes

Algorithm 14.1 is due to Hochbaum [125]. For a more sophisticated randomized rounding algorithm for set cover, see Srinivasan [244]. Theorem 14.5 is due to Nemhauser and Trotter [213].

15 Set Cover via the Primal–Dual Schema

As noted in Section 12.3, the primal–dual schema is the method of choice for designing approximation algorithms since it yields combinatorial algorithms with good approximation factors and good running times. We will ﬁrst present the central ideas behind this schema and then use it to design a simple f factor algorithm for set cover, where f is the frequency of the most frequent element. The primal–dual schema has its origins in the design of exact algorithms. In that setting, this schema yielded the most eﬃcient algorithms for some of the cornerstone problems in P, including matching, network ﬂow, and shortest paths. These problems have the property that their LP-relaxations have integral optimal solutions. By Theorem 12.3 we know that optimal solutions to linear programs are characterized by fact that they satisfy all the complementary slackness conditions. In fact, the primal–dual schema for exact algorithms is driven by these conditions. Starting with initial feasible solutions to the primal and dual programs, it iteratively starts satisfying complementary slackness conditions. When they are all satisﬁed, both solutions must be optimal. During the iterations, the primal is always modiﬁed integrally, so that eventually we get an integral optimal solution. Consider an LP-relaxation for an NP-hard problem. In general, the relaxation will not have an optimal solution that is integral. Does this rule out a complementary slackness condition driven approach? Interestingly enough, the answer is ‘no’. It turns out that the algorithm can be driven by a suitable relaxation of these conditions! This is the most commonly used way of designing primal–dual based approximation algorithms – but not the only way.

15.1 Overview of the schema

Let us consider the following primal program, written in standard form. n minimize j=1 cj xj

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Set Cover via the Primal–Dual Schema n subject to j=1 aij xj ≥ bi , xj ≥ 0,

i = 1, . . . , m j = 1, . . . , n

where aij , bi , and cj are speciﬁed in the input. The dual program is: m maximize i=1 m

bi y i aij yi ≤ cj , yi ≥ 0, i=1 subject to

j = 1, . . . , n i = 1, . . . , m

Most known approximation algorithms using the primal–dual schema run by ensuring one set of conditions and suitably relaxing the other. In the following description we capture both situations by relaxing both conditions. Eventually, if primal conditions are ensured, we set α = 1, and if dual conditions are ensured, we set β = 1. Primal complementary slackness conditions Let α ≥ 1. For each 1 ≤ j ≤ n: either xj = 0 or cj /α ≤

m i=1

aij yi ≤ cj .

Dual complementary slackness conditions Let β ≥ 1. n For each 1 ≤ i ≤ m: either yi = 0 or bi ≤ j=1 aij xj ≤ β · bi , Proposition 15.1 If x and y are primal and dual feasible solutions satisfying the conditions stated above then n m

cj xj ≤ α · β · j=1 i=1

bi y i .

Proof: n n m m

n j=1

aij xj yi (15.1)

cj xj ≤ α j=1 j=1 m i=1

aij yi bi y i . i=1 xj = α i=1 ≤ αβ

The ﬁrst and second inequalities follow from the primal and dual conditions, respectively. The equality follows by simply changing the order of summation. ✷

15.2

Primal–dual schema applied to set cover

127

The algorithm starts with a primal infeasible solution and a dual feasible solution; these are usually the trivial solutions x = 0 and y = 0. It iteratively improves the feasibility of the primal solution, and the optimality of the dual solution, ensuring that in the end a primal feasible solution is obtained and all conditions stated above, with a suitable choice of α and β, are satisﬁed. The primal solution is always extended integrally, thus ensuring that the ﬁnal solution is integral. The improvements to the primal and the dual go handin-hand: the current primal solution is used to determine the improvement to the dual, and vice versa. Finally, the cost of the dual solution is used as a lower bound on OPT, and by Proposition 15.1, the approximation guarantee of the algorithm is αβ.

15.2 Primal–dual schema applied to set cover

Let us obtain a factor f algorithm for the set cover problem using the primal– dual schema. For this algorithm, we will choose α = 1 and β = f . We will work with the primal and dual pair of LP’s given in (13.2) and (13.3), respectively. The complementary slackness conditions are: Primal conditions ∀S ∈ S : xS = 0 ⇒ e: e∈S

ye = c(S).

Set S will be said to be tight if e: e∈S ye = c(S). Since we will increment the primal variables integrally, we can state the conditions as: Pick only tight sets in the cover. Clearly, in order to maintain dual feasibility, we are not allowed to overpack any set. Dual conditions ∀e : ye = 0 ⇒

S: e∈S

xS ≤ f

Since we will ﬁnd a 0/1 solution for x, these conditions are equivalent to: Each element having a nonzero dual value can be covered at most f times. Since each element is in at most f sets, this condition is trivially satisﬁed for all elements. The two sets of conditions naturally suggest the following algorithm:

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15

Set Cover via the Primal–Dual Schema

Algorithm 15.2 (Set cover – factor f ) 1. Initialization: x ← 0; y ← 0 2. Until all elements are covered, do: Pick an uncovered element, say e, and raise ye until some set goes tight. Pick all tight sets in the cover and update x. Declare all the elements occurring in these sets as “covered”. 3. Output the set cover x. Theorem 15.3 Algorithm 15.2 achieves an approximation factor of f . Proof: Clearly there will be no uncovered elements and no overpacked sets at the end of the algorithm. Thus, the primal and dual solutions will both be feasible. Since they satisfy the relaxed complementary slackness conditions with α = f , by Proposition 15.1 the approximation factor is f . ✷ Example 15.4 A tight example for this algorithm is provided by the following set system:

1 e1 1 e2 ...

1 en-1 en

en+1

1+ε

Here, S consists of n − 1 sets of cost 1, {e1 , en }, . . . , {en−1 , en }, and one set of cost 1 + ε, {e1 , . . . , en+1 }, for a small ε > 0. Since en appears in all n sets, this set system has f = n. Suppose the algorithm raises yen in the ﬁrst iteration. When yen is raised to 1, all sets {ei , en }, i = 1, . . . , n − 1 go tight. They are all picked in the cover, thus covering the elements e1 , . . . , en . In the second iteration, yen+1 is raised to ε and the set {e1 , . . . , en+1 } goes tight. The resulting set cover has a cost of n + ε, whereas the optimum cover has cost 1 + ε. ✷

15.4

Notes

129

15.3 Exercises

15.1 How is the algorithm given in Exercise 2.11 for the weighted vertex cover problem related to Algorithm 15.2 for the case f = 2? 15.2 Remove the scaﬀolding of linear programming from Algorithm 15.2 to obtain a purely combinatorial factor f algorithm for set cover. Hint: See the algorithm in Exercise 2.11. 15.3 Let k be a ﬁxed constant, and consider instances of set cover whose maximum frequency, f , is bounded by k. Algorithm 15.2 shows that the integrality gap of LP (13.2) is upper bounded by k for these instances. Provide examples to show that this bound is essentially tight. Hint: Consider a regular hypergraph, G, on n vertices which has a hyperedge corresponding to each choice of k of the n vertices. Construct the set system as follows. It has an element corresponding to each hyperedge and a set corresponding to each vertex, with incidence deﬁning inclusion. 15.4 The following LP-relaxation is exact for the maximum weight matching problem (see deﬁnition in Exercise 12.8) in bipartite graphs but not in general graphs. Give a primal–dual algorithm, relaxing complementary slackness conditions appropriately, to show that the integrality gap of this LP is ≥ 1/2. What is the best upper bound you can place on the integrality gap? maximize e we xe xe ≤ 1, incident at xe ≥ 0, e: e v

(15.2) v∈V e∈E

subject to

15.5 (Chudak, Goemans, Hochbaum, and Williamson [46]) Interpret the layering-based algorithms obtained for set cover and feedback vertex set problems in Chapters 2 and 6 as primal–dual schema based algorithms. How are the complementary slackness conditions being relaxed?

15.4 Notes

Kuhn [179] gave the ﬁrst primal–dual algorithm – for the weighted bipartite matching problem – however, he used the name “Hungarian Method” to describe his algorithm. Dantzig, Ford, and Fulkerson [60] used this method

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Set Cover via the Primal–Dual Schema

for giving another means of solving linear programs and called it the primal– dual method. Although the schema was not very successful for solving linear programs, it soon found widespread use in combinatorial optimization. Algorithm 15.2 is due to Bar-Yehuda and Even [20]. Although it was not originally stated as a primal–dual algorithm, in retrospect, this was the ﬁrst use of the schema in approximation algorithms. The works of Agrawal, Klein, and Ravi [1] and Goemans and Williamson [105] revived the use of this schema in the latter setting, and introduced the powerful idea of growing duals in a synchronized manner (see Chapter 22). The mechanism of relaxing complementary slackness conditions was ﬁrst formalized in Williamson, Goemans, Mihail, and Vazirani [258]. For further historical information, see Goemans and Williamson [107].

16 Maximum Satisﬁability

The maximum satisﬁability problem has been a classical problem in approximation algorithms. More recently, its study has led to crucial insights in the area of hardness of approximation (see Chapter 29). In this chapter, we will use LP-rounding, with randomization, to obtain a 3/4 factor approximation algorithm. We will derandomize this algorithm using the method of conditional expectation. Problem 16.1 (Maximum satisﬁability (MAX-SAT)) Given a conjunctive normal form formula f on Boolean variables x1 , . . . , xn , and nonnegative weights, wc , for each clause c of f , ﬁnd a truth assignment to the Boolean variables that maximizes the total weight of satisﬁed clauses. Let C represent the set of clauses of f , i.e., f = c∈C c. Each clause is a disjunction of literals; each literal being either a Boolean variable or its negation. Let size(c) denote the size of clause c, i.e., the number of literals in it. We will assume that the sizes of clauses in f are arbitrary. For any positive integer k, we will denote by MAX-kSAT the restriction of MAX-SAT to instances in which each clause is of size at most k. MAXSAT is NP-hard; in fact, even MAX-2SAT is NP-hard (in contrast, 2SAT is in P). We will ﬁrst present two approximation algorithms for MAX-SAT, having guarantees of 1/2 and 1−1/e, respectively. The ﬁrst performs better if the clause sizes are large, and the seconds performs better if they are small. We will then show how an appropriate combination of the two algorithms achieves the promised approximation guarantee. In the interest of minimizing notation, let us introduce common terminology for all three algorithms. Random variable W will denote the total weight of satisﬁed clauses. For each clause c, random variable Wc denotes the weight contributed by clause c to W . Thus, W = c∈C Wc and E[Wc ] = wc · Pr[c is satisﬁed]. (Strictly speaking, this is abuse of notation, since the randomization used by the three algorithms is diﬀerent.)

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16.1 Dealing with large clauses

The ﬁrst algorithm is straightforward. Set each Boolean variable to be True independently with probability 1/2 and output the resulting truth assignment, say τ . For k ≥ 1, deﬁne αk = 1 − 2−k . Lemma 16.2 If size(c) = k, then E[Wc ] = αk wc . Proof: Clause c is not satisﬁed by τ iﬀ all its literals are set to False. The probability of this event is 2−k . ✷ For k ≥ 1, αk ≥ 1/2. By linearity of expectation, E[W ] = c∈C E[Wc ] ≥

1 2

wc ≥ c∈C 1 OPT, 2

where we have used a trivial upper bound on OPT – the total weight of clauses in C. Instead of converting this into a high probability statement, with a corresponding loss in guarantee, we show how to derandomize this procedure. The resulting algorithm deterministically computes a truth assignment such that the weight of satisﬁed clauses is ≥ E[W ] ≥ OPT/2. Observe that αk increases with k and the guarantee of this algorithm is 3/4 if each clause has two or more literals. (The next algorithm is designed to deal with unit clauses more eﬀectively.)

16.2 Derandomizing via the method of conditional expectation

We will critically use the self-reducibility of SAT (see Section A.5). Consider the self-reducibility tree T for formula f . Each internal node at level i corresponds to a setting for Boolean variables x1 , . . . , xi , and each leaf represents a complete truth assignment to the n variables. Let us label each node of T with its conditional expectation as follows. Let a1 , . . . , ai be a truth assignment to x1 , . . . , xi . The node corresponding to this assignment will be labeled with E[W |x1 = a1 , . . . , xi = ai ]. If i = n, this is a leaf node and its conditional expectation is simply the total weight of clauses satisﬁed by its truth assignment. Lemma 16.3 The conditional expectation of any node in T can be computed in polynomial time. Proof: Consider a node x1 = a1 , . . . , xi = ai . Let φ be the Boolean formula, on variables xi+1 , . . . , xn , obtained for this node via self-reducibility. Clearly,

16.2

Derandomizing via the method of conditional expectation

133

the expected weight of satisﬁed clauses of φ under a random truth assignment to the variables xi+1 , . . . , xn can be computed in polynomial time. Adding to this the total weight of clauses of f already satisﬁed by the partial assignment x1 = a1 , . . . , xi = ai gives the answer. ✷ Theorem 16.4 We can compute, in polynomial time, a path from the root to a leaf such that the conditional expectation of each node on this path is ≥ E[W ]. Proof: The conditional expectation of a node is the average of the conditional expectations of its two children, i.e., E[W |x1 = a1 , ..., xi = ai ] = E[W |x1 = a1 , ..., xi = ai , xi+1 = True]/2 + E[W |x1 = a1 , ..., xi = ai , xi+1 = False]/2. The reason, of course, is that xi+1 is equally likely to be set to True or False. As a result, the child with the larger value has a conditional expectation at least as large as that of the parent. This establishes the existence of the desired path. As a consequence of Lemma 16.3, it can be computed in polynomial time. ✷ The deterministic algorithm follows as a corollary of Theorem 16.4. We simply output the truth assignment on the leaf node of the path computed. The total weight of clauses satisﬁed by it is ≥ E[W ]. Let us show that the technique outlined above can, in principle, be used to derandomize more complex randomized algorithms. Suppose the algorithm does not set the Boolean variables independently of each other (for instance, see Remark 16.6). Now, E[W |x1 = a1 , ..., xi = ai ] = E[W |x1 = a1 , ..., xi = ai , xi+1 = True] · Pr[xi+1 = True|x1 = a1 , ..., xi = ai ]+ E[W |x1 = a1 , ..., xi = ai , xi+1 = False] · Pr[xi+1 = False|x1 = a1 , ..., xi = ai ]. The sum of the two conditional probabilities is again 1, since the two events are exhaustive. So, the conditional expectation of the parent is still a convex combination of the conditional expectations of the two children. If we can determine, in polynomial time, which of the two children has a larger value, we can again derandomize the algorithm. However, computing the conditional expectations may not be easy. Observe how critically independence was used in the proof of Lemma 16.3. It was because of independence that we could assume a random truth assignment on Boolean variables xi+1 , . . . , xn and thereby compute the expected weight of satisﬁed clauses of φ. In general, a randomized algorithm may pick from a larger set of choices and not necessarily with equal probability. But once again a convex combination of the conditional expectations of these choices, given by the probabilities

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Maximum Satisﬁability

of picking them, equals the conditional expectation of the parent. Hence there must be a choice that has at least as large a conditional expectation as the parent.

16.3 Dealing with small clauses via LP-rounding

+ Following is an integer program for MAX-SAT. For each clause c ∈ C, let Sc − (Sc ) denote the set of Boolean variables occurring nonnegated (negated) in c. The truth assignment is encoded by y. Picking yi = 1 (yi = 0) denotes setting xi to True (False). The constraint for clause c ensures that zc can be set to 1 only if at least one of the literals occurring in c is set to True, i.e., if clause c is satisﬁed by the picked truth assignment.

maximize c∈C wc zc ∀c ∈ C :

+ i∈Sc

(16.1) yi +

− i∈Sc

subject to

(1 − yi ) ≥ zc

∀c ∈ C : zc ∈ {0, 1} ∀i : yi ∈ {0, 1} The LP-relaxation is: maximize c∈C wc zc ∀c ∈ C :

+ i∈Sc

(16.2) yi +

− i∈Sc

subject to

(1 − yi ) ≥ zc

∀c ∈ C : 1 ≥ zc ≥ 0 ∀i : 1 ≥ yi ≥ 0 The algorithm is again straightforward. Solve LP (16.2). Let (y ∗ , z ∗ ) de∗ note the optimal solution. Independently set xi to True with probability yi , for 1 ≤ i ≤ n. Output the resulting truth assignment, say τ . We will use the random variables W and Wc deﬁned in Section 16.1. For k ≥ 1, deﬁne βk = 1 − 1 − 1 k k .

Lemma 16.5 If size(c) = k, then

∗ E[Wc ] ≥ βk wc zc .

16.3

Dealing with small clauses via LP-rounding

135

Proof: We may assume w.l.o.g. that all literals in c appear nonnegated (if xi appears negated, we can replace xi with xi throughout f and modify ∗ LP (16.2) accordingly without aﬀecting zc or Wc ). Further, by renaming variables, we may assume c = (x1 ∨ . . . ∨ xk ). Clause c is satisﬁed if x1 , . . . , xk are not all set to False. The probability of this event is k 1− i=1 (1 − yi ) ≥ 1 −

k i=1 (1

− yi ) k k

k

=1−

1−

k i=1

yi

k

k

≥1− 1−

∗ zc k

,

where the ﬁrst inequality follows from the arithmetic-geometric mean inequality which states that for nonnegative numbers a1 , . . . , ak , √ a1 + . . . + ak ≥ k a1 × . . . × ak . k The second inequality uses the constraint in LP (16.2) that y1 + . . . + yk ≥ zc .

g(z)

0

Deﬁne function g by: g(z) = 1 − 1 − z k k 1

z

.

This is a concave function with g(0) = 0 and g(1) = βk . Therefore, for ∗ z ∈ [0, 1], g(z) ≥ βk z. Hence, Pr[c is satisﬁed] ≥ βk zc . The lemma follows. ✷ Notice that βk is a decreasing function of k. Thus, if all clauses are of size at most k, E[W ] = c∈C E[Wc ] ≥ βk c∈C ∗ wc zc = βk OPTf ≥ βk OPT,

136

16

Maximum Satisﬁability

where OPTf is the optimal solution to LP (16.2). Clearly, OPTf ≥ OPT. This algorithm can also be derandomized using the method of conditional expectation (Exercise 16.3). Hence, for MAX-SAT instances with clause sizes at most k, it is a βk factor approximation algorithm. Since ∀k ∈ Z+ : 1− 1 k k >

1 , e

this is a 1 − 1/e factor algorithm for MAX-SAT.

16.4 A 3/4 factor algorithm

We will combine the two algorithms as follows. Let b be the ﬂip of a fair coin. If b = 0, run the ﬁrst randomized algorithm, and if b = 1, run the second randomized algorithm. Remark 16.6 Notice that we are eﬀectively setting xi to True with proba∗ bility 1 + 1 yi ; however, the xi ’s are not set independently! 4 2 ∗ Let z be the optimal solution of LP (16.2) on the given instance. Lemma 16.7 E[Wc ] ≥ 3 ∗ wc zc . 4

Proof: Let size(c) = k. By Lemma 16.2,

∗ E[Wc | b = 0] = αk wc ≥ αk wc zc , ∗ where we have used the fact that zc ≤ 1. By Lemma 16.5, ∗ E[Wc | b = 1] ≥ βk wc zc .

Combining we get E[Wc ] = 1 ∗ (αk + βk ) (E[Wc | b = 0] + E[Wc | b = 1]) ≥ wc zc . 2 2

Now, α1 +β1 = α2 +β2 = 3/2, and for k ≥ 3, αk +βk ≥ 7/8+(1−1/e) ≥ 3/2. The lemma follows. ✷ By linearity of expectation, E[W ] = c∈C E[Wc ] ≥

3 4

c∈C

∗ wc zc =

3 3 OPTf ≥ OPT, 4 4

(16.3)

where OPTf is the optimal solution to LP (16.2). Finally, consider the following deterministic algorithm.

16.5

Exercises

137

Algorithm 16.8 (MAX-SAT – factor 3/4) 1. Use the derandomized factor 1/2 algorithm to get a truth assignment, τ1 . 2. Use the derandomized factor 1 − 1/e algorithm to get a truth assignment, τ2 . 3. Output the better of the two assignments.

Theorem 16.9 Algorithm 16.8 is a deterministic factor 3/4 approximation algorithm for MAX-SAT. Proof: One of the two conditional expectations, E[W | b = 0] and E[W | b = 1], is at least as large as E[W ]. Hence, the total weight of clauses satisﬁed by the better of τ1 and τ2 is at least as large as E[W ]. ✷ By (16.3), E[W ] ≥ 3 OPTf . The weight of the integral solution produced 4 by Algorithm 16.8 is at least E[W ]. Therefore, the integrality gap of LP (16.2) is ≥ 3/4. Below we show that this is tight. Example 16.10 Consider the SAT formula f = (x1 ∨ x2 ) ∧ (x1 ∨ x2 ) ∧ (x1 ∨ x2 ) ∧ (x1 ∨ x2 ), where each clause is of unit weight. It is easy to see that setting yi = 1/2 and zc = 1 for all i and c is an optimal solution to LP (16.2) for any instance having size 2 clauses. Therefore OPTf = 4. On the other hand OPT = 3, and thus for this instance LP (16.2) has a integrality gap of 4/3. ✷ Example 16.11 Let us provide a tight example to Algorithm 16.8. Let f = (x ∨ y) ∧ (x ∨ y) ∧ (x ∨ z), and let the weights of these three clauses be 1, 1, and 2 + ε, respectively. By the remark made in Example 16.10, on this instance the factor 1 − 1/e algorithm will set each variable to True with probability 1/2 and so will be the same as the factor 1/2 algorithm. During derandomization, suppose variable x is set ﬁrst. The conditional expectations are E[W | x = True] = 3 + ε/2 and E[W | x = False] = 3 + ε. Thus, x will be set to False. But this leads to a total weight of 3 + ε, whereas by setting x to True we can get a weight of 4 + ε. Clearly, we can get an inﬁnite family of such examples by replicating these 3 clauses with new variables. ✷

16.5 Exercises

16.1 The algorithm of Section 16.1 achieves an approximation guarantee of αk if all clauses in the given instance have size at least k. Give a tight example of factor αk for this algorithm.

138

16

Maximum Satisﬁability

16.2 Show that the following is a factor 1/2 algorithm for MAX-SAT. Let τ be an arbitrary truth assignment and τ be its complement, i.e., a variable is True in τ iﬀ it is False in τ . Compute the weight of clauses satisﬁed by τ and τ , then output the better assignment. 16.3 Use the method of conditional expectation to derandomize the 1 − 1/e factor algorithm for MAX-SAT. 16.4 Observe that the randomization used in the 3/4 factor algorithm does not set Boolean variables independently of each other. As remarked in Section 16.2, the algorithm can still, in principle, be derandomized using the method of conditional expectation. Devise a way of doing so. Observe that the algorithm obtained is diﬀerent from Algorithm 16.8. 16.5 (Goemans and Williamson [104]) Instead of using the solution to LP ∗ (16.2), yi , as probability of setting xi to True, consider the more general ∗ scheme of using g(yi ), for a suitable function g. Can this lead to an improvement over the factor 1 − 1/e algorithm? 16.6 Consider the following randomized algorithm for the maximum cut problem, deﬁned in Exercise 2.1. After the initialization step of Algorithm 2.13, each of the remaining vertices is equally likely to go in sets A or B. Show that the expected size of the cut found is at least OPT/2. Show that the derandomization of this algorithm via the method of conditional expectation is precisely Algorithm 2.13. 16.7 Consider the following generalization of the maximum cut problem. Problem 16.12 (Linear equations over GF[2]) Given m equations over n GF[2] variables, ﬁnd an assignment for the variables that maximizes the number of satisﬁed equations. 1. Show that if m ≤ n, this problem is polynomial time solvable. 2. In general, the problem is NP-hard. Give a factor 1/2 randomized algorithm for it, and derandomize using the method of conditional expectation. 16.8 Consider the obvious randomized algorithm for the MAX k-CUT problem, Problem 2.14 in Exercise 2.3, which assigns each vertex randomly to one of the sets S1 , . . . , Sk . Show that the expected number of edges running between these sets is at least OPT/2. Show that the derandomization of this algorithm, via the method of conditional expectation, gives the greedy algorithm sought in Exercise 2.3. 16.9 Repeat Exercise 16.8 for the maximum directed cut problem, Problem 2.15 in Exercise 2.4, i.e., give a factor 1/4 randomized algorithm, and show that its derandomization gives a greedy algorithm.

16.6

Notes

139

16.6 Notes

The factor 1/2 algorithm, which was also the ﬁrst approximation algorithm for MAX-SAT, is due to Johnson [150]. The ﬁrst factor 3/4 algorithm was due to Yannakakis [261]. The (simpler) algorithm given here is due to Goemans and Williamson [104]. The method of conditional expectation is implicit in Erd¨s and Selfridge [74]. Its use for obtaining polynomial time algorithms o was pointed out by Spencer [243] (see Raghavan [225] and Alon and Spencer [6] for enhancements to this technique).

17 Scheduling on Unrelated Parallel Machines

LP-rounding has yielded approximation algorithms for a large number of NP-hard problems in scheduling theory (see Section 17.6). As a illustrative example, we present a factor 2 algorithm for the problem of scheduling on unrelated parallel machines. We will apply the technique of parametric pruning, introduced in Chapter 5, together with LP-rounding, to obtain the algorithm. Problem 17.1 (Scheduling on unrelated parallel machines) Given a set J of jobs, a set M of machines, and for each j ∈ J and i ∈ M , pij ∈ Z+ , the time taken to process job j on machine i, the problem is to schedule the jobs on the machines so as to minimize the makespan, i.e., the maximum processing time of any machine. We will denote the number of jobs by n and the number of machines by m. The reason for the name “unrelated” is that we have not assumed any relation between the processing times of a job on the diﬀerent machines. If each job j has the same running time, say pj , on each of the machines, then the machines are said to be identical. This problem was studied in Chapter 10 under the name minimum makespan scheduling, and we had derived a PTAS for it. A generalization of minimum makespan, that also admits a PTAS, is that of uniform parallel machines (see Exercise 17.5). In this case there is a speed si associated with each machine i, and the processing time for job j on machine i is pj /si .

17.1 Parametric pruning in an LP setting

An obvious integer program for this problem is the following. In this program xij is an indicator variable denoting whether job j is scheduled on machine i. The objective is to minimize t, the makespan. The ﬁrst set of constraints ensures that each job is scheduled on one of the machines, and the second set ensures that each machine has a processing time of at most t. minimize subject to i∈M t xij = 1, j∈J

(17.1)

17.2

Properties of extreme point solutions

141

xij pij ≤ t, j∈J i∈M i ∈ M, j ∈ J

xij ∈ {0, 1},

We show below that this integer program has unbounded integrality gap. Example 17.2 Suppose we have only one job, which has a processing time of m on each of the m machines. Clearly, the minimum makespan is m. However, the optimal solution to the linear relaxation is to schedule the job to the extent of 1/m on each machine, thereby leading to an objective function value of 1, and giving an integrality gap of m. ✷ This example is exploiting an “unfair” advantage that we have given to the linear relaxation. The integer program automatically sets xij to 0 if pij > t. On the other hand, the linear relaxation is allowed to set these variables to nonzero values, and thereby obtain a cheaper solution. The situation could be rectiﬁed if we could add the following constraint to the linear relaxation: ∀i ∈ M j ∈ J : if pij > t then xij = 0. However, this is not a linear constraint. We will use the technique of parametric pruning to get around this diﬃculty. The parameter will be T ∈ Z+ , which is our guess for a lower bound on the optimal makespan. The parameter will enable us to prune away all job–machine pairs such that pij > T . Deﬁne ST = {(i, j) | pij ≤ T }. We will deﬁne a family of linear programs, LP(T ), one for each value of parameter T ∈ Z+ . LP(T ) uses the variables xij for (i, j) ∈ ST only, and asks if there is a feasible, fractional schedule of makespan ≤ T using the restricted possibilities. xij = 1, i:(i,j)∈ST j∈J i∈M (i, j) ∈ ST

xij pij ≤ T, j:(i,j)∈ST xij ≥ 0,

17.2 Properties of extreme point solutions

Via an appropriate binary search, we will ﬁnd the smallest value of T such that LP(T ) has a feasible solution. Let T ∗ be this value. Clearly, T ∗ is a lower bound on OPT. The algorithm will round an extreme point solution to

142

17

Scheduling on Unrelated Parallel Machines

LP(T ∗ ) to ﬁnd a schedule having makespan ≤ 2T ∗ . Extreme point solutions to LP(T ) have several useful properties. Lemma 17.3 Any extreme point solution to LP(T ) has at most n + m nonzero variables. Proof: Let r = |ST | represent the number of variables on which LP(T ) is deﬁned. Recall that a feasible solution to LP(T ) is an extreme point solution iﬀ it corresponds to setting r linearly independent constraints of LP(T ) to equality. Of these r linearly independent constraints, at least r − (n + m) must be chosen from the third set of constraints (of the form xij ≥ 0). The corresponding variables are set to 0. So, any extreme point solution has at most n + m nonzero variables. ✷ Let x be an extreme point solution to LP(T ). We will say that job j is integrally set in x if it is entirely assigned to one machine. Otherwise, we will say that job j is fractionally set. Corollary 17.4 Any extreme point solution to LP(T ) must set at least n−m jobs integrally. Proof: Let x be an extreme point solution to LP(T ), and let α and β be the number of jobs that are integrally and fractionally set by x, respectively. Each job of the latter kind is assigned to at least 2 machines and therefore results in at least 2 nonzero entries in x. Hence we get α + β = n and α + 2β ≤ n + m. Therefore, β ≤ m and α ≥ n − m. ✷

The LP-rounding algorithm is based on several interesting combinatorial properties of extreme point solutions to LP(T ). Some of these are established in Section 17.4. Corresponding to an extreme point solution x to LP(T ), deﬁne G = (J, M, E) to be the bipartite graph on vertex set J ∪ M such that (j, i) ∈ E iﬀ xij = 0. Let F ⊂ J be the set of jobs that are fractionally set in x, and let H be the subgraph of G induced on vertex set F ∪ M . Clearly, (i, j) is an edge in H iﬀ 0 < xij < 1. A matching in H will be called a perfect matching if it matches every job j ∈ F . The rounding procedure uses the fact that graph H has a perfect matching (see Lemma 17.7).

17.3 The algorithm

The algorithm starts by computing the range in which it ﬁnds the right value of T . For this, it constructs the greedy schedule, in which each job is assigned to the machine on which it has the smallest processing time. Let α be the makespan of this schedule. Then the range is [α/m, α].

17.4

Additional properties of extreme point solutions

143

Algorithm 17.5 (Scheduling on unrelated parallel machines) 1. By a binary search in the interval [α/m, α], ﬁnd the smallest value of T ∈ Z+ for which LP(T ) has a feasible solution. Let this value be T ∗ . 2. Find an extreme point solution, say x, to LP(T ∗ ). 3. Assign all integrally set jobs to machines as in x. 4. Construct graph H and ﬁnd a perfect matching M in it (e.g., using the procedure of Lemma 17.7). 5. Assign fractionally set jobs to machines according to matching M.

17.4 Additional properties of extreme point solutions

We will say that a connected graph on vertex set V is a pseudo-tree if it contains at most |V | edges. Since the graph is connected, it must have at least |V | − 1 edges. So, it is either a tree or a tree plus a single edge. In the latter case it has a unique cycle. Let us say that a graph is a pseudo-forest if each of its connected components is a pseudo-tree. Recall that in Section 17.2 we deﬁned two graphs, G and H, corresponding to an extreme point solution x to LP(T ). Lemma 17.6 Graph G is a pseudo-forest. Proof: We will show that the number of edges in each connected component of G is bounded by the number of vertices in it. Hence, each connected component is a pseudo-tree. Consider a connected component Gc . Restrict LP(T ) and x to the jobs and machines of Gc only, to obtain LPc (T ) and xc . Let xc represent the rest of x. The important observation is that xc must be an extreme point solution to LPc (T ). Suppose that this is not the case. Then, xc is a convex combination of two feasible solutions to LPc (T ). Each of these, together with xc , form a feasible solution to LP(T ). Therefore, x is a convex combination of two feasible solutions to LP(T ), leading to a contradiction. Now, applying Lemma 17.3, we get that Gc is a pseudo-tree. ✷ Lemma 17.7 Graph H has a perfect matching. Proof: Each job that is integrally set in x has exactly one edge incident at it in G. Remove these jobs, together with their incident edges, from G. Clearly, the remaining graph is H. Since an equal number of edges and vertices were removed, H is also a pseudo-forest.

144

17

Scheduling on Unrelated Parallel Machines

J M M

M

J

M M J J J

M

J

J

M M M M

M

M

In H, each job has a degree of at least 2. So, all leaves in H must be machines. Keep matching a leaf with the job it is incident to, and remove them both from the graph. (At each stage all leaves must be machines.) In the end we will be left with even cycles (since we started with a bipartite graph). Match oﬀ alternate edges of each cycle. This gives a perfect matching in H. ✷ Theorem 17.8 Algorithm 17.5 achieves an approximation guarantee of factor 2 for the problem of scheduling on unrelated parallel machines. Proof: Clearly, T ∗ ≤ OPT, since LP(OPT) has a feasible solution. The extreme point solution, x, to LP(T ∗ ) has a fractional makespan of ≤ T ∗ . Therefore, the restriction of x to integrally set jobs has a (integral) makespan of ≤ T ∗ . Each edge (i, j) of H satisﬁes pij ≤ T ∗ . The perfect matching found in H schedules at most one extra job on each machine. Hence, the total makespan is ≤ 2T ∗ ≤ 2 · OPT. The algorithm clearly runs in polynomial time. ✷ Example 17.9 Let us provide a family of tight examples. The mth instance consists of m2 −m+1 jobs that need to be scheduled on m machines. The ﬁrst job has a processing time of m on all machines, and all the remaining jobs have unit processing time on each machine. The optimal schedule assigns the ﬁrst job to one machine, and m of the remaining jobs to each of the remaining m − 1 machines. Its makespan is m. It is easy to see that LP(T ) has no feasible solutions for T < m. Now suppose the following extreme point solution to LP(m) is picked. It assigns 1/m of the ﬁrst job and m − 1 other jobs to each of the m machines. Rounding will produce a schedule having a makespan of 2m − 1. ✷

17.5 Exercises

17.1 Give an alternative proof of Lemma 17.7 by using Hall’s Theorem. (This theorem states that a bipartite graph G = (U, V, E) has a matching

17.6

Notes

145

that matches all vertices of U iﬀ for every set U ⊆ U , the neighborhood of U is at least as large as U . The neighborhood of U is {v ∈ V | ∃u ∈ U with (u, v) ∈ E}.) Hint: For any set F ⊂ F , let M be its neighborhood. Show that the graph induced on F ∪ M must have at most |F | + |M | edges. On the other hand, since each vertex in F has a degree of at least 2, this graph must have at least 2|F | edges. 17.2 Prove that the solution given to LP(m) in Example 17.9 is an extreme point solution. 17.3 Does Algorithm 17.5 achieve a better factor than 2 for the special case that the machines are identical? 17.4 Prove the following strengthening of Lemma 17.6. There is an extreme point solution to LP(T ) such that its corresponding bipartite graph, G, is a forest. 17.5 (Hochbaum and Shmoys [129]) Give a PTAS for the problem of minimizing makespan on uniform parallel machines. In this problem there is a speed si associated with each machine i, and the processing time for job j on machine i is pj /si .

17.6 Notes

The result of this chapter is due to Lenstra, Shmoys, and Tardos [184]. For other LP-rounding based scheduling algorithms, see the survey by Hall [120].

18 Multicut and Integer Multicommodity Flow in Trees

The theory of cuts in graphs occupies a central place not only in the study of exact algorithms, but also approximation algorithms. We will present some key results in the next four chapters. This will also give us the opportunity to develop further the two fundamental algorithm design techniques introduced in Chapters 14 and 15. In Chapter 15 we used the primal–dual schema to derive a factor 2 algorithm for the weighted vertex cover problem. This algorithm was particularly easy to obtain because the relaxed dual complementary slackness conditions were automatically satisﬁed in any integral solution. In this chapter, we will use the primal–dual schema to obtain an algorithm for a generalization of this problem (see Exercise 18.1). This time, enforcing relaxed dual complementary slackness conditions will be a nontrivial part of the algorithm. Furthermore, we will introduce the procedure of reverse delete, which will be used in several other primal–dual algorithms.

18.1 The problems and their LP-relaxations

The following is an important generalization of the minimum s–t cut problem. In fact, it also generalizes the multiway cut problem (Problem 4.1). Problem 18.1 (Minimum multicut) Let G=(V, E) be an undirected graph with nonnegative capacity ce for each edge e ∈ E. Let {(s1 , t1 ), . . . , (sk , tk )} be a speciﬁed set of pairs of vertices, where each pair is distinct, but vertices in diﬀerent pairs are not required to be distinct. A multicut is a set of edges whose removal separates each of the pairs. The problem is to ﬁnd a minimum capacity multicut in G. The minimum s–t cut problem is the special case of multicut for k = 1. Problem 18.1 generalizes multiway cut because separating terminals s1 , . . . , sl is equivalent to separating all pairs (si , sj ), for 1 ≤ i < j ≤ l. This observation implies that the minimum multicut problem is NP-hard even for k = 3, since the multiway cut problem is NP-hard for the case of 3 terminals. In Chapter 20 we will obtain an O(log k) factor approximation algorithm for the minimum multicut problem. In this chapter, we will obtain a factor 2 algorithm for the special case when G is restricted to be a tree. Since G is

18.1

The problems and their LP-relaxations

147

a tree, there is a unique path between si and ti , and the multicut must pick an edge on this path to disconnect si from ti . Although the problem looks deceptively simple, Exercise 18.1 should convince the reader that this is not so. The minimum multicut problem is NP-hard even if restricted to trees of height 1 and unit capacity edges. Since we want to apply LP-duality theory to design the algorithm, let us ﬁrst give an integer programming formulation for the problem and obtain its LP-relaxation. Introduce a 0/1 variable de for each edge e ∈ E, which will be set to 1 iﬀ e is picked in the multicut. Let pi denote the unique path between si and ti in the tree. minimize e∈E ce de de ≥ 1, e∈pi subject to

i ∈ {1, . . . , k} e∈E

de ∈ {0, 1},

The LP-relaxation is obtained by replacing the constraint de ∈ {0, 1} by de ≥ 0. As in the derivation of LP (13.2), there is no need to add the constraint de ≤ 1 explicitly. minimize e∈E ce de de ≥ 1, e∈pi (18.1) i ∈ {1, . . . , k} e∈E

subject to

de ≥ 0,

We can now think of de as specifying the fractional extent to which edge e is picked. A solution to this linear program is a fractional multicut: on each path pi , the sum of fractions of edges picked is at least 1. In general, minimum fractional multicut may be strictly cheaper than minimum integral multicut. This is illustrated in Example 18.2. We will interpret the dual program as specifying a multicommodity ﬂow in G, with a separate commodity corresponding to each vertex pair (si , ti ). Dual variable fi will denote the amount of this commodity routed along the unique path from si to ti . k maximize i=1 fi fi ≤ ce , i: e∈pi

(18.2) e∈E i ∈ {1, . . . , k}

subject to

fi ≥ 0,

148

18

Multicut and Integer Multicommodity Flow in Trees

The commodities are routed concurrently. The object is to maximize the sum of the commodities routed, subject to the constraint that the sum of ﬂows routed through an edge is bounded by the capacity of the edge. Notice that the sum of ﬂows through an edge (u, v) includes ﬂow going in either direction, u to v and v to u. By the weak duality theorem, a feasible multicommodity ﬂow gives a lower bound on the minimum fractional multicut and hence also on the minimum integral multicut. By the LP-duality theorem, minimum fractional multicut equals maximum multicommodity ﬂow. Example 18.2 Consider the following graph with unit capacity edges and 3 vertex pairs: t1 ,s2

1/2

1/2

s1 ,t3

1/2

t2 ,s3

The arrows show how to send 3/2 units of ﬂow by sending 1/2 unit of each commodity. Picking each edge to the extent of 1/2 gives a multicut of capacity 3/2 as well. These must be optimal solutions to the primal and dual programs. On the other hand, any integral multicut must pick at least two of the three edges in order to disconnect all three pairs. Hence, minimum integral multicut has capacity 2. ✷ Finally, let us state one more problem. Problem 18.3 (Integer multicommodity ﬂow) Graph G and the source– sink pairs are speciﬁed as in the minimum multicut problem; however, the edge capacities are all integral. A separate commodity is deﬁned for each (si , ti ) pair. The object is to maximize the sum of the commodities routed, subject to edge capacity constraints and subject to routing each commodity integrally. Let us consider this problem when G is restricted to be a tree. If in (18.2), the variables are constrained to be nonnegative integers, we would get an integer programming formulation for this problem. Clearly, the objective function value of this integer program is bounded by that of the linear program (18.2).

18.2

Primal–dual schema based algorithm

149

Furthermore, the best fractional ﬂow may be strictly larger. For instance, in Example 18.2, maximum integral multicommodity ﬂow is 1, since sending 1 unit of any of the three commodities will saturate two of the edges. This problem is NP-hard, even for trees of height 3 (though the capacity has to be arbitrary).

18.2 Primal–dual schema based algorithm

We will use the primal–dual schema to obtain an algorithm that simultaneously ﬁnds a multicut and an integer multicommodity ﬂow that are within a factor of 2 of each other, provided the given graph is a tree. Hence, we get approximation algorithms for both problems, of factor 2 and 1/2, respectively. Let us deﬁne the multicut LP to be the primal program. An edge e is saturated if the total ﬂow through it equals its capacity. We will ensure primal complementary slackness conditions, i.e., α = 1, and relax the dual conditions with β = 2, where α and β are the parameters used in the general description of the primal–dual schema given in Chapter 15. Primal conditions: For each e ∈ E, de = 0 ⇒ i: e∈pi fi = ce . Equivalently, any edge picked in the multicut must be saturated. Relaxed dual conditions: For each i ∈ {1, . . . , k}, fi = 0 ⇒ e∈pi de ≤ 2. Equivalently, at most two edges can be picked from a path carrying nonzero ﬂow. (Clearly, we must pick at least one edge from each (si , ti ) path simply to ensure the feasibility of the multicut.) Let us root the tree G at an arbitrary vertex. Deﬁne the depth of vertex v to be the length of the path from v to the root; the depth of the root is 0. For two vertices u, v ∈ V , let lca(u, v) denote the lowest common ancestor of u and v, i.e., the minimum depth vertex on the path from u to v. Let e1 and e2 be two edges on a path from a vertex to the root. If e1 occurs before e2 on this path, then e1 is said to be deeper than e2 . The algorithm starts with an empty multicut and ﬂow, and iteratively improves the feasibility of the primal solution and the optimality of the dual solution. In an iteration, it picks the deepest unprocessed vertex, say v, and greedily routes integral ﬂow between pairs that have v as their lowest common ancestor. When no more ﬂow can be routed between these pairs, all edges that were saturated in this iteration are added to the list D in arbitrary order. When all the vertices have been processed, D will be a multicut; however, it may have redundant edges. To remove them, a reverse delete step is performed: edges are considered in the reverse of the order in which they were added to D, and if the deletion of edge e from D still gives a valid multicut, e is discarded from D.

150

18

Multicut and Integer Multicommodity Flow in Trees

Algorithm 18.4 (Multicut and integer multicommodity ﬂow in trees) 1. Initialization: f ← 0; D ← ∅. 2. Flow routing: For each vertex v, in nonincreasing order of depth, do: For each pair (si , ti ) such that lca(si , ti ) = v, greedily route integral ﬂow from si to ti . Add to D all edges that were saturated in the current iteration in arbitrary order. 3. Let e1 , e2 , . . . , el be the ordered list of edges in D. 4. Reverse delete: For j = l downto 1 do: If D − {ej } is a multicut in G, then D ← D − {ej }. 5. Output the ﬂow and multicut D. Lemma 18.5 Let (si , ti ) be a pair with nonzero ﬂow, and let lca(si , ti ) = v. At most one edge is picked in the multicut from each of the two paths, si to v and ti to v. Proof: The argument is the same for each path. Suppose two edges e and e are picked from the si –v path, with e being the deeper edge. Clearly, e must be in D all through reverse delete. Consider the moment during reverse delete when edge e is being tested. Since e is not discarded, there must be a pair, say (sj , tj ), such that e is the only edge of D on the sj –tj path. Let u be the lowest common ancestor of sj and tj . Since e does not lie on the sj –tj path, u must be deeper than e , and hence deeper than v. After u has been processed, D must contain an edge from the sj –tj path, say e . v s ❧ ❧ ❧ ❧ e ❧ ❧ us ❧ ❧ ❅ ❧ e ❅ ❧ ❅ ❧ ❅ ❧ ❧ ❅ ❧s ❧ ❅s s s sj tj si ti Since nonzero ﬂow has been routed from si to ti , e must be added during or after the iteration in which v is processed. Since v is an ancestor of u, e is added after e . So e must be in D when e is being tested. This contradicts the fact that at this moment e is the only edge of D on the sj –tj path. ✷

18.2

Primal–dual schema based algorithm

151

Theorem 18.6 Algorithm 18.4 achieves approximation guarantees of factor 2 for the minimum multicut problem and factor 1/2 for the maximum integer multicommodity ﬂow problem on trees. Proof: The ﬂow found at the end of Step 2 is maximal, and since at this point D contains all the saturated edges, D is a multicut. Since the reverse delete step only discards redundant edges, D is a multicut after this step as well. Thus, feasible solutions have been found for both the ﬂow and the multicut. Since each edge in the multicut is saturated, the primal conditions are satisﬁed. By Lemma 18.5, at most two edges have been picked in the multicut from each path carrying nonzero ﬂow. Therefore, the relaxed dual conditions are also satisﬁed. Hence, by Proposition 15.1, the capacity of the multicut found is within twice the ﬂow. Since a feasible ﬂow is a lower bound on the optimal multicut, and a feasible multicut is an upper bound on the optimal integer multicommodity ﬂow, the claim follows. ✷ Finally, we obtain the following approximate min–max relation from Theorem 18.6: Corollary 18.7 On trees with integer edge capacities, int. ﬂow F

max

|F | ≤

multicut C

min

c(C) ≤ 2 ·

int. ﬂow F

max

|F |,

where |F | represents the value of ﬂow function F and c(C) represents the capacity of multicut C. In Chapter 20 we will present an O(log k) factor algorithm for the minimum multicut problem in general graphs; once again, the lower bound used is an optimal fractional multicut. On the other hand, no nontrivial approximation algorithms are known for the integer multicommodity ﬂow problem in graphs more general than trees. As shown in Example 18.8, even for planar graphs, the integrality gap of an LP analogous to (18.2) is lower bounded by n/2, where n is the number of source–sink pairs speciﬁed. Example 18.8 Consider the following planar graph with n source–sink pairs. Every edge is of unit capacity. Any pair of paths between the ith and jth source–sink pairs intersect in at least one unit capacity edge. The magniﬁed part shows how this is arranged at each intersection. Thus, sending one unit of any commodity blocks all other commodities. On the other hand, half a unit of each commodity can be routed simultaneously.

152

18

Multicut and Integer Multicommodity Flow in Trees

s1 r s2 r s3 r

... r .... .... ❅r ❅ . .... .... ✫✪ ... .... .... ........ .... ............... ✐... ....

✬✩

sn−1 r sn r r t1 r t2 r t3 r r tn−1 tn ✷

18.3 Exercises

18.1 (Garg, Vazirani, and Yannakakis [98]) Give approximation factor preserving reductions between the following pairs of problems: (a) cardinality vertex cover and minimum multicut in trees of height 1 and unit capacity edges, (b) vertex cover with arbitrary weights and minimum multicut in trees of height 1 and arbitrary edge capacities. Hint: Given a vertex cover instance G, construct a height 1 tree that has a leaf corresponding to each vertex of G and a source–sink pair corresponding to each edge of G. 18.2 The following is a well-studied polynomial time solvable generalization of the maximum matching problem. Given an undirected graph G = (V, E) and a function b : V → Z+ , a b-matching is a set of edges, E ⊆ E, with associated multiplicities, m : E → Z+ , such that each vertex v ∈ V has at most b(v) edges incident at it, counting multiplicities. The size of this bmatching is the sum of multiplicities of edges in E . The maximum b-matching problem is that of ﬁnding a b-matching of maximum size. Show that the following pairs of problems are polynomial time equivalent: (a) maximum integer multicommodity ﬂow problem on trees of height 1 and unit capacity edges, and the maximum matching problem, (b) maximum integer multicommodity ﬂow problem on trees of height 1 and arbitrary capacity edges, and the maximum b-matching problem.

18.3

Exercises

153

18.3 (Garg, Vazirani, and Yannakakis [98]) Give a polynomial time algorithm for computing a maximum integer multicommodity ﬂow on unit capacity trees of arbitrary height. Hint: Apply dynamic programming, and use a subroutine for the maximum matching problem. 18.4 If Step 2 of Algorithm 18.4 is modiﬁed to include only one saturated edge after each iteration, show that the resulting set D may not even be a multicut. 18.5 If Step 4 in Algorithm 18.4 is removed, or is changed to a forward delete, show that its approximation factor is unbounded. 18.6 Modify step 4 in Algorithm 18.4 to: sort edges in D by decreasing capacity and remove redundant edges in this order. What factor can you prove for the modiﬁed algorithm? 18.7 Give tight examples for Algorithm 18.4 for both multicut and integer multicommodity ﬂow. 18.8 Prove that if e and e are both in D in Step 3 of Algorithm 18.4, and e is deeper than e , then e is added before or in the same iteration as e . 18.9 Find the best integral and fractional multicut and the best multicommodity ﬂow in the following graph. All capacities are 1, and the speciﬁed pairs are (s1 , t1 ), . . . , (s5 , t5 ). Notice that the optimal fractional multicut is not half integral. In contrast, the LP-relaxation of the multiway cut problem always has a half-integral optimal solution (see Chapter 19). t1 s3 s4 t2

t4

t3

s1

s2 s5 t5

154

18

Multicut and Integer Multicommodity Flow in Trees

18.4 Notes

Algorithm 18.4 is due to Garg, Vazirani, and Yannakakis [98]. For recent results on the integer multicommodity ﬂow problem, see Guruswami, Khanna, Rajaraman, Sheperd, and Yannakakis [118].

19 Multiway Cut

A simple combinatorial algorithm achieving an approximation factor of 2 − 2/k for the multiway cut problem, Problem 4.1, was presented in Chapter 4. In this chapter we will use LP-rounding to improve the factor to 3/2. In Chapter 14 we mentioned the remarkable property of half-integrality, possessed by LP-relaxations of certain NP-hard problems. The multiway cut problem and its generalization, the node multiway cut problem, possess this property. We will present a proof of this fact in Section 19.3. This is the only avenue known for obtaining a constant factor approximation algorithm for the latter problem.

19.1 An interesting LP-relaxation

The usual LP-relaxation for multiway cut has an integrality gap of 2 − 2/k (see Exercise 19.2). The key to an improved approximation guarantee is a clever LP-relaxation. Let ∆k denote the k −1 dimensional simplex. This is the k −1 dimensional convex polytope in Rk deﬁned by {x ∈ Rk | x ≥ 0 and i xi = 1}, where xi is the ith coordinate of point x. The simplex ∆3 is shown below.

(0,0,1)

1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 (0,1,0) 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000

(1,0,0)

The relaxation will map each vertex of G to a point in ∆k . Each of the k terminals will be mapped to a distinct vertex of this simplex, i.e., to a unit vector ei ∈ Rk . Let xv ∈ ∆k denote the point to which vertex v is mapped. The length of an edge (u, v) ∈ E will be deﬁned to be half the 1 distance between xu and xv . The entire relaxation is:

156

19

Multiway Cut

minimize

(u,v)∈E

c(u, v)d(u, v) 1 2 k i=1

(19.1)

subject to

d(u, v) = xv ∈ ∆k , xsi = ei ,

|xi − xi |, u v

(u, v) ∈ E v∈V si ∈ S

In Lemma 19.1 we show that this relaxation is really a linear program. An integral solution to this relaxation maps each vertex of G to a vertex of the simplex, respectively. Each edge (u, v) has length either 0 or 1, depending on whether u and v are mapped to the same or diﬀerent vertices of the simplex. Edges of length 1 form a multiway cut. The cost of this cut is the objective function value of this integral solution. Thus, an optimal integral solution corresponds to an optimal multiway cut. Lemma 19.1 Relaxation (19.1) can be expressed as a linear program. Proof: For each edge (u, v), replace the ﬁrst constraint with: xi ≥ xi − xi , 1 ≤ i ≤ k uv u v xi ≥ xi − xi , 1 ≤ i ≤ k uv v u d(u, v) = 1 2 k i=1

xi uv

Since the objective function is being minimized, an optimal solution must ✷ satisfy xi = |xi − xi |. The rest of the constraints are clearly linear. uv u v Example 19.2 In the example given below, the optimal fractional multiway cut is cheaper than than the optimal integral cut. The mapping of vertices to ∆3 in the optimal fractional solution is shown below; it achieves a cost of 7.5. On the other hand, the optimal integral solution costs 8. s1 s1 t t (1, 0, 0) ✂❇ ❙ ❙ ✂ ❇2 2 ❙ ✂ ❇ (.5, .5, 0) (.5, 0, .5) ❇ ❙ tv ✂ ut t u ✂ 1 ❇tv ❡ ✪❙ ✑ ✁◗ 2 ◗ ❡ ✪ ❙ ✑ ❆ 2 ◗ ✑ ❡ ❆ 1 ❙ 1 ✁ ✪ ◗ ✑ 2 2 (0, 1, 0) t ✑ ✁ t ❡t ❆t ◗t ❙t (0, 0, 1) ✪ s3 s3 w w s2 s2 (0, .5, .5) ✷

19.2

Randomized rounding algorithm

157

The following property will greatly simplify matters: Lemma 19.3 Let x be a feasible solution to relaxation (19.1). We may assume w.l.o.g. that for each edge (u, v) ∈ E, xu and xv diﬀer in at most two coordinates. Proof: We will divide edges by adding new vertices in such a way that this property holds and the cost of the solution remains unchanged. Suppose that (u, v) ∈ E and that xu and xv diﬀer in more than two coordinates. Replace this edge by two new edges (u, w) and (w, v), where w is a new vertex. Each of the new edges is of the same cost as c(u, v), thereby ensuring that the cost of the integral optimal solution is unchanged. We show below how to enforce d(u, v) = d(u, w) + d(w, v), thereby ensuring that the cost of the fractional solution remains unchanged. Consider the coordinates in which xu and xv diﬀer. Let i be the coordinate in which the diﬀerence is minimum. Without loss of generality, assume xi < u xi . Let α = xi − xi . There must be a coordinate j such that xj ≥ xj + α. v v u u v We will deﬁne point xw as follows. The ith and jth coordinates of xw are xi = xi and xj = xj + α. The remaining coordinates of xw are the same as w u w v those of xv . Clearly, xw ∈ ∆k and d(u, v) = d(u, w) + d(w, v). Notice that u and w diﬀer in two coordinates and w and v diﬀer in fewer coordinates than u and v. Therefore, each edge of E requires at most k − 2 such subdivisions to enforce the required property. ✷

19.2 Randomized rounding algorithm

Let x be an optimal solution to relaxation (19.1) satisfying the property stated in Lemma 19.3, and let OPTf denote its cost. Let Ei denote the subset of edges whose endpoints diﬀer in coordinate i, i.e., Ei = {(u, v) ∈ E | xi = xi }. Clearly, each edge e with d(e) > 0 will lie in two of these sets. u v Let Wi = e∈Ei c(e)d(e). Renumber the terminals so that Wk is the largest of W1 , . . . , Wk . For ρ ∈ (0, 1), deﬁne B(si , ρ) = {v ∈ V | xi ≥ ρ}. v Algorithm 19.4 operates as follows. It picks ρ at random in (0, 1) and σ at random from the two permutations (1, 2, . . . , k − 1, k) and (k − 1, k − 2, . . . , 1, k). It uses ρ and σ to construct a partition of V into k sets, V1 , . . . , Vk , ensuring that si ∈ Vi . Edges running between these sets will form the multiway cut. If σ is the ﬁrst (second) permutation, then these sets are constructed in the order V1 , V2 , . . . , Vk (Vk−1 , Vk−2 , . . . , V1 , Vk ). If ρ > 1/2, the sets B(si , ρ) are pairwise disjoint. Observe that in this case the partition is not aﬀected by σ,

158

19

Multiway Cut

because Vi is simply B(si , ρ) for 1 ≤ i ≤ k − 1, and Vk = V − (V1 ∪ · · · ∪ Vk−1 ). If ρ ≤ 1/2, the sets B(si , ρ) overlap and σ plays a role, as illustrated in the ﬁgure below for k = 3. sr 1 ✔❚ sr 1 ✔❚ sr 1 ✔❚

✔ ❚ ✔❚ ❚ V3 ✔ ❚ ❚ V2 ❚ s2 ✔ r ❚ rs3 ρ ≥ 1/2

✔ V1 ❚ ✔ ❚

❚ ❚ V3❚ V2 ❚ s2 ✔ r ❚rs3 ρ < 1/2, σ = (1, 2, 3) ✔

✔

✔

✔ ❚ ✔ V ❚ 1

❚

✔ ❚ ✔❚ V ❚ ✔ ❚ 1 ❚ ✔ V ❚ ❚ 2 ✔ ❚ V❚ ❚ 3 ❚ rs3 s2 ✔ r ρ < 1/2, σ = (2, 1, 3)

Algorithm 19.4 (Multiway cut) 1. Compute an optimal solution, x, to relaxation (19.1). 2. Renumber the terminals so that Wk is largest among W1 , . . . , Wk . 3. Pick uniformly at random ρ ∈ (0, 1) and σ ∈ {(1, 2, . . . , k − 1, k), (k − 1, k − 2, . . . , 1, k)}. 4. For i = 1 to k − 1: Vσ(i) ← B(si , ρ) − j<i Vσ(j) . 5. Vk ← V − i<k Vi . 6. Let C be the set of edges that run between sets in the partition V1 , . . . , Vk . Output C.

We will show that the expected cost of the multiway cut produced by the algorithm, E[c(C)], is at most (1.5 − 1/k) · OPTf . The following lemma will be critical. Lemma 19.5 If e ∈ E − Ek , Pr[e ∈ C] ≤ 1.5 d(e), and if e ∈ Ek , Pr[e ∈ C] ≤ d(e). Proof: Suppose e ∈ E −Ek . Let e = (u, v), and let i and j be the coordinates in which xu and xv diﬀer. There are two cases: the intervals [xi , xi ] and u v [xj , xj ] either overlap or they are disjoint. These two cases are shown below. u v Note that in either case the two intervals have the same length since xi −xi = v u xj − xj = d(e). Intervals α and β are deﬁned in the ﬁgure below for the two u v cases.

19.2

Randomized rounding algorithm

159

✛ 0 xi u

α

✲

✛

β

✲ xj u 1

xi xj v v

✛ 0 xi u

α

✲✛ xj xi v v

β

✲ xj u 1

Observe that the vertices u and v can end up in one of three sets, Vi , Vj , or Vk . Furthermore, if ρ ∈ [0, 1] − (α ∪ β), then both vertices will end up in the same set, and edge e will not be in the cut. Clearly, Pr[ρ ∈ (α ∪ β)] = |α| + |β| ≤ 2d(e). The critical observation that leads to the desired bound is that in the event ρ ∈ α and σ(j) < σ(i), u and v will both be put in the set Vj , and thus e will not be in the cut. Clearly, the probability of this event is |α|/2. Therefore Pr[e ∈ C] = |β| + |α|/2 ≤ 1.5 d(e). Next, suppose that e ∈ Ek , and that its endpoints diﬀer in coordinates i and k. In this case σ(i) < σ(k), and u and v will end up in diﬀerent sets only if ρ falls between xi and xi . The probability of this is d(e). ✷ u v Lemma 19.6 The multiway cut, C, output by Algorithm 19.4 satisﬁes E[c(C)] ≤ (1.5 − 1/k)OPTf . Proof: Clearly, C forms a multiway cut. Now, OPTf = e c(e)d(e). Since k each edge with nonzero length is in two of the sets Ei , i=1 Wi = 2 · OPTf . Since k was chosen so that Wk is the largest of these sets, Wk ≥ (2/k)·OPTf . Therefore E[c(C)] = e∈E k−1

c(e)Pr[e ∈ C] = Wi + Wk = 1.5 i=1 c(e)Pr[e ∈ C] + k c(e)Pr[e ∈ C]

e∈E−Ek

e∈Ek

≤ 1.5

Wi − 0.5Wk i=1 ≤ (1.5 − 1/k) · OPTf where the ﬁrst inequality follows from Lemma 19.5. ✷

160

19

Multiway Cut

Lemma 19.6 places an upper bound of 1.5 − 1/k on the integrality gap of relaxation 19.1 (see the notes in Section 19.5 for references to a slightly better 1 result). The worst lower bound know on the integrality gap is 8/(7 + k−1 ); Example 19.2 places a lower bound of 16/15. The bound on the expected weight of the multiway cut established in Lemma 19.6 can be converted into a high probability statement using standard techniques (see Exercises 1.10 and 19.4). Hence we get Theorem 19.7 There is a 3/2 factor randomized approximation algorithm for the multiway cut problem.

19.3 Half-integrality of node multiway cut

The following is a generalization of the multiway cut problem, in the sense that there is an approximation factor preserving reduction from the multiway cut problem to it (see Exercise 19.13). Problem 19.8 (Node multiway cut) Given a connected, undirected graph G = (V, E) with an assignment of costs to vertices, c : V → R+ , and a set of terminals S = {s1 , s2 , . . . , sk } ⊆ V that form an independent set in G, a node multiway cut is a subset of V − S whose removal disconnects the terminals from each other. The node multiway cut problem asks for the minimum cost such subset. We will show that the relaxation to the following integer program always has a half-integral optimal solution. A factor 2−2/k approximation algorithm will follow from this fact (see Exercise 19.11). In this program we have introduced a 0/1 variable dv for each vertex v ∈ V − S, which indicates whether vertex v has been picked. Let P denote the set of all paths running between distinct terminals. There is a constraint for each path p in P – it ensures that at least one vertex is picked from each path. minimize v∈V −S

cv dv dv ≥ 1, v∈p subject to

p∈P v ∈V −S

dv ∈ {0, 1},

The LP-relaxation is given below. As before, we will interpret dv ’s as distance labels. With respect to an assignment to these distance labels, let us deﬁne the length of a path to be the sum of distance labels of nonterminals on this path. The distance between a pair of vertices will be the length of the shortest path between them. A solution, d, is feasible only if the distance between every pair of terminals is at least 1.

19.3

Half-integrality of node multiway cut

161

minimize v∈V −S

cv dv dv ≥ 1, v∈p (19.2) p∈P v ∈V −S

subject to

dv ≥ 0,

As in Chapter 18, the dual will be interpreted as seeking a maximum multicommodity ﬂow. The commodities ﬂow between distinct terminals, and the constraint is that the total amount of ﬂow through a vertex be bounded by its cost. maximize p∈P fp fp ≤ ce , p:v∈p (19.3) v ∈V −S p∈P

subject to

fp ≥ 0,

Let d be an optimal solution to LP (19.2). We will show how to obtain, eﬃciently, a half-integral optimal solution from d . For the purposes of proof, let f be an optimal solution to the dual LP. Complementary slackness conditions give: Primal conditions: For each v ∈ V − S, if dv > 0 then v must be saturated. Dual conditions: For each path p, if fp > 0 then the length of p is exactly 1. Consider graph G with distance labels on vertices v ∈ V − S speciﬁed by d. For each terminal si , deﬁne its region Si to be the set of vertices reachable from si by paths of length zero (we will assume that si ∈ Si ). Deﬁne the boundary, Bi , of Si to be all vertices that are adjacent to Si , i.e., Bi = {v ∈ Si | for some u ∈ Si , (u, v) ∈ E}. The feasibility of d ensures that the k regions are disjoint and the boundaries do not contain any terminals. Claim 19.9 Suppose v ∈ Bi ∩ Bj for i = j. Then dv = 1. Proof: Clearly there is a path from si to sj on which v is the only vertex having a positive distance label. The claim follows from the feasibility of d. ✷ Let M = i=1 Bi be the set of boundary vertices. Partition this into two sets: M int being boundary vertices that occur in two or more boundary sets, and M disj being the rest; each vertex in M disj is in a unique boundary set. By Claim 19.9, each vertex in M int has distance label of 1. k 162

19

Multiway Cut

Lemma 19.10 Let p be a path between two distinct terminals such that fp > 0. Then, from the vertices in M , p uses either exactly one vertex of M int or exactly two vertices of M disj . Proof: By the dual complementary slackness condition, the length of p must be exactly 1. Thus, if p uses a vertex of M int , then it cannot have any other vertices of M on it. Suppose p uses three or more vertices of M disj . Assume that p runs from si to sj and that u and w are the ﬁrst and last vertices of M disj on p, respectively. Let v be any intermediate vertex of M disj on p. Since v ∈ M disj , v must be in a unique boundary, say Bk ; k = i or k = j are possible. sk v si u w sj

Let q be a path connecting v to sk via vertices in Sk ; such a path must exist since v ∈ Bk . Now consider the following two paths: the ﬁrst consists of the part of the path p from si to v followed by q, and the second consists of the reverse of q followed by the part of p from v to sj . At least one of these is a valid path running between distinct terminals (even if k = i or k = j). Moreover, since it is missing at least one of the positive distance label vertices of p, it must have length strictly less than 1. This contradicts the feasibility of d. The lemma follows. ✷ Let h be a solution to LP (19.2) that assigns distance labels of 1 to each vertex in M int , 1/2 to each vertex in M disj , and 0 to all remaining vertices. Lemma 19.11 h is an optimal solution to LP (19.2). Proof: Any valid path, p, from terminal si to sj must use vertices of both boundary sets Bi and Bj . Suppose it uses v ∈ Bi ∩Bj . By deﬁnition v ∈ M int , and so hv = 1. Otherwise, it uses two vertices of M disj . In either case the length of p is at least 1, thus showing that h is a feasible solution. Next we will show that the objective function value of h is the same as that of ﬂow f , thereby showing that h is optimal. Partition paths carrying nonzero ﬂow in f into two sets: P1 consists of paths that use one vertex of M int and P2 consists of paths that use two vertices of M disj . By Lemma 19.10 these are the only two possibilities. By the primal complementary slackness conditions and the optimality of d, each vertex in M is saturated by f .

19.4

Exercises

163

Therefore, the total ﬂow carried by paths in P1 is in P2 is 1 v∈M disj cv . Hence the total ﬂow is 2 cv + v∈M int

v∈M int cv

and by paths

1 2

cv = v∈M disj v∈V −S

hv cv . ✷

This proves the lemma.

Clearly h can be obtained from an optimal solution, d, to LP (19.2) in polynomial time. This gives: Theorem 19.12 LP (19.2) always has a half-integral solution. Moreover, any optimal solution can be converted into such a solution in polynomial time.

19.4 Exercises

In Chapter 4 we presented a 2 − 2/k factor algorithm for the minimum multiway cut problem by comparing the solution found to the integral optimal solution. In the next two exercises we develop an algorithm with the same guarantee using LP-duality. 19.1 Given terminals s1 , . . . , sk , consider the multicommodity ﬂow problem in which each pair of terminals can form a source–sink pair. Thus there are k 2 commodities. Give an LP for maximizing this multicommodity ﬂow and obtain the dual LP. The dual seeks a distance label assignment for edges satisfying the triangle inequality and ensures that the distance between any two terminals is at least 1. An optimal solution to the dual can be viewed as a fractional multiway cut. 19.2 Consider the following algorithm for ﬁnding a multiway cut. Solve the dual LP to obtain an optimal fractional multiway cut. This gives a distance label assignment, say d. Pick ρ at random in the interval [0, 1 ]. An edge 2 (u, v) is picked iﬀ for some terminal s, d(u, s) ≤ ρ ≤ d(v, s). Prove that the expected cost of the cut picked is at most twice the optimal fractional multiway cut. Derandomize this algorithm, and give a modiﬁcation to make it a factor 2 − 2/k algorithm. Hint: Show that for each edge (u, v), the probability that it is picked is bounded by 2 · d(u, v). 19.3 In an attempt to improve the factor of the previous algorithm, suppose we choose ρ at random in the interval [0, 1]. What goes wrong? How is this rectiﬁed in Algorithm 19.4?

164

19

Multiway Cut

19.4 Derive Theorem 19.7 from Lemma 19.6. Hint: Lemma 19.6 implies that Pr[c(C) ≤ 1.5 · OPTf ] ≥ 2/k ≥ 2/n. Run Algorithm 19.4 polynomially many times and output the best cut. 19.5 How does the approximation guarantee of the algorithm change if σ is picked to be a random permutation from Sk ? 19.6 (Y. Rabani) For the case k = 3, replace the randomized rounding procedure of Algorithm 19.4 with the following. Pick ρ1 and ρ2 independently and uniformly from (0, 1). Pick one of the three dimensions at random, say i. Merge with si all nonterminals v satisfying xi ≥ ρ1 . Arbitrarily pick one of v the remaining two dimensions, say j, and denote the third dimension by k. Merge with sj all remaining nonterminals v satisfying xj +xi /2 ≥ ρ2 . Finally, v v merge with sk all remaining nonterminals. Show that this modiﬁed algorithm achieves an approximation guarantee of 7/6 for the 3-way cut problem. 19.7 We present another relaxation for the multiway cut problem for which the worst integrality gap known is no worse than that for LP (19.1) (see also Chapter 30). Given an undirected graph G = (V, E) with costs on edges, obtain the directed graph H by replacing each edge (u, v) of G by two directed edges (u → v) and (v → u), each having the same cost as (u, v). Assign a 0/1 indicator variable de to each edge e in H. Suppose the terminals are numbered s1 , . . . , sk in some order. Let P be the collection of all simple paths from a lower-numbered terminal to a higher-numbered terminal. Consider the following bidirected integer programming formulation for the multiway cut problem. minimize e∈H c(e)de de ≥ 1, e∈p (19.4) p∈P e∈H

subject to

de ∈ {0, 1},

1. Show that an optimal solution to IP (19.4) yields an optimal solution to the multiway cut problem. 2. Obtain the LP-relaxation and dual program. Give a good physical interpretation of the dual. 3. Show that the graph given in Example 19.2 has an integrality gap of 16/15 for this relaxation as well (by showing a primal and dual solution of cost 7.5). 4. Show that the cost of the optimal solution to the integer program and the relaxation is independent of the ordering imposed on the terminals. 19.8 Consider Algorithm 4.3 for the multiway cut problem. Show that the analogous algorithm for the node multiway cut problem, based on isolating

19.4

Exercises

165

cuts, does not achieve a constant factor. What is the best factor you can prove for this algorithm? 19.9 The multiway cut problem also possesses the half-integrality property. Give an LP for the multiway cut problem similar to LP (19.2), and prove this fact. 19.10 Show that the lower bound on OPT given by LP (19.2) can be smaller by a factor of 2 − 2/k by giving a graph in which the optimal node multiway cut is 2 − 2/k times bigger than the maximum ﬂow. 19.11 Theorem 19.12 leads directly to a factor 2 approximation algorithm for the node multiway cut problem, by rounding up the halves in a halfintegral solution. Obtain a factor 2 − 2/k algorithm, and give a family of tight examples for this algorithm. Hint: Not all vertices of M disj are required for obtaining a multiway cut. For the tight example, consider the following graph.

s3 s2 2 2 k+ ε 2 .. .

s1

2

sk

19.12 Consider the following problem. Problem 19.13 (Directed multiway cut) Given a directed graph G = (V, E) with an assignment of capacities to edges, c : E → R+ , and a set of terminals S = {s1 , s2 , . . . , sk } ⊆ V , a directed multiway cut is a set of edges whose removal ensures that the remaining graph has no path from si to sj for each pair of distinct terminals si and sj . The directed multiway cut problem asks for the minimum cost such set. Obtain an LP-relaxation for this problem similar to LP (19.2). The dual can be interpreted as a directed multicommodity ﬂow LP. Find the optimal fractional directed multiway cut and ﬂow in the following example:

166

19

Multiway Cut

s1

s2

Notice that unlike LP (19.2), this relaxation does not always have an optimal half-integral solution. 19.13 Let us deﬁne the following two problems: Problem 19.14 (Subset feedback edge set) Given a connected, undirected graph G = (V, E) with an assignment of weights to edges, w : E → R+ , and a set of special vertices S = {s1 , s2 , . . . , sk } ⊆ V , a subset feedback edge set is a set of edges whose removal ensures that the remaining graph has no cycle containing a special vertex. The subset feedback edge set problem asks for the minimum weight such set. Problem 19.15 (Subset feedback vertex set) Given a connected, undirected graph G = (V, E) with an assignment of weights to vertices, w : V → R+ , and a set of special vertices S = {s1 , s2 , . . . , sk } ⊆ V , a subset feedback vertex set is a subset of V − S whose removal ensures that the remaining graph has no cycle containing a special vertex. The subset feedback vertex set problem asks for the minimum weight such set. These and previously introduced problems are related by approximation factor preserving reductions given in the following ﬁgure (each arrow represents such a reduction). Give these reductions. For a deﬁnition of such reductions, see Section A.3.1. ✟ ✟✟ ✙ ✟

Vertex Cover

❍❍

Multiway Cut

Multicut in Trees

Node Multiway Cut

❍❍ ❥

✟ ✙ ✟✟

✟ ❍❍ ❍❍ ❥

Subset Feedback Edge Set

Directed Multiway Cut

❄

❄

Feedback Vertex Set

◗

Subset Feedback Vertex Set

◗ s ◗

✠

19.5

Notes

167

The current best factors known for multiway cut and subset feedback vertex set are 1.34 and 8, respectively. For the rest of the problems, the current best factor is 2.

19.5 Notes

Algorithm 19.4 is due to Calinescu, Karloﬀ, and Rabani [36]. The current best guarantee known for the multiway cut problem is 1.3438, due to Karger, Klein, Stein, Thorup, and Young [157]. This is also the best upper bound known on the integrality gap of the relaxation used. Freund and Karloﬀ 1 [87] give a family of instances achieving a lower bound of 8/(7 + k−1 ) on the integrality gap; Example 19.2 is from their paper. Theorem 19.12 is due to Garg, Vazirani, and Yannakakis [96]. For currently best approximation algorithms known for directed multiway cut, subset feedback edge set, and subset feedback vertex set, see Naor and Zosin [210], Even, Naor, Schieber, and Zosin [76], and Even, Naor, and Zosin [77], respectively.

20 Multicut in General Graphs

The importance of min–max relations to combinatorial optimization was mentioned in Chapter 1. Perhaps the most useful of these is the celebrated max-ﬂow min-cut theorem. Indeed, much of ﬂow theory, and the theory of cuts in graphs, has been built around this theorem. It is not surprising, therefore, that a concerted eﬀort was made to obtain generalizations of this theorem to the case of multiple commodities. There are two such generalizations. In the ﬁrst one, the objective is to maximize the sum of the commodities routed, subject to ﬂow conservation and capacity constraints. In the second generalization, a demand dem(i) is speciﬁed for each commodity i, and the objective is to maximize f , called throughput, such that for each i, f · dem(i) amount of commodity i can be routed simultaneously. We will call these sum multicommodity ﬂow and demands multicommodity ﬂow problems, respectively. Clearly, for the case of a single commodity, both problems are the same as the maximum ﬂow problem. Each of these generalizations is associated with a fundamental NP-hard cut problem, the ﬁrst with the minimum multicut problem, Problem 18.1, and the second with the sparsest cut problem, Problem 21.2. In each case an approximation algorithm for the cut problem gives, as a corollary, an approximate max-ﬂow min-cut theorem. In this chapter we will study the ﬁrst generalization; the second is presented in Chapter 21. We will obtain an O(log k) factor approximation algorithm for the minimum multicut problem, where k is the number of commodities. A factor 2 algorithm for the special case of trees was presented in Chapter 18.

20.1 Sum multicommodity ﬂow

Problem 20.1 (Sum multicommodity ﬂow) Let G = (V, E) be an undirected graph with nonnegative capacity ce for each edge e ∈ E. Let {(s1 , t1 ), . . . , (sk , tk )} be a speciﬁed set of pairs of vertices where each pair is distinct, but vertices in diﬀerent pairs are not required to be distinct. A separate commodity is deﬁned for each (si , ti ) pair. For convenience, we will think of si as the source and ti as the sink of this commodity. The objective

20.1

Sum multicommodity ﬂow

169

is to maximize the sum of the commodities routed. Each commodity must satisfy ﬂow conservation at each vertex other than its own source and sink. Also, the sum of ﬂows routed through an edge, in both directions combined, should not exceed the capacity of this edge. Let us ﬁrst give a linear programming formulation for this problem. For each commodity i, let Pi denote the set of all paths from si to ti in G, and k let P = i=1 Pi . The LP will have a variable fp for each p ∈ P , which will denote the ﬂow along path p. The endpoints of this path uniquely specify the commodity that ﬂows on this path. The objective is to maximize the sum of ﬂows routed on these paths, subject to edge capacity constraints. Notice that ﬂow conservation constraints are automatically satisﬁed in this formulation. The program has exponentially many variables; however, that is not a concern since we will use it primarily to obtain a clean formulation of the dual program. maximize p∈P fp fp ≤ ce , p:e∈p (20.1) e∈E p∈P

subject to

fp ≥ 0,

Let us obtain the dual of this program. For this, let de be the dual variable associated with edge e. We will interpret these variables as distance labels of edges. minimize e∈E ce de de ≥ 1, e∈p (20.2) p∈P e∈E

de ≥ 0,

The dual program tries to ﬁnd a distance label assignment to edges so that on each path p ∈ P , the distance labels of edges add up to at least 1. Equivalently, a distance label assignment is feasible iﬀ for each commodity i, the shortest path from si to ti has length at least 1. Notice that the programs (18.2) and (18.1) are special cases of the two programs presented above for the restriction that G is a tree. The following remarks made in Chapter 18 hold for the two programs presented above as well: an optimal integral solution to LP (20.2) is a minimum multicut, and an optimal fractional solution can be viewed as a minimum fractional multicut. By the LP-duality theorem, minimum fractional multicut equals maximum multicommodity ﬂow and, as shown in Example 18.2, it may be strictly smaller than minimum integral multicut.

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This naturally raises the question whether the ratio of minimum multicut and maximum multicommodity ﬂow is bounded. Equivalently, is the integrality gap of LP (20.2) bounded? In the next section we present an algorithm for ﬁnding a multicut within an O(log k) factor of the maximum ﬂow, thereby showing that the gap is bounded by O(log k).

20.2 LP-rounding-based algorithm

First notice that the dual program (20.2) can be solved in polynomial time using the ellipsoid algorithm, since there is a simple way of obtaining a separation oracle for it: simply compute the length of a minimum si –ti path, for each commodity i, w.r.t. the current distance labels. If all these lengths are ≥ 1, we have a feasible solution. Otherwise, the shortest such path provides a violated inequality. Alternatively, the LP obtained in Exercise 20.1 can be solved in polynomial time. Let de be the distance label computed for each edge e, and let F = e∈E ce de . Our goal is to pick a set of edges of small capacity, compared to F , that is a multicut. Let D be the set of edges with positive distance labels, i.e., D = {e | de > 0}. Clearly, D is a multicut; however, its capacity may be very large compared to F (Exercises 20.3 and 20.4). How do we pick a small capacity subset of D that is still a multicut? Since the optimal fractional multicut is the most cost-eﬀective way of disconnecting all source–sink pairs, edges with large distance labels are more important than those with small distance labels for this purpose. The algorithm described below indirectly gives preference to edges with large distance labels. The algorithm will work on graph G = (V, E) with edge lengths given by de . The weight of edge e is deﬁned to be ce de . Let dist(u, v) denote the length of the shortest path from u to v in this graph. For a set of vertices S ⊂ V , δ(S) denotes the set of edges in the cut (S, S), c(S) denotes the capacity of this cut, i.e., the total capacity of edges in δ(S), and wt(S) denotes the weight of set S, which is roughly the sum of weights of all edges having both endpoints in S (a more precise deﬁnition is given below). The algorithm will ﬁnd disjoint sets of vertices, S1 , . . . , Sl , l ≤ k, in G, called regions, such that: • No region contains any source–sink pair, and for each i, either si or ti is in one of the regions. • For each region Si , c(Si ) ≤ ε wt(Si ), where ε is a parameter that will be deﬁned below. By the ﬁrst condition, the union of the cuts of these regions, i.e., M = δ(S1 ) ∪ δ(S2 ) ∪ . . . ∪ δ(Sl ), is a multicut, and by the second condition, its capacity c(M ) ≤ εF . (When we give the precise deﬁnition of wt(S), this inequality will need to be modiﬁed slightly.)

20.2

LP-rounding-based algorithm

171

20.2.1

Growing a region: the continuous process

The sets S1 , . . . , Sl are found through a region growing process. Let us ﬁrst present a continuous process to clarify the issues. For the sake of time eﬃciency, the algorithm itself will use a discrete process (see Section 20.2.2). Each region is found by growing a set starting from one vertex, which is the source or sink of a pair. This will be called the root of the region. Suppose the root is s1 . The process consists of growing a ball around the root. For each radius r, deﬁne S(r) to be the set of vertices at a distance ≤ r from s1 , i.e., S(r) = {v | dist(s1 , v) ≤ r}. S(0) = {s1 }, and as r increases continuously from 0, at discrete points, S(r) grows by adding vertices in increasing order of their distance from s1 . Lemma 20.2 If the region growing process is terminated before the radius becomes 1/2, then the set S that is found contains no source–sink pair. Proof: The distance between any pair of vertices in S(r) is ≤ 2r. Since for ✷ each commodity i, dist(si , ti ) ≥ 1, the lemma follows. For technical reasons that will become clear in Lemma 20.3 (see also Exercises 20.5 and 20.6), we will assign a weight to the root, wt(s1 ) = F/k. The weight of S(r) is the sum of wt(s1 ) and the sum of the weights of edges, or parts of edges, in the ball of radius r around s1 . Let us state this formally. For edges e having at least one endpoint in S(r), let qe denote the fraction of edge e that is in S(r). If both endpoints of e are in S(r), then qe = 1. Otherwise, suppose e = (u, v) with u ∈ S(r) and v ∈ S(r). For such edges, qe = r − dist(s1 , u) . dist(s1 , v) − dist(s1 , u)

Deﬁne the weight of region S(r), wt(S(r)) = wt(s1 ) + ce de qe ,

where the sum is over all edges having at least one endpoint in S(r). We want to ﬁx ε so that we can guarantee that we will encounter the condition c(S(r)) ≤ ε wt(S(r)) for r < 1/2. The important observation is that at each point the rate at which the weight of the region is growing is at least c(S(r)). Until this condition is encountered, d wt(S(r)) ≥ c(S(r)) dr > ε wt(S(r)) dr. Exercise 20.5 will help the reader gain some understanding of such a process. Lemma 20.3 Picking ε = 2 ln(k + 1) suﬃces to ensure that the condition c(S(r)) ≤ ε wt(S(r)) will be encountered before the radius becomes 1/2.

172

20

Multicut in General Graphs

Proof: The proof is by contradiction. Suppose that throughout the region growing process, starting with r = 0 and ending at r = 1/2, c(S(r)) > ε wt(S(r)). At any point the incremental change in the weight of the region is d wt(S(r)) = e ce de dqe .

Clearly, only edges having one endpoint in S(r) will contribute to the sum. Consider such an edge e = (u, v) such that u ∈ S(r) and v ∈ S(r). Then, ce de dqe = ce de dr. dist(s1 , v) − dist(s1 , u)

Since dist(s1 , v) ≤ dist(s1 , u) + de , we get de ≥ dist(s1 , v) − dist(s1 , u), and hence ce de dqe ≥ ce dr. This gives d wt(S(r)) ≥ c(S(r)) dr > ε wt(S(r)) dr. As long as the terminating condition is not encountered, the weight of the region increases exponentially with the radius. The initial weight of the region is F/k and the ﬁnal weight is at most F + F/k. Integrating we get

F+F k

F k

1 d wt(S(r)) > wt(S(r))

1 2

0

ε dr.

Therefore, ln(k + 1) > 1 ε. However, this contradicts the assumption that 2 ε = 2 ln(k + 1), thus proving the lemma. ✷ 20.2.2 The discrete process

The discrete process starts with S = {s1 } and adds vertices to S in increasing order of their distance from s1 . Essentially, it involves executing a shortest path computation from the root. Clearly, the sets of vertices found by both processes are the same. The weight of region S is redeﬁned for the discrete process as follows: wt(S) = wt(s1 ) + e ce de ,

where the sum is over all edges that have at least one endpoint in S, and wt(s1 ) = F/k. The discrete process stops at the ﬁrst point when c(S) ≤ ε wt(S), where ε is again 2 ln(k + 1). Notice that for the same set S, wt(S) in the discrete process is at least as large as that in the continuous process.

20.2

LP-rounding-based algorithm

173

Therefore, the discrete process cannot terminate with a larger set than that found by the continuous process. Hence, the set S found contains no source– sink pair. 20.2.3 Finding successive regions

The ﬁrst region is found in graph G, starting with any one of the sources as the root. Successive regions are found iteratively. Let G1 = G and S1 be the region found in G1 . Consider a general point in the algorithm when regions S1 , . . . , Si−1 have already been found. Now, Gi is deﬁned to be the graph obtained by removing vertices S1 ∪ . . . ∪ Si−1 , together with all edges incident at them from G. If Gi does not contain a source–sink pair, we are done. Otherwise, we pick the source of such a pair, say sj , as the root, deﬁne its weight to be F/k, and grow a region in Gi . All deﬁnitions, such as distance and weight, are w.r.t. graph Gi . We will denote these with a subscript of Gi . Also, for a set of vertices S in Gi , cGi (S) will denote the total capacity of edges incident at S in Gi , i.e., the total capacity of edges in δGi (S). As before, the value of ε is 2 ln(k +1), and the terminating condition is cGi (Si ) ≤ ε wtGi (Si ). Notice that in each iteration the root is the only vertex that is deﬁned to have nonzero weight.

S1

t4

S2

s1

t2

s3

S3 s2 s4 t1

δG (S1 )

1

t3

δG (S2 )

2

δG (S3 )

2

In this manner, we will ﬁnd regions S1 , . . . , Sl , l ≤ k, and will output the set M = δG1 (S1 ) ∪ . . . ∪ δGl (Sl ). Since edges of each cut are removed from the graph for successive iterations, the sets in this union are disjoint, and c(M ) = i cGi (Si ). The algorithm is summarized below. Notice that while a region is growing, edges with large distance labels will remain in its cut for a longer time, and

174

20

Multicut in General Graphs

thus are more likely to be included in the multicut found. (Of course, the precise time that an edge remains in the cut is given by the diﬀerence between the distances from the root to the two endpoints of the edge.) As promised, the algorithm indirectly gives preference to edges with large distance labels. Algorithm 20.4 (Minimum multicut) 1. Find an optimal solution to the LP (20.2), thus obtaining distance labels for edges of G. 2. ε ← 2 ln (k + 1) , H ← G, M ← ∅; 3. While ∃ a source–sink pair in H do: Pick such a source, say sj ; Grow a region S with root sj until cH (S) ≤ ε wtH (S); M ← M ∪ δH (S); H ← H with vertices of S removed; 4. Output M . Lemma 20.5 The set M found is a multicut. Proof: We need to prove that no region contains a source–sink pair. In each iteration i, the sum of weights of edges of the graph and the weight deﬁned on the current root is bounded by F + F/k. By the proof of Lemma 20.3, the continuous region growing process is guaranteed to encounter the terminating condition before the radius of the region becomes 1/2. Therefore, the distance between a pair of vertices in the region, Si , found by the discrete process is also bounded by 1. Notice that we had deﬁned these distances w.r.t. graph Gi . Since Gi is a subgraph of G, the distance between a pair of vertices in G cannot be larger than that in Gi . Hence, Si contains no source–sink pair. ✷ Lemma 20.6 c(M ) ≤ 2εF = 4 ln(k + 1)F . Proof: In each iteration i, by the terminating condition we have cGi (Si ) ≤ ε wtGi (Si ). Since all edges contributing to wtGi (Si ) will be removed from the graph after this iteration, each edge of G contributes to the weight of at most one region. The total weight of all edges of G is F . Since each iteration helps disconnect at least one source–sink pair, the number of iterations is bounded by k. Therefore, the total weight attributed to source vertices is at most F . Summing gives: c(M ) = i cGi (Si ) ≤ ε i wtGi (Si )

≤ε

k

F + k

ce de e = 2εF. ✷

20.3

A tight example

175

Theorem 20.7 Algorithm 20.4 achieves an approximation guarantee of O(log k) for the minimum multicut problem. Proof: The proof follows from Lemmas 20.5 and 20.6, and from the fact that the value of the fractional multicut, F , is a lower bound on the minimum multicut. ✷ Exercise 20.6 justiﬁes the choice of wt(s1 ) = F/k. Corollary 20.8 In an undirected graph with k source–sink pairs, max |F | ≤ min |C| ≤ O(log k) max |F | ,

m/c ﬂow F

multicut C

m/c ﬂow F

where |F | represents the value of multicommodity ﬂow F , and |C| represents the capacity of multicut C.

20.3 A tight example

Example 20.9 We will construct an inﬁnite family of graphs for which the integrality gap for LP (20.2) is Ω(log k), thereby showing that our analysis of Algorithm 20.4 and the approximate max-ﬂow min-multicut theorem presented in Corollary 20.8 are tight within constant factors. The construction uses expander graphs. An expander is a graph G = (V, E) in which every vertex has the same degree, say d, and for any nonempty subset S ⊂ V , |E(S, S)| > min(|S|, |S|), where E(S, S) denotes the set of edges in the cut (S, S), i.e., edges that have one endpoint in S and the other in S. Standard probabilistic arguments show that almost every constant degree graph, with d ≥ 3, is an expander (see Section 20.6). Let H be such a graph containing k vertices. Source–sink pairs are designated in H as follows. Consider a breadth ﬁrst search tree rooted at some vertex v. The number of vertices within distance α−1 of vertex v is at most 1+d+d2 +. . .+dα−1 < dα . Picking α = logd k/2 ensures that at least k/2 vertices are at a distance ≥ α from v. Let us say that a pair of vertices are a source–sink pair if the distance between them is at least α. Therefore, we have chosen Θ(k 2 ) pairs of vertices as source–sink pairs. Each edge in H is of unit capacity. Thus, the total capacity of edges of H is O(k). Since the distance between each source–sink pair is Ω(log k), any ﬂow path carrying a unit of ﬂow uses up Ω(log k) units of capacity. Therefore,

176

20

Multicut in General Graphs

the value of maximum multicommodity ﬂow in H is bounded by O(k/log k). Next we will prove that a minimum multicut in H, say M , has capacity Ω(k), thereby proving the claimed integrality gap. Consider the connected components obtained by removing M from H. Claim 20.10 Each connected component has at most k/2 vertices. Proof: Suppose a connected component has strictly more than k/2 vertices. Pick an arbitrary vertex v in this component. By the argument given above, the number of vertices that are within distance α − 1 of v in the entire graph H is < dα ≤ k/2. Thus, there is a vertex u in the component such that the distance between u and v is at least α, i.e., u and v form a source–sink pair. Thus removal of M has failed to disconnect a source–sink pair, leading to a contradiction. ✷ By Claim 20.10, and the fact that H is an expander, each component S has |δ(S)| ≥ |S|. Since each vertex of H is in one of the components, S |δ(S)| ≥ k, where the sum is over all connected components. Since an edge contributes to the cuts of at most two components, the number of edges crossing components is Ω(k). This gives the desired lower bound on the minimum multicut. Next, let us ensure that the number of source–sink pairs deﬁned in the graph is not related to the number of vertices in it. Notice that replacing an edge of H by a path of unit capacity edges does not change the value of maximum ﬂow or minimum multicut. Using this operation we can construct from H a graph G having n vertices, for arbitrary n ≥ k. The integrality gap of LP (20.2) for G is Ω(log k). ✷

20.4 Some applications of multicut

We will obtain an O(log n) factor approximation algorithm for the following problem by reducing to the minimum multicut problem. See Exercise 20.7 for further applications. Problem 20.11 (2CNF≡ clause deletion) A 2CNF≡ formula consists of a set of clauses of the form (u ≡ v), where u and v are literals. Let F be such a formula, and wt be a function assigning nonnegative rational weights to its clauses. The problem is to delete a minimum weight set of clauses of F so that the remaining formula is satisﬁable. Given a 2CNF≡ formula F on n Boolean variables, let us deﬁne graph G(F ) with edge capacities as follows: The graph has 2n vertices, one corresponding to each literal. Corresponding to each clause (p ≡ q) we include the two edges (p, q) and (p, q), each having capacity equal to the weight of the clause (p ≡ q).

20.5

Exercises

177

Notice that the two clauses (p ≡ q) and (p ≡ q) are equivalent. We may assume w.l.o.g. that F does not contain two such equivalent clauses, since we can merge their weights and drop one of these clauses. With this assumption each clause corresponds to two distinct edges in G(F ). Lemma 20.12 Formula F is satisﬁable iﬀ no connected component of G(F ) contains a variable and its negation. Proof: If (p, q) is an edge in G(F ) then the literals p and q must take the same truth value in every satisfying truth assignment. Thus, all literals of a connected component of G(F ) are forced to take the same truth value. Therefore, if F is satisﬁable, no connected component in G(F ) contains a variable and its negation. Conversely, notice that if literals p and q occur in the same connected component, then so do their negations. If no connected component contains a variable and its negation, the components can be paired so that in each pair, one component contains a set of literals and the other contains the complementary literals. For each pair, set the literals of one component to true and the other to false to obtain a satisfying truth assignment. ✷ For each variable and its negation, designate the corresponding vertices in G(F ) to be a source–sink pair, thus deﬁning n source–sink pairs. Let M be a minimum multicut in G(F ) and C be a minimum weight set of clauses whose deletion makes F satisﬁable. In general, M may have only one of the two edges corresponding to a clause. Lemma 20.13 wt(C) ≤ c(M ) ≤ 2 · wt(C). Proof: Delete clauses corresponding to edges of M from F to get formula F . The weight of clauses deleted is at most c(M ). Since G(F ) does not contain any edges of M , it does not have any component containing a variable and its negation. By Lemma 20.12, F is satisﬁable, thus proving the ﬁrst inequality. Next, delete from G(F ) the two edges corresponding to each clause in C. This will disconnect all source–sink pairs. Since the capacity of edges deleted is 2wt(C), this proves the second inequality. ✷ Since we can approximate minimum multicut to within an O(log n) factor, we get: Theorem 20.14 There is an O(log n) factor approximation algorithm for Problem 20.11.

20.5 Exercises

20.1 By deﬁning for each edge e and commodity i a ﬂow variable fe,i , give an LP that is equivalent to LP (20.1) and has polynomially many variables.

178

20

Multicut in General Graphs

Obtain the dual of this program and show that it is equivalent to LP (20.2); however, unlike LP (20.2), it has only polynomially many constraints. 20.2 Let d be an optimal solution to LP (20.2). Show that d must satisfy the triangle inequality. 20.3 Intuitively, our goal in picking a multicut is picking edges that are bottlenecks for multicommodity ﬂow. In this sense, D is a very good starting point: prove that D is precisely the set of edges that are saturated in every maximum multicommodity ﬂow. Hint: Use complementary slackness conditions. 20.4 Give an example to show that picking all of D gives an Ω(n) factor for multicut. 20.5 Consider the following growth process. W (t) denotes the weight at time t. Assume that the initial weight is W (0) = W0 , and that at each point the rate of growth is proportional to the current weight, i.e., dW (t) = εW (t)dt. Give the function W (t). Next, assume that W0 = F/k and that W (1/2) = F + F/k. What is ε? Hint: W (t) = W0 eεt and ε = 2 ln(k + 1). 20.6 This exercise justiﬁes the choice of wt(s1 ), which was ﬁxed to be F/k. Suppose we ﬁx it at W0 . Clearly, ε is inversely related to W0 (see Lemma 20.3). However, the approximation factor of the algorithm is given by ε(F + kW0 ) (see Lemma 20.6). For what value of W0 is the approximation factor minimized? 20.7 Consider the following problem, which has applications in VLSI design. Problem 20.15 (Graph bipartization by edge deletion) Given an edge weighted undirected graph G = (V, E), remove a minimum weight set of edges to leave a bipartite graph. Obtain an O(log n) factor approximation algorithm for this problem by reducing it to Problem 20.11. 20.8 (Even, Naor, Schieber, and Rao [75]) This exercise develops an O(log2 n) factor algorithm for the following problem. Problem 20.16 (Minimum length linear arrangement) Given an undirected graph G = (V, E), ﬁnd a numbering of its vertices from 1 to n, h : V → {1, . . . , n}, so as to minimize

20.6

Notes

179

|h(u) − h(v)|.

(u,v)∈E

1. Show that the following is an LP-relaxation of this problem. This LP has a variable de for each edge e ∈ E, which we will interpret as a distance label. For any distance label assignment d to the edges of G, deﬁne distd (u, v) to be the length of the shortest path from u to v in G. Give a polynomial time separation oracle for this LP, thereby showing that it can be solved in polynomial time. minimize e∈E de distd (u, v) ≥ u∈S (20.3) 1 (|S|2 − 1), 4 S ⊆ V, v ∈ S e∈E

de ≥ 0,

2. Let d be an optimal solution to LP (20.3). Show that for any S ⊆ V, v ∈ S there is a vertex u ∈ S such that distd (u, v) ≥ (|S| + 1)/4. 3. For S ⊆ V , deﬁne wt(S) to be the sum of distance labels of all edges having both endpoints in S. Also, deﬁne c(S, S) to be the number of edges in the cut (S, S). Give a region growing process similar to that described in Section 20.2.1 that ﬁnds a cut (S, S) in G with wt(S) ≤ wt(S) such that c(S, S) is O(wt(S)(log n)/n). 4. Show that a divide-and-conquer algorithm that recursively ﬁnds a numbering for vertices in S from 1 to |S|, and a numbering for vertices in S from |S| + 1 to n achieves an approximation guarantee of O(log2 n). Hint: Assuming each edge in the cut (S, S) is of length n − 1, write a suitable recurrence for the cost incurred by the algorithm.

20.6 Notes

Theorem 20.7 and its corollary are due to Garg, Vazirani, and Yannakakis [97]. Problem 20.11 was introduced in Klein, Rao, Agrawal, and Ravi [173]. For showing existence of expanders via a probabilistic argument, see Pinsker [220].

21 Sparsest Cut

In this chapter we will obtain an approximation algorithm for the sparsest cut problem using an interesting LP-rounding procedure that employs results on low distortion embeddings of metrics in 1 spaces. As mentioned in Chapter 20, we will get as a corollary an approximate max-ﬂow min-cut theorem for the demands version of multicommodity ﬂow. Approximation algorithms for several other important problems will also follow.

21.1 Demands multicommodity ﬂow

Problem 21.1 (Demands multicommodity ﬂow) Let G = (V, E) be an undirected graph with a nonnegative capacity ce for each edge e ∈ E. Let {(s1 , t1 ), . . . , (sk , tk )} be a speciﬁed set of pairs of vertices, where each pair is distinct, but vertices in diﬀerent pairs are not required to be distinct. A separate commodity is deﬁned for each (si , ti ) pair; for convenience, we will think of si as the source and ti as the sink of this commodity. For each commodity i, a nonnegative demand, dem(i), is also speciﬁed. The objective is to maximize f , called throughput, such that for each commodity i, f · dem(i) units of this commodity can be routed simultaneously, subject to ﬂow conservation and capacity constraints, i.e., each commodity must satisfy ﬂow conservation at each vertex other than its own source and sink, and the sum of ﬂows routed through an edge, in both directions combined, should not exceed the capacity of this edge. We will denote the optimal throughput by f ∗. Consider a cut (S, S) in G. Let c(S) denote the capacity of edges in this cut and dem(S) denote the total demand separated by this cut, i.e., dem(S) = i: |{si ,ti }∩S|=1

dem(i).

Clearly, the ratio of these quantities places an upper bound on the throughput, i.e., f ∗ ≤ c(S) . This motivates: dem(S) Problem 21.2 (Sparsest cut) Let G = (V, E) be an undirected graph with capacities, source–sink pairs, and demands deﬁned as in Problem 21.1.

21.2

Linear programming formulation

181

The sparsity of cut (S, S) is given by c(S) . The problem is to ﬁnd a cut dem(S) of minimum sparsity. We will denote the sparsity of this cut by α∗ . Among all cuts, α∗ puts the most stringent upper bound on f ∗ . Is this upper bound tight? Example 21.3 shows that it is not. However, minimum sparsity cannot be arbitrarily larger than maximum throughput; we will show that their ratio is bounded by O(log k). Example 21.3 Consider the bipartite graph K3,2 with all edges of unit capacity and a unit demand between each pair of nonadjacent vertices – a total of four commodities.

1/2

1

1/2

1

1

1/2 1/2

1

1/2 1/2

It is easy to check that a sparsest cut of K3,2 has sparsity 1. This graph can be viewed as the union of two stars K3,1 (the centers of the stars are the vertices on the right side of the bipartition), and, as in Example 18.2, we get the unique way of routing one unit of each of the three commodities having source and sink on the left side of the bipartition. However, this saturates all edges, making it impossible to route the fourth commodity. Hence, throughput is strictly smaller than 1. ✷

21.2 Linear programming formulation

We start by giving a linear programming formulation of the problem of maxi imizing throughput, f . Let Pi = {qj } denote the set of all paths between si i and ti . Introduce variable fj to denote the ﬂow of commodity i sent along i path qj . The ﬁrst set of constraints ensures that the demand of each commodity is met (with factor f ), and the second set are edge capacity constraints. maximize subject to j i fj ≤ ce , i e∈qj

f i fj ≥ f · dem(i),

(21.1) i = 1, . . . , k e∈E

f ≥0 i fj ≥ 0

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21

Sparsest Cut

Deﬁne the graph H with vertex set VH = {si , ti |1 ≤ i ≤ k} and edge set EH = {(si , ti )|1 ≤ i ≤ k} to be the demand graph. For each edge e = (si , ti ) of H, let dem(e) = dem(i). We will show that the dual to LP (21.1) yields a metric (V, d) satisfying: Theorem 21.4 Let f ∗ denote the optimal throughput. Then, f∗ = min e∈G ce de e∈H

metric d

dem(e)de

.

Let li and de be dual variables associated with the ﬁrst and second set of inequalities of LP (21.1). We will interpret de ’s as distance label assignments to the edges of G. The ﬁrst set of inequalities ensures that for each commodity i, li is upper bounded by the length of any path from si to ti w.r.t. the distance label assignment. minimize e∈E ce de d e ≥ li , i e∈qj

(21.2) i qj ∈ Pi , i = 1, . . . , k

subject to k li dem(i) ≥ 1 de ≥ 0, li ≥ 0, i=1 e∈E i = 1, . . . , k

Example 21.5 For the instance given in Example 21.3, the optimal throughput is f ∗ = 3/4; this corresponds to routing the four commodities as follows:

3/8 1/4 1/4 1/4 3/8 3/8 3/8 3/8

3/8

The optimal dual solution is: de = 1/8 for each edge e and li = 1/4 for each commodity i. It would be instructive for the reader to verify feasibility and optimality of these solutions. ✷

21.3

Metrics, cut packings, and

1 -embeddability

183

Claim 21.6 There is an optimal distance label assignment d for the dual program (21.2) that is a metric on V . Furthermore, for each commodity i, li = d(si ,ti ) , and the second inequality holds with equality, i.e., i d(si ,ti ) dem(i) = 1. Proof: If for some three points u, v, and w, duw > duv + dvw , then decrease duw to duv + dvw . Since this does not decrease the shortest path between any si –ti pair, the solution still remains feasible. Moreover, the objective function value cannot increase by this process. Continuing in this manner, we will obtain a metric on V . Now, the length of a shortest path from si to ti is given by the distance label d(si ,ti ) . Setting li = d(si ,ti ) does not change the feasibility or the objective function value of the solution. Finally, if the second inequality holds strictly, then we can scale down all distance labels without violating feasibility, thus contradicting the optimality of d. ✷ By Claim 21.6, the dual program yields a metric (V, d) that minimizes e∈G ce de e∈H

dem(e)de

.

By the LP-duality theorem, this equals the optimal throughput. This proves Theorem 21.4.

21.3 Metrics, cut packings, and

1 -embeddability

In Section 21.3.1, we will deﬁne the notion of a cut packing for a metric and will show that the question of ﬁnding a good approximation to the sparsest cut for graph G reduces to that of ﬁnding a “good” cut packing for the metric obtained in Theorem 21.4. The latter question is reduced, in Section 21.3.2, to the question of ﬁnding a “good” 1 -embedding for the metric. Eventually, Section 21.4 deals with ﬁnding the embedding itself. 21.3.1 Cut packings for metrics

Let us think of a metric (V, d) as deﬁning the lengths of edges of the complete graph on V . Let En denote the set of all edges in the complete graph on n vertices. Let y be a function assigning nonnegative values to subsets of V , i.e., y : 2V → R+ . We will denote the value of y on set S by yS . As before, let us say that edge e feels yS if e is in the cut (S, S). The amount of cut that edge e feels is S:e∈δ(S) y(S). Function y is called a cut packing for metric (V, d) if no edge feels more cut than its length, i.e., for each edge e ∈ En , S:e∈δ(S) y(S) ≤ de . If this inequality holds with equality for each edge e ∈ En , then y is said to be an exact cut packing. The reason for the

184

21

Sparsest Cut

name “cut packing” is that equivalently, we can think of y as assigning value y(S) + y(S) to each cut (S, S). As shown below, in general, there may not be an exact cut packing for metric (V, d). Let us relax this notion by allowing edges to be underpacked up to a speciﬁed extent. For β ≥ 1, y is said to be a β-approximate cut packing if the amount of cut felt by any edge is at least 1/β fraction of its length, i.e., for each edge e ∈ En , de /β ≤ S:e∈δ(S) y(S) ≤ de . Clearly, the smaller β is, the better the cut packing. The following theorem shows the importance of ﬁnding a good cut packing for (V, d). Theorem 21.7 Let (V, d) be the metric obtained in Theorem 21.4, and let y be a β-approximate cut packing for (V, d). Among cuts with y(S) = 0, let (S , S ) be the sparsest. Then, the sparsity of this cut is at most β · f ∗ . Proof: Let y be a β-approximate cut packing for metric (V, d). Then, f∗ = = e∈G ce de e∈H S

dem(e)de

≥

e∈G ce e∈H

S:e∈δ(S)

y(S) βy(S)

dem(e)

S:e∈δ(S)

β 1 ≥ · β

y(S)c(S) y(S)dem(S) S c(S ) . dem(S )

The ﬁrst inequality follows using both the upper bound and the lower bound on the amount of cut felt by an edge; the former in the numerator and the latter in the denominator. The equality after that follows by changing the order of summation. The last inequality follows from the well known result stated below. ✷ Proposition 21.8 For any nonnegative reals a1 , . . . , an and positive reals b1 , . . . , bn and α1 , . . . , αn , i αi ai ai ≥ min . i bi i αi bi

Moreover, this inequality holds with equality iﬀ the n values ai /bi are all equal. Corollary 21.9 If there is an exact cut packing for metric (V, d), then every cut (S, S) with yS = 0 has sparsity f ∗ and thus is a sparsest cut in G. Proof: By Theorem 21.7, the minimum sparsity cut with yS = 0 has sparsity at most f ∗ (since β = 1). Since the sparsity of any cut upper bounds f ∗ , the sparsity of this cut equals f ∗ , and this is a sparsest cut in G. But then all

21.3

Metrics, cut packings, and

1 -embeddability

185

inequalities in the proof of Theorem 21.7 must hold with equality. Now, by the second statement in Proposition 21.8, we get that every cut (S, S) with yS = 0 has sparsity f ∗ . ✷ The sparsest cut in the instance speciﬁed in Example 21.3 has sparsity strictly larger than f ∗ . By Corollary 21.9, the optimal metric for this instance does not have an exact cut packing. However, it turns out that every metric has an O(log n)-approximate cut packing – we will show this using the notion of 1 -embeddability of metrics. 21.3.2

1 -embeddability

of metrics

A norm on the vector space Rm is a function · : Rm → R+ , such that for any x, y ∈ Rm , and λ ∈ R: • x = 0 iﬀ x = 0, • λx = |λ| · x , • x+y ≤ x + y . For p ≥ 1, the x p p -norm

is deﬁned by

1 p

=

1≤k≤m

|xk |p . p -metric,

The associated

denoted by d p , is deﬁned by p d p (x, y) = x − y

for all x, y ∈ Rm . In this section, we will only consider the 1 -norm. Let σ be a mapping, σ : V → Rm for some m. Let us say that σ(u) − σ(v) 1 is the 1 length of edge (u, v) under σ. We will say that σ is an isometric 1 -embedding for metric (V, d) if it preserves the 1 lengths of all edges, i.e., ∀u, v ∈ V, d(u, v) = σ(u) − σ(v) 1 . As shown below, in general, the metric computed by solving the dual program may not be isometrically 1 -embeddable. Thus, we will relax this notion – we will ensure that the mapping does not stretch any edge, but we will allow it to shrink edges up to a speciﬁed factor. For β ≥ 1, we will say that σ is a β-distortion 1 -embedding for metric (V, d) if ∀u, v ∈ V : 1 d(u, v) ≤ σ(u) − σ(v) β

1

≤ d(u, v).

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21

Sparsest Cut

Next, we show that the question of ﬁnding an approximate cut packing for a metric is intimately related to that of ﬁnding a low distortion 1 embedding for it. Lemma 21.10 Let σ : V → Rm be a mapping. There is a cut packing y : 2V → R+ such that each edge feels as much cut under y as its 1 length under σ. Moreover, the number of nonzero yS ’s is at most m(n − 1). Proof: First consider the case when m = 1. Let the n vertices of V be mapped to u1 ≤ u2 ≤ · · · ≤ un . Assume w.l.o.g. that the vertices are also numbered in this order. For each i, 1 ≤ i ≤ n − 1, let y{v1 ,...,vi } = ui+1 − ui . Clearly, this cut packing satisﬁes the required condition. For arbitrary m, we observe that since the 1 -norm is additive, we can deﬁne a cut packing for each dimension independently, and the sum of these packings satisﬁes the required condition. ✷ Lemma 21.11 Let y : 2V → R+ be a cut packing with m nonzero yS ’s. There is a mapping σ : V → Rm such that for each edge, its 1 length under σ is the same as the amount of cut it feels under y. Proof: We will have a dimension corresponding to each set S ⊆ V such that yS = 0. For vertices in S, this coordinate will be 0, and for vertices in S, this coordinate will be yS . Thus, this dimension contributes exactly as much to the 1 length of an edge as the amount of cut felt by this edge due to yS . Hence this mapping satisﬁes the required condition. ✷ Lemmas 21.10 and 21.11 give: Theorem 21.12 There exists a β-distortion 1 -embedding for metric (V, d) iﬀ there exists a β-approximate cut packing for it. Moreover, the number of nonzero cuts and the dimension of the 1 -embedding are polynomially related. Corollary 21.13 Metric (V, d) is isometrically an exact cut packing for it.

1 -embeddable

iﬀ there exists

We have already shown that the metric obtained for the instance in Example 21.3 does not have an exact cut packing. Therefore, it is not isometrically 1 -embeddable. However, we will show that any metric has an O(log n)-distortion 1 -embedding; this fact lies at the heart of the approximation algorithm for the sparsest cut problem.

21.4 Low distortion

1 -embeddings

for metrics

First consider the following one-dimensional embedding for metric (V, d): pick a set S ⊆ V , and deﬁne the coordinate of vertex v to be σ(v) = mins∈S d(s, v),

21.4

Low distortion

1 -embeddings

for metrics

187

i.e., the length of the shortest edge from v to S. This mapping does not stretch any edge: Lemma 21.14 For the one-dimensional embedding given above, ∀u, v ∈ V, |σ(u) − σ(v)| ≤ d(u, v). Proof: Let s1 and s2 be the closest vertices of S to u and v, respectively. Assume w.l.o.g. that d(s1 , u) ≤ d(s2 , v). Then, |σ(u) − σ(v)| = d(s2 , v) − d(s1 , u) ≤ d(s1 , v) − d(s1 , u) ≤ d(u, v). The last inequality follows by the triangle inequality. ✷ More generally, consider the following m-dimensional embedding: Pick m subsets of V , S1 , . . . , Sm , and deﬁne the ith coordinate of vertex v to be σi (v) = mins∈Si d(s, v)/m; notice the scaling factor of m used. The additivity of 1 metric, together with Lemma 21.14, imply that this mapping also does not stretch any edge. 21.4.1 Ensuring that a single edge is not overshrunk

The remaining task is to choose the sets in such a way that no edge shrinks by a factor of more than O(log n). It is natural to use randomization for picking the sets. Let us ﬁrst ensure that a single edge (u, v) is not overshrunk. For this purpose, deﬁne the expected contribution of set Si to the 1 length of edge (u, v) to be E[|σi (u) − σi (v)|]. For simplicity, assume that n is a power of 2; let n = 2l . For 2 ≤ i ≤ l + 1, set Si is formed by picking each vertex of V with probability 1/2i . The embedding w.r.t. these sets works for the single edge (u, v) with high probability. The proof of this fact involves cleverly taking into consideration the expected contribution of each set. For diﬀerent metrics, diﬀerent sets have a large contribution. In order to develop intuition for the proof, we ﬁrst illustrate this through a series of examples. Example 21.15 In the following three metrics, d(u, v) = 1, and the n vertices are placed as shown in the ﬁgure below. u s n/2 u s 1 u√s n s n−2 s√ n−2 n sv n/2 sv 1 √s v n

188

21

Sparsest Cut

For each metric, the expected contribution of one of the sets is Ω(d(u, v)/l). For the ﬁrst metric, this set is Sl , since it will be a singleton with constant probability. For the second metric, this set is S2 , since it will contain exactly one of u and v with constant probability. For the third metric, this set is S l/2 √ since with constant probability, it will contain exactly one vertex of , ✷ the 2 n vertices bunched up with u and v. In the next lemma, we encapsulate the basic mechanism for establishing a lower bound on the expected contribution of a set Si . For any vertex x and nonnegative real r, let B(x, r) denote the ball of radius r around x, i.e., B(x, r) = {s ∈ V |d(x, s) ≤ r}. Lemma 21.16 If for some choice of r1 ≥ r2 ≥ 0, and constant c, Pr[(Si ∩ B(u, r1 ) = ∅) and (Si ∩ B(v, r2 ) = ∅)] ≥ c, then the expected contribution of Si is ≥ c(r1 − r2 )/l. Proof: Under the event described, d(u, Si ) ≥ r1 and d(v, Si ) ≤ r2 . If so, σi (u) ≥ r1 /l and σi (v) ≤ r2 /l. Therefore, |σi (u) − σi (v)| ≥ (r1 − r2 )/l, and the lemma follows. ✷ The remaining task is to deﬁne suitable radii r1 and r2 for each set Si such that the probabilistic statement of Lemma 21.16 holds. We will need the following simple probabilistic fact: Lemma 21.17 For 1 ≤ t ≤ l − 1, let A and B be disjoint subsets of V , such that |A| < 2t and |B| ≥ 2t−1 . Form set S by picking each vertex of V independently with probability p = 1/(2t+1 ). Then, Pr[(S ∩ A = ∅) and (S ∩ B = ∅)] ≥ (1/2)(1 − e−1/4 ). Proof: Pr[S ∩ A = ∅] = (1 − p)|A| ≥ (1 − p|A|) ≥ 1 , 2

where the ﬁrst inequality follows by taking the ﬁrst two terms of the binomial expansion. Pr[S ∩ B = ∅] = (1 − p)|B| ≤ e−p|B| ≤ e−1/4 , where we have used the inequality 1 − x ≤ e−x . Therefore, Pr[S ∩ B = ∅] = 1 − (1 − p)|B| ≥ 1 − e−1/4 .

21.4

Low distortion

1 -embeddings

for metrics

189

Finally, observe that since A and B are disjoint, the two events [S ∩ A = ∅] and [S ∩ B = ∅] are independent. The lemma follows. ✷ For convenience, let c = (1/2)(1 − e−1/4 ). For 0 ≤ t ≤ l, deﬁne ρt = min{ρ ≥ 0 : |B(u, ρ)| ≥ 2t and |B(v, ρ)| ≥ t 2 }, i.e., ρt is the smallest radius such that the ball around u and the ball around v each has at least 2t vertices. Clearly, ρ0 = 0 and ρl ≥ d(u, v). Let ˆ ˆ t = max{t : ρt < d(u, v)/2}; clearly, t ≤ l − 1. Finally, for any vertex x and ◦ nonnegative real r, let B (x, r) denote the open ball of radius r around x, i.e., B ◦ (x, r) = {s ∈ V |d(x, s) < r}. ˆ Lemma 21.18 For 1 ≤ t ≤ t, the expected contribution of St+1 is at most ρt −ρt−1 ˆ + 1, the expected contribution of St+1 is at most c· , and for t = t l c l

·

d(u,v) 2

− ρt−1 .

ˆ Proof: First consider t such that 1 ≤ t ≤ t. By the deﬁnition of ρt , for at least one of the two vertices u and v, the open ball of radius ρt contains fewer than 2t vertices. Assume w.l.o.g. that this happens for vertex u, i.e., |B ◦ (u, ρt )| < 2t . Again, by deﬁnition, |B(v, ρt−1 )| ≥ 2t−1 . Since ρt−1 < ρt < d(u, v)/2, the two sets B ◦ (u, ρt ) and B(v, ρt−1 ) are disjoint. Thus, by Lemma 21.17, the probability that St+1 is disjoint from the ﬁrst set and intersects the second is least c. Now, the ﬁrst claim follows from Lemma 21.16. ˆ ˆ Next, let t = t + 1. By the deﬁnition of t, for at least one of the two vertices u and v, the open ball of radius d(u, v)/2 contains fewer than 2t vertices. As before, w.l.o.g. assume this happens for vertex u, i.e., |B ◦ (u, d(u, v)/2)| < 2t . Clearly, |B(v, ρt−1 )| ≥ 2t−1 . Since ρt−1 < d(u, v)/2, the two sets B ◦ (u, d(u, v)/2) and B(v, ρt−1 ) are disjoint. The rest of the reasoning is the same as before. ✷ Lemma 21.19 The expected contribution of all sets S2 , . . . , Sl+1 is at most c d(u,v) . 2 · l Proof: By Lemma 21.18, the expected contribution of all sets S2 , . . . , Sl+1 is at least the following telescoping sum: c · (ρ1 − ρ0 ) + (ρ2 − ρ1 ) + . . . + l d(u, v) − ρt ˆ 2 = c d(u, v) · . 2 l ✷ Lemma 21.20 Pr contribution of all sets is ≥ c d(u,v) 4l

≥

c/2 . 2 − c/2

Proof: Denote the probability in question by p. Clearly, the total contribution of all sets S2 , . . . , Sl+1 to the 1 length of edge (u, v) is at most d(u, v)/2l. This fact and Lemma 21.19 give:

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21

Sparsest Cut

p·

c d(u, v) d(u, v) d(u, v) + (1 − p) · ≥ . l 4l l c/2 2−c/2 .

Therefore, p ≥ 21.4.2

✷

Ensuring that no edge is overshrunk

The above embedding does not overshrink edge (u, v) with constant probability. In order to ensure that no edge is overshrunk, we will ﬁrst enhance this probability. The key idea is to repeat the entire process several times independently and use Chernoﬀ bounds to bound the error probability. We will use the following statement of the Chernoﬀ bound: Let X1 , . . . , Xn be indepenn dent Bernoulli trials with Pr[Xi = 1] = p, 0 < p < 1, and let X = i=1 Xi ; clearly, E[X] = n p. Then, for 0 < δ ≤ 1, Pr[X < (1 − δ) n p] < exp(−δ 2 n p/2). Pick sets S2 , . . . , Sl+1 using probabilities speciﬁed above, independently j N = O(log n) times each. Call the sets so obtained Si , 2 ≤ i ≤ l + 1, 1 ≤ j ≤ N . Consider the N · l = O(log2 n) dimensional embedding of metric (V, d) w.r.t. these N · l sets. We will prove that this is an O(log n)-distortion 1 -embedding for metric (V, d). Lemma 21.21 For N = O(log n), this embedding satisﬁes: Pr[ σ(u) − σ(v) 1 ] ≥ where p = c/(2 − c). Proof: We will think of the process of picking sets S2 , . . . , Sl+1 once as a single Bernoulli trial; thus, we have N such trials. A trial succeeds if the contribution of all its sets is ≥ (c d(u, v))/2l. By Lemma 21.20, the probability of success is at least p. Using the Chernoﬀ bound with δ = 1/2, the probability that at most N p/2 of these trials succeed is at most exp(N p/8). Clearly, this is bounded by 1/2n2 for N = O(log n). If at least N p/2 trials succeed, the 1 length of edge (u, v) will be at least p c d(u, v)/4l = d(u, v)/O(log n). The lemma follows. ✷ Adding the error probabilities for all n(n − 1)/2 edges, we get: Theorem 21.22 The N l = O(log2 n) dimensional embedding given above is an O(log n)-distortion 1 -embedding for metric (V, d), with probability at least 1/2. 1 p c d(u, v) ] ≥ 1 − 2, 4l 2n

21.5

LP-rounding-based algorithm

191

21.5 LP-rounding-based algorithm

The reader can verify that Claim 21.6 and Theorems 21.7, 21.12, and 21.22 lead to an O(log n) factor approximation algorithm for the sparsest cut problem. In this section, we will improve the approximation guarantee to O(log k) where k is the number of source–sink pairs speciﬁed. For this purpose, notice that Theorem 21.7 holds even for the following less stringent approximate cut packing: no edge is allowed to be overpacked, and the edges of the demand graph are not under-packed by more than a β factor (the rest of the edges are allowed to be under-packed to any extent). In turn, such a cut packing can be obtained from an 1 -embedding that does not overshrink edges of the demand graph only. Since these are only O(k 2 ) in number, where k is the number of source–sink pairs, we can ensure that these edges are not shrunk by a factor of more than O(log k), thus enabling an improvement in the approximation guarantee. Let V ⊆ V be the set of vertices that are sources or sinks, |V | ≤ 2k. For simplicity, assume |V | is a power of 2; let |V | = 2l . The sets S2 , . . . , Sl+1 will be picked from V , and it is easy to verify from the proof of Lemma 21.21 that N = O(log k) will suﬃce to ensure that none of the O(k 2 ) edges of the demand graph is shrunk by more than a factor of O(log k). The complete algorithm is: Algorithm 21.23 (Sparsest cut) 1. Solve the dual LP (21.2) to obtain metric (V, d). j j 2. Pick sets Si , 2 ≤ i ≤ l + 1, 1 ≤ j ≤ N , where set Si is formed by picking each vertex of V independently with probability 1/2i . 3. Obtain an 1 -embedding of (V, d) in O(log2 k)-dimensional space w.r.t. these sets. 4. Obtain an approximate cut packing for (V, d) from the 1 -embedding. 5. Output the sparsest cut used by the cut packing. Theorem 21.24 Algorithm 21.23 achieves an approximation guarantee of O(log k) for the sparsest cut problem. Corollary 21.25 For a demands multicommodity ﬂow instance with k source– sink pairs, 1 O(log k)

S ⊂ V dem(S)

min

c(S)

≤

throughput f

max

f

≤ min

c(S)

S ⊂ V dem(S)

.

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21

Sparsest Cut

21.6 Applications

We present below a number of applications of the sparsest cut problem. 21.6.1 Edge expansion

Expander graphs have numerous applications; for instance, see Example 20.9. We will obtain an O(log n) factor algorithm for the problem of determining the edge expansion of a graph: Problem 21.26 (Edge expansion) Given an undirected graph G = (V, E), the edge expansion of a set S ⊂ V with |S| ≤ n/2, is deﬁned to be |δ(S)|, i.e., the number of edges in the cut (S, S). The problem is to ﬁnd a minimum expansion set. Consider the special case of demands multicommodity ﬂow in which we have n(n − 1)/2 distinct commodities, one for each pair of vertices. This is called the uniform multicommodity ﬂow problem. For this problem, the sparsity of any cut (S, S) is given by c(S) . |S| · |S| Let (S, S), with |S| ≤ |S|, be the cut found by Algorithm 21.23 when run on G with uniform demands. Notice that |S| is known within a factor of 2, since n/2 ≤ |S| ≤ n. Thus, S has expansion within an O(log n) factor of the minimum expansion set in G. Clearly, the generalization of this problem to arbitrary edge costs also has an O(log n) factor approximation algorithm. 21.6.2 Conductance

The conductance of a Markov chain characterizes its mixing rate, i.e., the number of steps needed to ensure that the probability distribution over states is suﬃciently close to its stationary distribution. Let P be the transition matrix of a discrete-time Markov chain on a ﬁnite state space X, and let π denote the stationary probability distribution of this chain. We will assume that the chain is aperiodic, connected, and that it satisﬁes the detailed balance condition, i.e., π(x)P (x, y) = π(y)P (y, x) ∀x, y ∈ X. Deﬁne undirected graph G = (X, E) on vertex set X such that (x, y) ∈ E iﬀ π(x)P (x, y) = 0. The edge weights are deﬁned to be w(x, y) = π(x)P (x, y). The conductance of this chain is given by Φ= min w(S, S) , π(S)

S⊂X,0<π(S)≤1/2

21.6

Applications

193

where w(S, S) is the sum of weights of all edges in the cut (S, S). For any set S, the numerator of the quotient deﬁned above is the probability that the chain in equilibrium escapes from set S to S in one step. Thus the quotient gives the conditional probability of escape, given that the chain is initially in S and Φ measures the ability of the chain to not get trapped in any small region of the state space. Theorem 21.24 leads to an O(log n) factor approximation algorithm for computing conductance. First, observe that it suﬃces to approximate the following symmetrized variant of Φ: Φ = min w(S, S) , π(S)π(S) (21.3)

S⊂X,0<π(S)≤1

since Φ and Φ are within a factor of 2 of each other (notice that if 0 < π(S) ≤ 1/2, then 1/2 ≤ π(S) < 1). Next, let us show that computing Φ is really a special case of the sparsest cut problem. Consider graph G = (X, E) with edge weights as deﬁned above. For each pair of vertices x, y ∈ X, deﬁne a distinct commodity with a demand of π(x)π(y). It is easy to see that the sparsity of a cut (S, S) for this instance is simply the quotient deﬁned in (21.3). Hence, the sparsity of the sparsest cut is Φ . 21.6.3 Balanced cut

The following problem ﬁnds applications in partitioning problems, such as circuit partitioning in VLSI design. Furthermore, it can be used to perform the “divide” step of the divide-and-conquer algorithms for certain problems; for instance, see the algorithm for Problem 21.29 below. Problem 21.27 (Minimum b-balanced cut) Given an undirected graph G = (V, E) with nonnegative edge costs and a rational b, 0 < b ≤ 1/2, ﬁnd a minimum capacity cut (S, S) such that b · n ≤ |S| < (1 − b) · n. A b-balanced cut for b = 1/2 is called a bisection cut, and the problem of ﬁnding a minimum capacity such cut is called the minimum bisection problem. We will use Theorem 21.24 to obtain a pseudo-approximation algorithm for Problem 21.27 – we will ﬁnd a (1/3)-balanced cut whose capacity is within an O(log n) factor of the capacity of a minimum bisection cut (see the notes in Section 21.8 for a true approximation algorithm). For V ⊂ V , let GV denote the subgraph of G induced by V . The algorithm is: Initialize U ← ∅ and V ← V . Until |U | ≥ n/3, ﬁnd a minimum expansion set in GV , say W , then set U ← U ∪W and V ← V −W . Finally, let S ← U , and output the cut (S, V − S). Claim 21.28 The cut output by the algorithm is a (1/3)-balanced cut whose capacity is within an O(log n) factor of the capacity of a minimum bisection cut in G.

194

21

Sparsest Cut

Proof: At the end of the penultimate iteration, |U | < n/3. Thus, at the beginning of the last iteration, |V | ≥ 2n/3. At most half of these vertices are added to U in the last iteration. Therefore, |V − S| ≥ n/3 and n/3 ≤ |S| < n/3. Hence, (S, V − S) is a (1/3)-balanced cut. Let (T, T ) be a minimum bisection cut in G. Since at the beginning of each iteration, |V | ≥ 2n/3, each of the sets T ∩ V and T ∩ V has at least n/6 vertices. Thus, the expansion of a minimum expansion set in GV c(T ) in each iteration is at most (n/6) . Since the algorithm ﬁnds a set having expansion within a factor of O(log n) of optimal in any iteration, the set U found satisﬁes: c(U ) c(T ) ≤ O(log n) · . |U | n/6 Since the ﬁnal set S has at most 2n/3 vertices, summing up we get c(S) ≤ O(log n) · c(T )(2n/3) , n/6 ✷

thereby giving c(S) ≤ O(log n) · c(T ). 21.6.4 Minimum cut linear arrangement

Problem 21.29 (Minimum cut linear arrangement) Given an undirected graph G = (V, E) with nonnegative edge costs, for a numbering of its vertices from 1 to n, deﬁne Si to be the set of vertices numbered at most i, for 1 ≤ i ≤ n − 1; this deﬁnes n − 1 cuts. The problem is to ﬁnd a numbering that minimizes the capacity of the largest of these n−1 cuts, i.e., it minimizes max{c(Si )| 1 ≤ i ≤ (n − 1)}. Using the pseudo-approximation algorithm obtained above for the (1/3)balanced cut problem, we will obtain a true O(log2 n) factor approximation algorithm for this problem. A key observation is that in any arrangement, Sn/2 is a bisection cut, and thus the capacity of a minimum bisection cut in G, say β, is a lower bound on the optimal arrangement. The reason we get a true approximation algorithm is that the (1/3)-balanced cut algorithm compares the cut found to β. The algorithm is recursive: ﬁnd a (1/3)-balanced cut in GV , say (S, S), and recursively ﬁnd a numbering of S in GS using numbers from 1 to |S| and a numbering of S in GS using numbers from |S| + 1 to n. Of course, the recursion ends when the set is a singleton, in which case the prescribed number is assigned to this vertex. Claim 21.30 The algorithm given above achieves an O(log2 n) factor for the minimum cut linear arrangement problem.

21.7

Exercises

195

Proof: The following binary tree T (not necessarily complete) encodes the outcomes of the recursive calls made by the algorithm: Each recursive call corresponds to a node of the tree. Suppose recursive call α ends with two further calls, α1 and α2 , where the ﬁrst call assigns smaller numbers and the second assigns larger numbers. Then, α1 will be made the left child of α in T and α2 will be made the right child of α. If recursive call α was made with a singleton, then α will be a leaf of the tree. To each nonleaf, we will assign the set of edges in the cut found during this call, and to each leaf we will assign its singleton vertex. Thus, the left to right ordering of leaves gives the numbering assigned by the algorithm to the vertices. Furthermore, the edge sets associated with nonleaf nodes deﬁne a partitioning of all edges of G. The cost of edges associated with any nonleaf is O(log n)β by Claim 21.28. Since each recursive call ﬁnds a (1/3)-balanced cut, the depth of recursion, and hence the depth of T , is O(log n). Following is a crucial observation: Consider any edge (u, v) in G. Let α be the lowest common ancestor of leaves corresponding to u and v in T . Then, (u, v) belongs to the set of edges associated with node α. With respect to the numbering found by the algorithm, consider a cut (Si , Si ), 1 ≤ i ≤ n − 1. Any edge in this cut connects vertices numbered j and k with j ≤ i and k ≥ i + 1. Thus, such an edge must be associated with a node that is a common ancestor of the leaves numbered i and i + 1. Since the depth of T is O(log n), there are O(log n) such common ancestors. Since the cost of edges associated with any node in T is O(log n)β, the cost of cut (Si , Si ) is bounded by O(log2 n)β. The claim follows since we have already argued that β is a lower bound on the optimal arrangement. ✷

21.7 Exercises

21.1 For each of the three metrics given in Example 21.15, one of the sets S2 , . . . , Sl+1 has an expected contribution of Ω(d(u, v)/l). Give a metric for which each set has an expected contribution of Θ(d(u, v)/l2 ). 21.2 Show that n points embedded in 1 space can be an isometric embedding in (a higher dimensional) 2 space. 2 Hint: Since 1 and 2 are both additive across dimensions, ﬁrst show that 2 it is suﬃcient to consider n points in one dimension. Sort these points, and renumber, say x1 , . . . , xn . Now embed these in (Rn−1 , 2 ) as follows. Let 2 √ √ αi = xi+1 − xi . Map point xi to ( α1 , . . . , αi−1 , 0, . . . , 0). 21.3 Why can’t the pseudo-approximation algorithm given at the beginning of Section 21.6.3 be converted to a true approximation algorithm, i.e., so that in the end, we compare the (1/3)-balanced cut found to the optimal (1/3)-balanced cut?

196

21

Sparsest Cut

Hint: Construct graphs for which the capacity of a minimum bisection cut is arbitrarily higher than that of a (1/3)-balanced cut. 21.4 Show that the above algorithm extends to ﬁnding a b-balanced cut that is within an O(log n) factor of the best b -balanced cut for b ≤ 1/3 and b < b . Where in the argument is the restriction b ≤ 1/3 used? 21.5 Give an approximation factor preserving reduction from the problem of ﬁnding a minimum b-balanced cut, for b < 1/2, to the minimum bisection problem. 21.6 (Linial, London and Rabinovich [190]) Extend Theorem 21.22 to show that for any p ≥ 1, there is an O(log n) distortion p -embedding for metric (V, d) in O(log2 n)-dimensional space. Hint: Map point v to d(v,Si ) , for i = 1, . . . , Q, where Q is the dimension Q1/p of the embedding. Use the fact that |d(u, Si ) − d(v, Si )| ≤ d(u, v) and the monotonicity of p -norm. 21.7 (Feige [79]) Consider the following algorithm for: Problem 21.31 (Bandwidth minimization) Given an undirected graph G = (V, E), number the vertices with distinct integers from 1 to n so that the spread of the longest edge is minimized, where the spread of edge (u, v) is the absolute value of the diﬀerence of the numbers assigned to u and v. Algorithm 21.32 (Bandwidth minimization) 1. Deﬁne metric (V, d), where duv is the length of the shortest path from u to v in G. 2. Obtain an O(log n)-distortion 2 -embedding of (V, d). 3. Pick a line from a spherically symmetric distribution, and project the n points onto . 4. Number the vertices from 1 to n according to their ordering on . 5. Output the numbering. Remark 21.33 Lemma 26.7 gives an algorithm for picking . 1. Show that the expected number of pairs of vertices that are within a distance of 1 of each other on is bounded by O(log n u,v 1 ). duv

21.8

Notes

197

2. Show that 1 = O(n log n · OPT). duv

u,v

Hint: Use the fact that in G, the number of vertices within a distance of k of a vertex v is bounded by 2k · OPT. 3. Show that √ high probability, the spread of the numbering output is with √ at most O( nOPT log n), i.e., this is an O( n log n) factor algorithm. Hint: If the spread of the output numbering is s, then the number of pairs of vertices that are within a distance of 1 of each other on is at least s2 .

21.8 Notes

The seminal work of Leighton and Rao [182] gave the ﬁrst approximate maxﬂow min-cut theorem, for the case of uniform multicommodity ﬂow. They also gave a factor O(log n) approximation algorithm for the associated special case of sparsest cut and a pseudo-approximation algorithm for the b-balanced cut problem. The general version of demands multicommodity ﬂow was ﬁrst considered by Klein, Agarwal, Ravi, and Rao [173]. Theorem 21.22 is due to Linial, London, and Rabinovich [190], based on a result of Bourgain [32] who showed the existence of such an embedding and gave an exponential time algorithm for ﬁnding it. The application of this theorem to the sparsest cut problem, Theorem 21.24, was independently given by Aumann and Rabani [16], and Linial, London, and Rabinovich [190]. An O(log2 n) factor algorithm for the minimum bisection problem, and hence for the minimum b-balanced cut problem (see Exercise 21.5), was given by Feige and Krauthgamer [83]. The application of sparsest cut to computing conductance is due to Sinclair [241], and the application of balanced cuts to the minimum cut linear arrangement problem is due to Bhatt and Leighton [26]. See Exercise 26.9 for a semideﬁnite program for ﬁnding an optimal distortion 2 -embedding of n points. 2

22 Steiner Forest

We will obtain a factor 2 approximation algorithm for the Steiner forest problem by enhancing the primal–dual schema with the idea of growing duals in a synchronized manner. The Steiner forest problem generalizes the metric Steiner tree problem, for which a factor 2 algorithm was presented in Chapter 3. Recall, however, that we had postponed giving the lower bounding method behind that algorithm; we will clarify this as well. As in the Steiner tree problem (Theorem 3.2), the main case of the Steiner forest problem is also the metric case (see Exercise 22.2). However, the primal–dual algorithm remains the same for both cases, so we don’t impose this restriction. Problem 22.1 (Steiner forest) Given an undirected graph G = (V, E), a cost function on edges c : E → Q+ , and a collection of disjoint subsets of V , S1 , . . . Sk , ﬁnd a minimum cost subgraph in which each pair of vertices belonging to the same set Si is connected. Let us restate the problem; this will also help generalize it later. Deﬁne a connectivity requirement function r that maps unordered pairs of vertices to {0, 1} as follows: r(u, v) = 1 if u and v belong to the same set Si 0 otherwise

Now, the problem is to ﬁnd a minimum cost subgraph F that contains a u–v path for each pair (u, v) with r(u, v) = 1. In general, the solution will be a forest.

22.1 LP-relaxation and dual

In order to give an integer programming formulation for this problem, let us deﬁne a function on all cuts in G, f : 2V → {0, 1}, which speciﬁes the minimum number of edges that must cross each cut in any feasible solution. f (S) = 1 if ∃ u ∈ S and v ∈ S such that r(u, v) = 1 0 otherwise

22.2

Primal–dual schema with synchronization

199

Let us also introduce a 0/1 variable xe for each edge e ∈ E; xe will be set to 1 iﬀ e is picked in the subgraph. The integer program is: minimize e∈E ce xe xe ≥ f (S), e: e∈δ(S)

(22.1) S⊆V e∈E

subject to

xe ∈ {0, 1},

where δ(S) denotes the set of edges crossing the cut (S, S). Following is the LP-relaxation of (22.1); once again, we have dropped the redundant conditions xe ≤ 1. minimize e∈E ce xe xe ≥ f (S), e: e∈δ(S)

(22.2) S⊆V e∈E

subject to xe ≥ 0, The dual program is: maximize

S⊆V

f (S) · yS yS ≤ ce ,

S: e∈δ(S)

(22.3) e∈E S⊆V

subject to yS ≥ 0,

Notice that the primal and dual programs form a covering and packing pair of LPs (see Section 13.1 for deﬁnitions).

22.2 Primal–dual schema with synchronization

We will introduce a new idea in the primal–dual schema for approximation algorithms, setting it apart from the way this schema is used for designing exact algorithms. The later algorithms work on demand – in each iteration, we pick one unsatisﬁed complementary slackness condition, and satisfy it by modifying the primal and dual solutions suitably. The new idea is that of raising duals in a synchronized manner. The algorithm is not trying to rectify a speciﬁc condition. Instead, it tries many possibilities simultaneously, one of which leads to primal improvement.

200

22

Steiner Forest

Some ﬁgurative terminology will help describe the algorithm more easily. Let us say that edge e feels dual yS if yS > 0 and e ∈ δ(S). Say that set S has been raised in a dual solution if yS > 0. Clearly, raising S or S has the same eﬀect. Sometimes we will also say that we have raised the cut (S, S). Further, there is no advantage in raising set S with f (S) = 0, since this does not contribute to the dual objective function. Thus, we may assume that such cuts are never raised. Say that edge e is tight if the total amount of dual it feels equals its cost. The dual program is trying to maximize the sum of the dual variables yS subject to the condition that no edge feels more dual than its cost, i.e., no edge is overtight. Next, let us state the primal and relaxed dual complementary slackness conditions. The algorithm will pick edges integrally only. Deﬁne the degree of set S to be the number of picked edges crossing the cut (S, S). Primal conditions: For each e ∈ E, xe = 0 ⇒ Equivalently, every picked edge must be tight. i: e∈δ(S)

yS = ce .

Relaxed dual conditions: The following relaxation of the dual conditions would have led to a factor 2 algorithm: for each S ⊆ V, yS = 0 ⇒ e: e∈δ(S) xe ≤ 2 · f (S), i.e., every raised cut has degree at most 2. However, we do not know how to ensure this condition. Interestingly enough, we can still obtain a factor 2 algorithm – by relaxing this condition further! Raised sets will be allowed to have high degree; however, we will ensure that on average, raised duals have degree at most 2. The exact deﬁnition of “on average” will be given later. The algorithm starts with null primal and dual solutions. In the spirit of the primal–dual schema, the current primal solution indicates which cuts need to be raised, and in turn, the current dual solution indicates which edge needs to be picked. Thus, the algorithm iteratively improves the feasibility of the primal, and the optimality of the dual, until a feasible primal is obtained. Let us describe what happens in an iteration. At any point, the picked edges form a forest. Say that set S is unsatisﬁed if f (S) = 1, but there is no picked edge crossing the cut (S, S). Set S is said to be active if it is a minimal (w.r.t. inclusion) unsatisﬁed set in the current iteration. Clearly, if the currently picked primal solution is infeasible, there must an unsatisﬁed set and therefore an active set w.r.t. it. Lemma 22.2 Set S is active iﬀ it is a connected component in the currently picked forest and f (S) = 1. Proof: Let S be an active set. Now, S cannot contain part of a connected component because otherwise there will already be a picked edge in the cut (S, S). Thus, S is a union of connected components. Since f (S) = 1, there is a vertex u ∈ S and v ∈ S such that r(u, v) = 1. Let S be the connected component containing u. Clearly, S is also unsatisﬁed, and by the minimality ✷ of S, S = S .

22.2

Primal–dual schema with synchronization

201

By the characterization of active sets given in Lemma 22.2, it is easy to ﬁnd all active sets in the current iteration. The dual variables of these sets are raised in a synchronized manner, until some edge goes tight. Any one of the newly tight edges is picked, and the current iteration terminates. When a primal feasible solution is found, say F , the edge augmentation step terminates. However, F may contain redundant edges, which need to be pruned for achieving the desired approximation factor; this is illustrated in Example 22.4. Formally, edge e ∈ F is said to be redundant if F − {e} is also a feasible solution. All redundant edges can be dropped simultaneously from F . Equivalently, only nonredundant edges are retained. This algorithm is presented below. We leave its eﬃcient implementation as an exercise. Algorithm 22.3 (Steiner forest) 1. (Initialization) F ← ∅; for each S ⊆ V , yS ← 0. 2. (Edge augmentation) while there exists an unsatisﬁed set do: simultaneously raise yS for each active set S, until some edge e goes tight; F ← F ∪ {e}. 3. (Pruning) return F = {e ∈ F | F − {e} is primal infeasible}

Example 22.4 Consider a star in which all edges have cost 1, except one edge whose cost is 3. r r ❚ 1✔ 1❚ ✔ ❚✉ ✔ 1 ✔❚ 1 ✔1 ❚ ✔ ❚r r

r

3

✉

The only requirement is to connect the end vertices of the edge of cost 3. The algorithm will add to F all edges of cost 1 before adding the edge of cost 3. Clearly, at this point, F is not within twice the optimal. However, this will be corrected in the pruning step when all edges of cost 1 will be removed. ✷ Let us run the algorithm on a nontrivial example to illustrate its ﬁner points. Example 22.5 Consider the following graph. Costs of edges are marked, and the only nonzero connectivity requirements are r(u, v) = 1 and r(s, t) = 1. The thick edges indicate an optimal solution of cost 45.

202

22

Steiner Forest

u

20

v

6 16 a 12 9 b 12

6 19

s

t

In the ﬁrst iteration, the following four singleton sets are active: {s}, {t}, {u}, and {v}. When their dual variables are raised to 6 each, edges (u, a) and (v, b) go tight. One of them, say (u, a) is picked, and the iteration ends. In the second iteration, {u, a} replaces {u} as an active set. However, in this iteration there is no need to raise duals, since there is already a tight edge, (v, b). This edge is picked, and the iteration terminates. The primal and dual solutions at this point are shown below, with picked edges marked thick:

6 u 20 6 16 a 12 9 b 12 6 19 v 6

s 6

t 6

In the third iteration, {v, b} replaces {v} as an active set. When the active sets are raised by 2 each, edge (u, s) goes tight and is picked. In the fourth iteration, the active sets are {u, s, a}, {v} and {t}. When they are raised by 1 each, edge (b, t) goes tight and is picked. The situation now is:

22.2

Primal–dual schema with synchronization

203

2 6 u

1

3 6 20 v 6 9 16 12 a b

6

19 12 8 s t

9

In the ﬁfth iteration, the active sets are {a, s, u} and {b, v, t}. When they are raised by 1 each, (u, v) goes tight, and we now have a primal feasible solution:

2 6 u 20 v 2

3 6

1

6 9 16 12 a b

6

19 12 8 9 t

s

In the pruning step, edge (u, a) is deleted, and we obtain the following solution of cost 54:

204

22

Steiner Forest

u

20 6 6 9 a 12 b 12

v

16

19

s

t

✷

22.3 Analysis

In Lemma 22.6 we will show that simultaneously deleting all redundant edges still leaves us with a primal feasible solution, i.e., it is never the case that two edges e and f are both redundant individually, but on deletion of e, f becomes nonredundant. Lemma 22.6 At the end of the algorithm, F and y are primal and dual feasible solutions, respectively. Proof: At the end of Step 2, F satisﬁes all connectivity requirements. In each iteration, dual variables of connected components only are raised. Therefore, no edge running within the same component can go tight, and so F is acyclic, i.e., it is a forest. Therefore, if r(u, v) = 1, there is a unique u–v path in F . Thus, each edge on this path in nonredundant and is not deleted in Step 3. Hence, F is primal feasible. When an edge goes tight, the current iteration ends and active sets are redeﬁned. Therefore, no edge is overtightened. Hence, y is dual feasible. ✷ Let degF (S) denote the number of edges of F crossing the cut (S, S). The characterization of degrees of satisﬁed components established in the next lemma will be crucial in proving the approximation guarantee of the algorithm. Lemma 22.7 Consider any iteration of the algorithm, and let C be a component w.r.t. the currently picked edges. If f (C) = 0 then degF (C) = 1. Proof: Suppose degF (C) = 1, and let e be the unique edge of F crossing the cut (C, C). Since e is nonredundant (every edge in F is nonredundant),

22.3

Analysis

205

there is a pair of vertices, say u, v, such that r(u, v) = 1 and e lies on the unique u–v path in F . Since this path crosses the cut (C, C) exactly once, one of these vertices must lie in C and the other in C. Now, since r(u, v) = 1, we get that f (C) = 1, thus leading to a contradiction. ✷ Lemma 22.8 e∈F ce ≤ 2

S⊆V

yS

Proof: Since every picked edge is tight, ce = e∈F e∈F

yS .

S: e∈δ(S)

Changing the order of summation we get: ce = e∈F S⊆V

yS = e∈δ(S)∩F S⊆V

degF (S) · yS .

Thus, we need to show that degF (S) · yS ≤ 2

S⊆V S⊆V

yS .

(22.4)

We will prove the following stronger claim. In each iteration, the increase in the left-hand side of inequality (22.4) is bounded by the increase in the right-hand side. Consider an iteration, and let ∆ be the extent to which active sets were raised in this iteration. Then, we need to show: ∆×

S

active degF (S) ≤ 2∆ × (# of active sets).

Notice that the degree w.r.t. F of any active set S is due to edges that will be picked during or after the current iteration. Let us rewrite this inequality as follows:

S active degF (S) ≤ 2. # of active sets

(22.5)

Thus, we need to show that in this iteration, the average degree of active sets w.r.t. F is at most 2. The mechanics of the argument lies in the fact that in a tree, or in general in a forest, the average degree of vertices is at most 2.

206

22

Steiner Forest

Let H be a graph on vertex set V and edge set F . Consider the set of connected components w.r.t. F at the beginning of the current iteration. In H, shrink the set of vertices of each of these components to a single node to obtain graph H (we will call the vertices of H nodes for clarity). Notice that in going from H to H , all edges picked in F before the current iteration have been shrunk. Clearly, the degree of a node in H is equal to the degree of the corresponding set in H. Let us say that a node of H corresponding to an active component is an active node; any other node will be called inactive. Each active node of H has nonzero degree (since there must be an edge incident to it to satisfy its requirement), and H is a forest. Now, remove all isolated nodes from H . The remaining graph is a forest with average degree at most 2. By Lemma 22.7 the degree of each inactive node in this graph is at least 2, i.e., the forest has no inactive leaves. Hence, the average degree of active nodes is at most 2. ✷ Observe that the proof given above is essentially a charging argument: for each active node of degree greater than 2, there must be correspondingly many active nodes of degree 1, i.e., leaves, in the forest. The exact manner in which the dual conditions have been relaxed must also be clear now: in each iteration, the duals being raised have average degree at most 2. Lemmas 22.6 and 22.8 give: Theorem 22.9 Algorithm 22.3 achieves an approximation guarantee of factor 2 for the Steiner forest problem. The tight example given for the metric Steiner tree problem, Example 3.4, is also a tight example for this algorithm. Algorithm 22.3 places an upper bound of 2 on the integrality gap of LP-relaxation (22.2) for the Steiner forest problem. Example 22.10 places a lower bound of (essentially) 2 on this LP, even if restricted to the minimum spanning tree problem. Let us run Algorithm 22.3 on an instance of the metric Steiner tree problem. If the edge costs satisfy the strict triangle inequality, i.e., for any three vertices u, v, w, c(u, v) < c(u, w) + c(v, w), then it is easy to see that the algorithm will ﬁnd a minimum spanning tree on the required vertices, i.e., it is essentially the algorithm for the metric Steiner tree problem presented in Chapter 3. Even if the triangle inequality is not strictly satisﬁed, the cost of the solution found is the same as the cost of an MST. Furthermore, if among multiple tight edges, the algorithm always prefers picking edges running between required vertices, it will ﬁnd an MST. This clariﬁes the lower bound on which that algorithm was based. The MST problem is a further special case: every pair of vertices need to be connected. Observe that when run on such an instance, Algorithm 22.3 essentially executes Kruskal’s algorithm, i.e., in each iteration, it picks the cheapest edge running between two connected components. Hence it ﬁnds an optimal MST. However, as shown in Example 22.10, the dual found may be as small as half the primal.

22.4

Exercises

207

Example 22.10 Consider a cycle on n vertices, with all edges of cost 1. The cost of an optimal MST is n − 1. the dual found is n/2. Algorithm 22.3 ﬁnds a dual of value n/2: 1/2 around each vertex. Indeed, this is an optimal dual solution, since there is a fractional primal solution of the same value: pick each edge to the extent of half. This places a lower bound of (essentially) 2 on the integrality gap of LP (22.2), even if restricted to the minimum spanning tree problem. ✷

22.4 Exercises

22.1 Show, using the max-ﬂow min-cut theorem, that a subgraph of G has all the required paths iﬀ it does not violate any of the cut requirements in IP (22.1). Use this fact to show that IP (22.1) is an integer programming formulation for the Steiner forest problem. 22.2 Show that there is an approximation-factor-preserving reduction from the Steiner forest problem to the metric Steiner forest problem. Show that there is no loss of generality in requiring that the edge costs satisfy the triangle inequality for the Steiner network problem. Hint: The reasoning is the same as that for the Steiner tree problem. 22.3 How does the feasibility and approximation guarantee of the solution found change if 1. the pruning step of Algorithm 22.3 is replaced with the reverse delete step of Algorithm 18.4. 2. the reverse delete step of Algorithm 18.4 is replaced by the pruning step of Algorithm 22.3. 22.4 Give an example for which some cut raised by Algorithm 22.3 has degree at least 3 w.r.t. the primal solution found. 22.5 Run Algorithm 22.3 on an instance of the minimum spanning tree problem. Pick an arbitrary vertex as the root, and throw away all raised duals containing this vertex. Show that the cost of the tree found is twice the sum of the remaining duals. Hint: Show that in an iteration which starts with k connected components, and lasts for time ∆, the total increase to the left-hand side of inequality (22.4) is precisely 2(k − 1)∆. 22.6 Let us think of running Step 2 of Algorithm 22.3 continuously in time. Thus, in unit time, a dual grows a unit amount. Consider an instance of the

208

22

Steiner Forest

Steiner forest problem, (G = (V, E), c, S1 , . . . , Sk ), and its modiﬁcation in which one of the vertices from V − (S1 ∪ . . . ∪ Sk ) is added to one of the sets. Run Algorithm 22.3 on both these instances. Call these runs R1 and R2 , respectively. 1. Show that if k = 1, i.e., the starting instance was a Steiner tree instance, then the following holds. If at time t two vertices u, v ∈ S1 are connected by a tight path in run R1 , then they are connected by a tight path at time t in run R2 as well. 2. Give a counterexample to the previous claim in case k > 1. 22.7 (Goemans and Williamson [105]) Algorithm 22.3 actually works for a general class of problems that includes the Steiner forest problem as a special case. A function f : 2V → {0, 1} is said to be proper if it satisﬁes the following properties: 1. f (V ) = 0; 2. f (S) = f (S); 3. If A and B are two disjoint subsets of V and f (A∪B) = 1, then f (A) = 1 or f (B) = 1. Notice that function f deﬁned for the Steiner forest problem is a proper function. Consider the integer program (22.1) with f restricted to be a proper function. Show that Algorithm 22.3 is in fact a factor 2 approximation algorithm for this class of integer programs. 22.8 (Goemans and Williamson [105]) Consider the following problem. Problem 22.11 (Point-to-point connection) Given a graph G = (V, E), a cost function on edges c : E → Q+ (not necessarily satisfying the triangle inequality) and two disjoint sets of vertices, S and T , of equal cardinality, ﬁnd a minimum cost subgraph that has a path connecting each vertex in S to a unique vertex in T . 1. Give a factor 2 approximation algorithm for this problem. Hint: Show that this can be formulated as an integer program using (22.1), with f being a proper function. 2. Relax the problem to requiring that each vertex in S be connected to some vertex in T (not necessarily unique). Give a factor 2 approximation algorithm for this problem as well. Hint: Reduce to the Steiner tree problem. 22.9 (Goemans and Williamson [105]) Consider the following variant of the metric Steiner tree problem. Problem 22.12 (Prize-collecting Steiner tree) We are given a complete undirected graph G = (V, E) and a special vertex r ∈ V . Function cost : E →

22.4

Exercises

209

Q+ satisﬁes the triangle inequality, and π : V → Q+ is the penalty function for vertices. The problem is to ﬁnd a tree containing r which minimizes the sum of the costs of the edges in the tree and the penalties of vertices not in the tree. 1. Consider the following integer program for this problem. It has a variable, xe , for each edge e and a variable, ZT , for each set T of vertices not containing r. ZT is set to 1 for the set T of vertices that are not included in the optimal tree. Obtain the LP-relaxation and dual for this LP. The dual will have a variable for each set S of vertices not containing r. Let’s call this variable yS . minimize e∈E ce xe +

T ⊆V ;r∈T

ZT v∈T πv

(22.6) S ⊆ V ;r ∈ S e∈E T ⊆ V ;r ∈ T

subject to e∈δ(S) xe +

T ⊇S

ZT ≥ 1,

xe ∈ {0, 1}, ZT ∈ {0, 1},

2. The following primal–dual algorithm for this problem is along the lines of Algorithm 22.3. Initialize as follows. Each vertex v = r is a singleton active set, with a charge of πv . Ordered list F is set to ∅. The dual variables of all active sets are grown in a synchronized manner. As a dual grows, its charge decreases by the same amount. If set S runs out of charge, it is declared dead and all of its unmarked vertices are marked with “S”. When an edge e goes tight, it is added to F . The rest of the action depends on the following cases. • If e connects active set S to r: Set S is deactivated and is declared connected to r. All unmarked vertices of S are marked “r”. • If e connects active set S to a set that is connected to r: Same action as in previous case. • If e connects sets S and S which are either both active or one is active and one is dead: The active sets among S and S are deactivated. S ∪S is declared active and is given the sum of the leftover charges of S and S. When there are no more active sets, the algorithm performs a dynamic reverse delete on F . This is an enhanced reverse delete procedure in which requirements change dynamically. All vertices marked “r” are labeled Required. We will say that F is feasible if there is a path from each Required vertex to r using edges of F . Let e = (u, v) ∈ F . Then the maximal dead set w.r.t. e, containing v is the maximal set S such that v ∈ S, u ∈ S and S was declared dead by the algorithm. If there is no set /

210

22

Steiner Forest

satisfying these conditions, then the maximal dead set w.r.t. e, containing v is deﬁned to be ∅. Edges e ∈ F are considered in the reverse order in which they were inserted in F . For each edge e, if F − e is feasible, then e is removed from F . Otherwise, suppose e = (u, v), and let S be the maximal dead set w.r.t. e, containing v. If S = ∅, then declare all vertices marked “S” Required. Repeat for the maximal dead set w.r.t. e containing u. 3. Show that at the beginning of the reverse delete step, F is feasible. Also, show that F is never infeasible, even if the set of Required vertices grows. 4. Show that at the end, F is a tree containing r and satisfying a) all vertices marked “r” are included in F , and b) if a vertex marked “S” is included in F , then all vertices marked “T ”, where T ⊇ S, are also included in F . 5. The primal solution is constructed as follows. For each edge e in F , set xe to 1, and set the remaining xe ’s to 0. For each maximal dead set T , none of whose vertices are in F , set ZT to 1. For all the remaining sets, set their Z variable to 0. Prove that this is a factor 2 approximation algorithm for the prizecollecting Steiner tree problem, by showing that the primal and dual solutions produced satisfy ce xe + 2 · e∈E T ⊆V ;r∈T

ZT v∈T πv

≤2·

S⊆V ;r∈T

yS .

Hint: If ZT = 1, the sum of penalties of vertices in T equals the total dual contained in T . Twice the rest of the duals pays for the cost of the tree. Show the latter by proving, similar to Lemma 22.7, that at any point in the algorithm, there is at most one inactive set of degree 1 (the one containing r). 22.10 Consider the following generalization of the Steiner forest problem to higher connectivity requirements: the speciﬁed connectivity requirement function r maps pairs of vertices to {0, . . . , k}, where k is part of the input. Assume that multiple copies of any edge can be used; each copy of edge e will cost c(e). Using Algorithm 22.3 as a subroutine, give a factor 2 · ( log2 k + 1) algorithm for the problem of ﬁnding a minimum cost graph satisfying all connectivity requirements. 22.11 We give below the bidirected cut relaxation for the Steiner tree problem. This is believed to have a smaller integrality gap than the undirected relaxation (22.2), though there is no proof of this fact yet. From graph G, obtain directed graph H by replacing each edge (u, v) by the two edges (u → v) and (v → u), each of the same cost as (u, v). Designate an arbitrary required vertex, say r, as the root. Say that S ⊂ V is valid if it contains a required

22.4

Exercises

211

vertex and r ∈ S. Let xe be an indicator variable for each edge e ∈ H. The integer program is: minimize e∈E ce xe xe ≥ 1, e: e∈δ(S)

(22.7) valid set S e∈H

subject to

xe ∈ {0, 1},

1. Show that the optimal solution to this integer program is an optimal Steiner tree. 2. Obtain the LP-relaxation and dual for IP (22.7). 3. Show that the cost of the optimal solution to (22.7) and its relaxation is independent of the root chosen. 4. Show that the integrality gap of the relaxation is bounded by 2. 5. (Rajagopalan and Vazirani [226]) Show that the integrality gap of this relaxation for the following graph is 10/9. In this graph, the bold vertices are required and the remaining vertices are Steiner. t ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙t t ❙ ❙ ❙ ❙ t ❙ 6. (M. Goemans) The following family of graphs puts a lower bound of essentially 8/7 on the integrality gap of relaxation (22.7). This is currently the worst example known. am i 2 bm i 1 2

m cij

1

2

m aj

2

m dij

1

2

✚ ✚ m ✚ a0

✚ ✚2

✚ ✚

bm j

212

22

Steiner Forest

Graph Gn has n+1 required vertices a0 , a1 , . . . , an , and n2 Steiner vertices b1 , . . . , bn and cij and dij for 1 ≤ i < j ≤ n. The ﬁgure above gives edges and costs. Verify that the optimal Steiner tree has cost 4n and the optimal solution to relaxation (22.7) has cost 7n + 1/2. 7. Construct other graphs for which this relaxation has a gap (it is not easy!). 8. (Edmonds [70]) Consider the special case that there are no Steiner vertices, i.e., we want to ﬁnd a minimum spanning tree in G. Give a primal– dual algorithm that uses this relaxation to ﬁnd a tree and a dual of the same cost, thereby showing that this relaxation is exact, i.e., always has an integral optimal solution, for the minimum spanning tree problem. (In contrast, the undirected relaxation has an integrality gap of 2 even for the minimum spanning tree problem.) 22.12 (Pr¨mel and Steger [223]) This exercise develops an algorithm for o the Steiner tree problem using the weighted matroid parity problem and the following structural fact. Let us say that a Steiner tree is 3-restricted if every Steiner vertex used in this tree has exactly three neighbors, all of which are required vertices. The cost of an optimal 3-restricted Steiner tree is within 5/3 of the cost of an optimal Steiner tree (Zelikovsky [262]). Show that an optimal 3-restricted Steiner tree can be found in polynomial time, given an oracle for the weighted matroid parity problem. The latter problem is neither known to be in P nor is it known to be NP-hard. However, a randomized polynomial time algorithm is known for the case of unary weights. Use this fact, and scaling, to obtain a 5/3 + ε factor algorithm for the Steiner tree problem for any ε > 0. The weighted matroid parity problem is the following. Let (S, I) be a matroid, where S is the ground set and I is the collection of independent sets. Nonnegative weights are provided for elements of S. Further, a partition of S into pairs (x1 , x2 ), . . . , (x2n−1 , x2n ) is also provided. The problem is to pick a maximum weight collection of pairs so that the picked elements form an independent set.

22.5 Notes

This chapter is based on the work of Goemans and Williamson [105]. The ﬁrst factor 2 approximation algorithm for the Steiner forest problem was given by Agrawal, Klein, and Ravi [1]. See also the survey by Goemans and Williamson [107].

23 Steiner Network

The following generalization of the Steiner forest problem to higher connectivity requirements has applications in network design and is also known as the survivable network design problem. In this chapter, we will give a factor 2 approximation algorithm for this problem by enhancing the LP-rounding technique to iterated rounding. A special case of this problem was considered in Exercise 22.10. Problem 23.1 (Steiner network) We are given an undirected graph G = (V, E), a cost function on edges c : E → Q+ (not necessarily satisfying the triangle inequality), a connectivity requirement function r mapping unordered pairs of vertices to Z+ , and a function u : E → Z+ ∪ {∞} stating an upper bound on the number of copies of edge e we are allowed to use; if ue = ∞, there is no upper bound for edge e. The problem is to ﬁnd a minimum cost multigraph on vertex set V that has r(u, v) edge disjoint paths for each pair of vertices u, v ∈ V . Each copy of edge e used for constructing this graph will cost c(e).

23.1 The LP-relaxation and half-integrality

In order to give an integer programming formulation for this problem, we will ﬁrst deﬁne a cut requirement function, f : 2V → Z+ , as we did for the metric Steiner forest problem. For every S ⊆ V , f (S) is deﬁned to be the largest connectivity requirement separated by the cut (S, S), i.e., f (S) = max{r(u, v)|u ∈ S and v ∈ S}. minimize e∈E ce xe xe ≥ f (S), e: e∈δ(S)

(23.1) S⊆V e ∈ E and ue = ∞ e ∈ E and ue = ∞

subject to

xe ∈ Z+ , xe ∈ {0, 1, . . . , ue }, The LP-relaxation is:

214

23

Steiner Network

minimize e∈E ce xe xe ≥ f (S), e: e∈δ(S)

(23.2) S⊆V e ∈ E and ue = ∞ e ∈ E and ue = ∞

subject to xe ≥ 0,

ue ≥ xe ≥ 0,

Since LP (23.2) has exponentially many constraints, we will need the ellipsoid algorithm for ﬁnding an optimal solution. Exercise 23.1 develops a polynomial-sized LP. As shown in Chapters 14 and 19, certain NP-hard problems, such as vertex cover and node multiway cut, admit LP-relaxations having the remarkable property that they always have a half-integral optimal solution. Rounding up all halves to 1 in such a solution leads to a factor 2 approximation algorithm. Does relaxation (23.2) have this property? The following lemma shows that the answer is “no”. Lemma 23.2 Consider the Petersen graph (see Section 1.2) with a connectivity requirement of 1 between each pair of vertices and with each edge of unit cost. Relaxation (23.2) does not have a half-integral optimal solution for this instance. Proof: Consider the fractional solution xe = 1/3 for each edge e. Since the Petersen graph is 3-edge connected (in fact, it is 3-vertex connected as well), this is a feasible solution. The cost of this solution is 5. In any feasible solution, the sum of edge variables incident at any vertex must be at least 1, to allow connectivity to other vertices. Therefore, any feasible solution must have cost at least 5 (since the Petersen graph has 10 vertices). Hence, the solution given above is in fact optimal. Any solution with xe = 1 for some edge e must have cost exceeding 5, since additional edges are required to connect the endpoints of e to the rest of the graph. Therefore, any half-integral solution of cost 5 would have to pick, to the extent of one half each, the edges of a Hamiltonian cycle. Since the Petersen graph has no Hamiltonian cycles, there is no half-integral optimal solution. ✷ Let us say that an extreme point solution, also called a vertex solution or a basic feasible solution, for an LP is a feasible solution that cannot be written as the convex combination of two feasible solutions. The solution xe = 1/3, for each edge e, is not an extreme point solution. An extreme optimal solution is shown in the ﬁgure below; thick edges are picked to the extent of 1/2, thin edges to the extent of 1/4, and the missing edge is not picked.

23.1

The LP-relaxation and half-integrality

215

The isomorphism group of the Petersen graph is edge-transitive, and there are 15 related extreme point solutions; the solution xe = 1/3 for each edge e is the average of these. Notice that although the extreme point solution is not half-integral, it picks some edges to the extent of half. We will show below that in fact this is a property of any extreme point solution to LP (23.2). We will obtain a factor 2 algorithm by rounding up these edges and iterating. Let H be the set of edges picked by the algorithm at some point. Then, the residual requirement of cut (S, S) is f (S) = f (S) − |δH (S)|, where δH (S) represents the set of edges of H crossing the cut (S, S). In general, the residual cut requirement function, f , may not correspond to the cut requirement function for any set of connectivity requirements. We will need the following deﬁnitions to characterize it: Function f : 2V → Z+ is said to be submodular if f (V ) = 0, and for every two sets A, B ⊆ V , the following two conditions hold: • f (A) + f (B) ≥ f (A ∩ B) + f (A ∪ B) • f (A) + f (B) ≥ f (A − B) + f (B − A). Remark 23.3 Sometimes submodularity is deﬁned only with the ﬁrst condition. We will need to work with the stronger deﬁnition given above. Two subsets of V , A and B, are said to cross if each of the sets, A − B, B − A, and A ∩ B, is nonempty. If A and B don’t cross then either they are disjoint or one of these sets is contained in the other. Lemma 23.4 For any graph G on vertex set V , the function |δG (.)| is submodular.

216

23

Steiner Network

Proof: If sets A and B do not cross, then the two conditions given in the deﬁnition of submodular functions hold trivially. Otherwise, edges having one endpoint in A ∩ B and the other in A ∪ B (edge e1 in the ﬁgure below) contribute to δ(A) and δ(B) but not to δ(A − B) or δ(B − A). Similarly, edge e2 below does not contribute to δ(A ∩ B) or to δ(A ∪ B). The remaining edges contribute equally to both sides of both conditions. ✷

e1

A

e2

B

Function f : 2V → Z is said to be weakly supermodular if f (V ) = 0, and for every two sets A, B ⊆ V , at least one of the following conditions holds: • f (A) + f (B) ≤ f (A − B) + f (B − A) • f (A) + f (B) ≤ f (A ∩ B) + f (A ∪ B). It is easy to check that the original cut requirement function is weakly supermodular; by Lemma 23.5, so is the residual cut requirement function. Lemma 23.5 Let H be a subgraph of G. If f : 2V (G) → Z+ is a weakly supermodular function, then so is the residual cut requirement function f . Proof: Suppose f (A) + f (B) ≤ f (A − B) + f (B − A); the proof of the other case is similar. By Lemma 23.4, |δH (A)|+|δH (B)| ≥ |δH (A−B)|+|δH (B−A)|. Subtracting, we get f (A) + f (B) ≤ f (A − B) + f (B − A). ✷ We can now state the central polyhedral fact needed for the factor 2 algorithm in its full generality. Theorem 23.6 For any weakly supermodular function f , any extreme point solution, x, to LP (23.2) must pick some edge to the extent of at least a half, i.e., xe ≥ 1/2 for at least one edge e.

23.2

The technique of iterated rounding

217

23.2 The technique of iterated rounding

In this section, we will give an iterated rounding algorithm for the Steiner network problem, using Theorem 23.6. Algorithm 23.7 (Steiner network) 1. Initialization: H ← ∅: f ← f . 2. While f ≡ 0, do: Find an extreme optimal solution, x, to LP (23.2) with cut requirements given by f . For each edge e such that xe ≥ 1/2, include xe copies of e in H, and decrement ue by this amount. Update f : for S ⊆ V , f (S) ← f (S) − |δH (S)|. 3. Output H. The algorithm presented above achieves an approximation guarantee of factor 2 for an arbitrary weakly supermodular function f . Establishing a polynomial running time involves showing that an extreme optimal solution to LP (23.2) can be found eﬃciently. We do not know how to do this for an arbitrary weakly supermodular function f . However, if f is the original cut requirement function for some connectivity requirements, then a polynomial time implementation follows from the existence of a polynomial time separation oracle for each iteration. For the ﬁrst iteration, a separation oracle follows from a max-ﬂow subroutine. Given a solution x, construct a graph on vertex set V with capacity xe for each edge e. Then, for each pair of vertices u, v ∈ V , check if this graph admits a ﬂow of at least r(u, v) from u to v. If not, we will get a violated cut, i.e., a cut (S, S) such that δx (S) < f (S), where δx (S) = e: e∈δ(S)

xe .

Let f be the cut requirement function of a subsequent iteration. Given a solution to LP (23.2) for this function, say x , deﬁne x as follows: for each edge e, xe = xe + eH , where eH is the number of copies of edge e in H. The following lemma shows that a separation oracle for the original function f leads to a separation oracle for f . Furthermore, this lemma also shows that there is no need to update f explicitly after each iteration. Lemma 23.8 A cut (S, S) is violated by solution x under cut requirement function f iﬀ it is violated by solution x under cut requirement function f . Proof: Notice that δx (S) = δx (S) + |δH (S)|. Since f (S) = f (S) + |δH (S)|, ✷ δx (S) ≥ f (S) iﬀ δx (S) ≥ f (S).

218

23

Steiner Network

Lemma 23.8 implies that solution x is feasible for the cut requirement function f iﬀ solution x is feasible for f . Assuming Theorem 23.6, whose proof we will provide below, let us show that Algorithm 23.7 achieves an approximation guarantee of 2. Theorem 23.9 Algorithm 23.7 achieves an approximation guarantee of 2 for the Steiner network problem. Proof: By induction on the number of iterations executed by the algorithm when run with a weakly supermodular cut requirement function f , we will prove that the cost of the integral solution obtained is within a factor of two of the cost of the optimal fractional solution. Since the latter is a lower bound on the cost of the optimal integral solution, the claim follows. For the base case, if f requires one iteration, the claim follows, since the algorithm rounds up only edges e with xe ≥ 1/2. For the induction step, assume that x is the extreme optimal solution ˆ obtained in the ﬁrst iteration. Obtain x from x by zeroing out components ˆ that are strictly smaller than 1/2. By Theorem 23.6, x = 0. Let H be the set of edges picked in the ﬁrst iteration. Since H is obtained by rounding up nonzero ˆ components of x and each of these components is ≥ 1/2, cost(H) ≤ 2·cost(ˆ ). x Let f be the residual requirement function after the ﬁrst iteration and H be the set of edges picked in subsequent iterations for satisfying f . The ˆ key observation is that x − x is a feasible solution for f , and thus by the ˆ induction hypothesis, cost(H ) ≤ 2 · cost(x − x). Let us denote by H + H the edges of H together with those of H . Clearly, H + H satisﬁes f . Now, cost(H + H ) ≤ cost(H) + cost(H ) ˆ ≤ 2 · cost(ˆ ) + 2 · cost(x − x) ≤ 2 · cost(x). x Corollary 23.10 The integrality gap of LP (23.2) is bounded by 2. Notice that previous algorithms obtained using LP-rounding solved the relaxation once and did the entire rounding based on this solution. These algorithms did not exploit the full power of rounding – after part of the solution is rounded, the remaining fractional solution may not be the best solution to continue the rounding process. It may be better to assume integral values for the rounded variables and recompute fractional values for the remaining variables, as is done above. We will call this technique iterated rounding. Example 23.11 The tight example given for the metric Steiner tree problem, Example 3.4, is also a tight example for this algorithm. Observe that after including a subset of edges of the cycle, an extreme optimal solution to the resulting problem picks the remaining edges of the cycle to the extent of one half each. The algorithm ﬁnds a solution of cost (2 − ε)(n − 1), whereas the cost of the optimal solution is n. ✷ ✷

23.3

Characterizing extreme point solutions

219

23.3 Characterizing extreme point solutions

From polyhedral combinatorics we know that a feasible solution for a set of linear inequalities in Rm is an extreme point solution iﬀ it satisﬁes m linearly independent inequalities with equality. Extreme solutions of LP (23.2) satisfy an additional property which leads to a proof of Theorem 23.6. We will assume that the cut requirement function f in LP (23.2) is an arbitrary weakly supermodular function. Given a solution x to this LP, we will say that an inequality is tight if it holds with equality. If this inequality corresponds to the cut requirement of a set S, then we will say that set S is tight. Let us make some simplifying assumptions. If xe = 0 for some edge e, this edge can be removed from the graph, and if xe ≥ 1, xe copies of edge e can be picked and the cut requirement function be updated accordingly. We may assume without loss of generality that an extreme point solution x satisﬁes 0 < xe < 1, for each edge e in graph G. Therefore, each tight inequality corresponds to a tight set. Let the number of edges in G be m. We will say that a collection, L, of subsets of V forms a laminar family if no two sets in this collection cross. The inequality corresponding to a set S deﬁnes a vector in Rm : the vector has a 1 corresponding to each edge e ∈ δG (S), and 0 otherwise. We will call this the incidence vector of set S, and will denote it by AS . Theorem 23.12 Corresponding to any extreme point solution to LP (23.2) there is a collection of m tight sets such that • their incidence vectors are linearly independent, and • collection of sets forms a laminar family. Example 23.13 The extreme point solution for the Peterson graph assigns nonzero values to 14 of the 15 edges. By Theorem 23.12, there should be 14 tight sets whose incidence vectors are linearly independent. These are marked in ﬁgure. ✷ Fix an extreme point solution, x, to LP (23.2). Let L be a laminar family of tight sets whose incidence vectors are linearly independent. Denote by span(L) the vector space generated by the set of vectors {AS |S ∈ L}. Since x is an extreme point solution, the span of the collection of all tight sets is m. We will show that if span(L) < m, then there is a tight set S whose addition to L does not violate laminarity and also increases the span. Continuing in this manner, we will obtain m tight sets as required in Theorem 23.12. We begin by studying properties of crossing tight sets. Lemma 23.14 Let A and B be two crossing tight sets. Then, one of the following must hold: • A − B and B − A are both tight and AA + AB = AA−B + AB−A • A ∪ B and A ∩ B are both tight and AA + AB = AA∪B + AA∩B .

220

23

Steiner Network

Proof: Since f is weakly supermodular, either f (A) + f (B) ≤ f (A − B) + f (B − A) or f (A) + f (B) ≤ f (A ∪ B) + f (A ∩ B). Let us assume the former holds; the proof for the latter is similar. Since A and B are tight, we have δx (A) + δx (B) = f (A) + f (B). Since A − B and B − A are not violated, δx (A − B) + δx (B − A) ≥ f (A − B) + f (B − A). Therefore, δx (A) + δx (B) ≤ δx (A − B) + δx (B − A). As argued in Lemma 23.4 (which established the submodularity of function |δG (.)|), edges having one endpoint in A ∪ B and the other in A ∩ B can contribute only to the left-hand side of this inequality. The rest of the edges must contribute equally to both sides. So, this inequality must be satisﬁed with equality. Furthermore, since xe > 0 for each edge e, G cannot have any edge having one endpoint in A ∪ B and the other in A ∩ B. Therefore, AA + AB = AA−B + AB−A . ✷ For any set S ⊆ V , deﬁne its crossing number to be the number of sets of L that S crosses. Lemma 23.15 Let S be a set that crosses set T ∈ L. Then, each of the sets S − T, T − S, S ∪ T and S ∩ T has a smaller crossing number than S. Proof: The ﬁgure below illustrates the three ways in which a set T ∈ L can cross one of these four sets without crossing T itself (T is shown dotted). In all cases, T crosses S as well. In addition, T crosses S but not any of the four sets. ✷

S

T

23.4

A counting argument

221

Lemma 23.16 Let S be a tight set such that AS ∈ span(L) and S crosses some set in L. Then, there is a tight set S having a smaller crossing number than S and such that AS ∈ span(L). Proof: Let S cross T ∈ L. Suppose the ﬁrst possibility established in Lemma 23.14 holds; the proof of the second possibility is similar. Then, S −T and T − S are both tight sets and AS + AT = AS−T + AT −S . This linear dependence implies that AS−T and AT −S cannot both be in span(L), since otherwise AS ∈ span(L). By Lemma 23.15, S − T and T − S both have a smaller crossing number than S. The lemma follows. ✷ Corollary 23.17 If span(L) = Rm , then there is a tight set S such that AS ∈ span(L) and L ∪ {S} is a laminar family. By Corollary 23.17, if L is a maximal laminar family of tight sets with linearly independent incidence vectors, then |L| = m. This establishes Theorem 23.12.

23.4 A counting argument

The characterization of extreme point solutions given in Theorem 23.12 will yield Theorem 23.6 via a counting argument. Let x be an extreme point solution and L be the collection of tight sets established in Theorem 23.12. The number of sets in L equals the number of edges in G, i.e., m. The proof is by contradiction. Suppose that for each edge e, xe < 1/2. Then, we will show that G has more than m edges. Since L is a laminar family, it can be viewed as a forest of trees if its elements are ordered by inclusion. Let us make this precise. For S ∈ L, if S is not contained in any other set of L, then we will say that S is a root set. If S is not a root set, we will say that T is the parent of S if T is a minimal set in L containing S; by laminarity of L, T is unique. Further, S will be called a child of T . Let the relation descendent be the reﬂexive transitive closure of the relation “child”. Sets that have no children will be called leaves. In this manner, L can be partitioned into a forest of trees, each rooted at a root set. For any set S, by the subtree rooted at S we mean the set of all descendents of S. Edge e is incident at set S if e ∈ δG (S). The degree of S is deﬁned to be |δG (S)|. Set S owns endpoint v of edge e = (u, v) if S is the smallest set of L containing v. The subtree rooted at set S owns endpoint v of edge e = (u, v) if some descendent of S owns v. Since G has m edges, it has 2m endpoints. Under the assumption that ∀e, xe < 1/2, we will prove that for any set S, the endpoints owned by the subtree rooted at S can be redistributed in such a way that S gets at least 3 endpoints, and each of its proper descendents gets 2 endpoints. Carrying

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out this procedure for each of the root sets of the forest, the total number of endpoints in the graph must exceed 2m, leading to a contradiction. We have assumed that ∀e : 0 < xe < 1/2. For edge e, deﬁne ye = 1/2−xe , the halves complement of e. Clearly, 0 < ye < 1/2. For S ∈ L deﬁne its corequirement to be coreq(S) = e∈δ(S) ye =

1 |δG (S)| − f (S). 2

Clearly, 0 < coreq(S) < |δG (S)|/2. Furthermore, since |δG (S)| and f (S) are both integral, coreq(S) is half-integral. Let us say that coreq(S) is semiintegral if it is not integral, i.e., if coreq(S) ∈ {1/2, 3/2, 5/2, . . .}. Since f (S) is integral, coreq(S) is semi-integral iﬀ |δG (S)| is odd. Sets having a corequirement of 1/2 play a special role in this argument. The following lemma will be useful in establishing that certain sets have this corequirement. Lemma 23.18 Suppose S has α children and owns β endpoints, where α + β = 3. Furthermore, each child of S, if any, has a corequirement of 1/2. Then, coreq(S) = 1/2. Proof: Since each child of S has corequirement of 1/2, it has odd degree. Using this and the fact that α + β = 3, one can show that S must have odd degree (see Exercise 23.3). Therefore the corequirement of S is semi-integral. Next, we show that coreq(S) is strictly smaller than 3/2, thereby proving the lemma. Clearly, coreq(S) = e∈δ(S) ye ≤

S

coreq(S ) + e ye ,

where the ﬁrst sum is over all children S of S, and the second sum is over all edges e having an endpoint in S. Since ye is strictly smaller than 1/2, if β > 0, then coreq(S) < 3/2. If β = 0, all edges incident at the children of S cannot also be incident at S, since otherwise the incidence vectors of these four sets will be linearly dependent. Therefore, coreq(S) <

S

coreq(S ) = 3/2. ✷

The next two lemmas place lower bounds on the number of endpoints owned by certain sets. Lemma 23.19 If set S has only one child, then it must own at least two endpoints.

23.4

A counting argument

223

Proof: Let S be the child of S. If S has no endpoint incident at it, the set of edges incident at S and S must be the same. But then AS = AS , leading to a contradiction. S cannot own exactly one endpoint, because then δx (S) and δx (S ) will diﬀer by a fraction, contradicting the fact that both these sets are tight and have integral requirements. The lemma follows. ✷ Lemma 23.20 If set S has two children, one of which has a corequirement of 1/2, then it must own at least one endpoint. Proof: Let S and S be the two children of S, with coreq(S ) = 1/2. Suppose S does not own any endpoints. Since the three vectors AS , AS , and AS are linearly independent, the set of edges incident at S cannot all be incident at S or all be incident at S . Let a denote the sum of ye ’s of all edges incident at S and S, and let b denote the sum of ye ’s of all edges incident at S and S . Thus, a > 0, b > 0, and a + b = coreq(S) = 1/2. Since S has a semi-integral corequirement, it must have odd degree. Therefore, the degrees of S and S have diﬀerent parities, and these two sets have diﬀerent corequirements. Furthermore, coreq(S) = coreq(S ) + a − b. Therefore, coreq(S) − coreq(S ) = a − b. But −1/2 < a − b < 1/2. Therefore, S and S must have the same corequirement, leading to a contradiction. ✷ Lemma 23.21 Consider a tree T rooted at set S. Under the assumption that ∀e, xe < 1/2, the endpoints owned by T can be redistributed in such a way that S gets at least 3 endpoints, and each of its proper descendents gets 2 endpoints. Furthermore, if coreq(S) = 1/2, then S must get at least 4 endpoints. Proof: The proof is by induction on the height of tree T . For the base case, consider a leaf set S. S must have degree at least 3, because otherwise an edge e incident at it will have xe ≥ 1/2. If it has degree exactly 3, coreq(S) is semi-integral. Further, since coreq(S) < |δG (S)|/2 = 3/2, the corequirement of S is 1/2. Since S is a leaf, it owns an endpoint of each edge incident at it. Therefore, S has the required number of endpoints. Let us say that a set has a surplus of 1 if 3 endpoints have been assigned to it and a surplus of 2 if 4 endpoints have been assigned to it. For the induction step, consider a nonleaf set S. We will prove that by moving the surplus of the children of S and considering the endpoints owned by S itself, we can assign the required number of endpoints to S. There are four cases: 1. If S has 4 or more children, we can assign the surplus of each child to S, thus assigning at least 4 endpoints to S. 2. Suppose S has 3 children. If at least one of them has a surplus of 2, or if S owns an endpoint, we can assign 4 endpoints to S. Otherwise, each child must have a corequirement of half, and by Lemma 23.18, coreq(S) = 1/2 as well. Thus, assigning S the surplus of its children suﬃces.

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3. Suppose S has two children. If each has a surplus of 2, we can assign 4 endpoints to S. If one of them has surplus 1, then by Lemma 23.20, S must own at least one endpoint. If each child has a surplus of 1 and S owns exactly one endpoint, then we can assign 3 endpoints to S, and this suﬃces by Lemma 23.18. Otherwise, we can assign 4 endpoints to S. 4. If S has one child, say S , then by Lemma 23.19, S owns at least 2 endpoints. If S owns exactly 2 endpoints and S has surplus of exactly 1, then we can assign 3 endpoints to S; by Lemma 23.18, coreq(S) = 1/2, so this suﬃces. In all other cases, we can assign 4 endpoints to S. ✷

23.5 Exercises

23.1 Give an LP-relaxation for the Steiner network problem, having polynomially many constraints over polynomially many variables. Hint: Pick a minimum cost set of edges so as to route n independent 2 commodities, one for each pair of vertices. Each ﬂow should be at least as large as the connectivity requirement of this pair. The extent to which an edge is picked bounds the amount of each commodity that can ﬂow through this edge. 23.2 Show that a function f : 2V → Z+ satisfying the following conditions is submodular: f (V ) = 0, f is symmetric, i.e., for any set A ⊆ V f (A) = f (V −A), and for every two sets A, B ⊆ V f (A)+f (B) ≥ f (A∩B)+f (A∪B). 23.3 Prove that set S in Lemma 23.18 must have odd degree. (Consider the following possibilities: S owns endpoint v of edge (u, v) that is incident at S, S owns endpoint v of edge (u, v) that is incident at a child of S, and an edge is incident at two children of S.) 23.4 Prove that there must be a set in L that has degree at most 3, and thus some edge must have xe ≥ 1/3. The counting argument required for this is much simpler. Notice that this fact leads to a factor 3 algorithm. (The counting argument requires the use of Lemma 23.19.) The next two exercises develop a factor 2Hk algorithm for the Steiner network problem using the primal–dual schema, where k is the largest connectivity requirement speciﬁed in the instance. For simplicity, assume that the upper bounds, ue , are 1 for each edge e. 23.5 (Williamson, Goemans, Mihail, and Vazirani [258]) Say that a function h : 2V → {0, 1} is uncrossable if h(V ) = 0, and for any two sets A, B ⊂ V , if

23.5

Exercises

225

h(A) = h(B) = 1 then h(A−B) = h(B −A) = 1 or h(A∩B) = h(A∪B) = 1. Exercise 22.7 asked for a factor 2 approximation algorithm for IP (22.1) for the case that f was a proper function. In this exercise, we will extend this further to the case that f is an uncrossable function. Now, we need to enhance the last step of Algorithm 22.3; the pruning step needs to be done using reverse delete. Again, F denotes the forest of edges picked by the algorithm. Let us say that a set A ⊂ V is unsatisﬁed w.r.t. the picked edges F if h(A) = 1 and δF (A) = ∅. A minimal unsatisﬁed set will be said to be active. The algorithm is as follows. Algorithm 23.22 (Uncrossable function) 1. (Initialization) F ← ∅; for each S ⊆ V , yS ← 0. 2. (Edge augmentation) while there exists an unsatisﬁed set do: simultaneously raise yS for each active set S, until some edge e goes tight; F ← F ∪ {e}. 3. Let e1 , e2 , . . . , el be the ordered list of edges in F . 4. (Reverse delete) For j = l downto 1 do: If F − {ej } satisﬁes h, then F ← F − {ej }. 5. Return F .

Show that in each iteration, active sets must be disjoint. Assuming that active sets can be eﬃciently found, show that Algorithm 23.22 ﬁnds a primal solution of cost at most twice the dual, i.e., ce ≤ 2 e∈F S

yS .

Hint: Corresponding to each edge e ∈ F , there must be a set A ⊂ V such that h(A) = 1 and δF (A) = {e}. Call such a set a witness for e. A family C consisting of a witness for each e ∈ F is called a witness family. Include V in this family. Show, by uncrossing, that C can be assumed to be laminar and therefore can be viewed as a tree. Use this to prove that in each iteration, the average degree of active sets is at most two, as in Lemma 22.8. 23.6 Give an example to show that if reverse delete is replaced by a forward delete, then the approximation factor for Algorithm 23.22 can be unbounded for some uncrossable function. 23.7 (Goemans, Goldberg, Plotkin, Shmoys, Tardos, and Williamson [102]) We will solve the Steiner network problem in k phases, numbered 0, 1, . . . , k − 1. In each phase, we will pick a forest from the remaining graph. The solution will be the union of the k forests. Let Fp−1 denote the set of edges picked

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in phases numbered 0, 1, . . . , p − 1. At the beginning of the pth phase, deﬁne the deﬁciency of set S ⊂ V to be max{f (S) − |δFp−1 (A)|, 0}. The ﬁrst p − 1 phases ensure that every set has deﬁciency at most k − p. In the pth phase, deﬁne function h as h(S) = 1 if deﬁciency(S) = k − p 0 otherwise

Show that h is an uncrossable function. Show that Algorithm 23.22 can be implemented in polynomial time for this uncrossable function, i.e., active sets can be found in polynomial time. Let F be the set of edges picked by Algorithm 23.22 from E −Fp−1 , and y be the dual solution constructed when run with function h. Construct the dual program to LP (23.2), and show that there is a feasible solution, say d, to this program such that ce ≤ 2 e∈F S

yS ≤

2 g(d), k−p

where g(d) is the objective function value of dual solution d. Adding over all k phases leads to the required factor. Hint: Use a max-ﬂow algorithm for ﬁnding active sets. The dual program will have a variable ze for each edge e. For edges e ∈ Fp−1 , set ze = S: e∈δ(S) yS , for constructing a dual feasible solution. 23.8 Give an inﬁnite family of graphs to show that the performance guarantee of the algorithm in Exercise 23.7 is tight within constant factors. The following deﬁnitions will be useful for the next three exercises. These notions are connected to the theme of this chapter, i.e., small subgraphs with speciﬁed numbers of disjoint paths, via Menger’s theorem (see Exercise 12.5. An undirected graph is said to be k-vertex (k-edge) connected if it has at least k+1 vertices, and the removal of any set of at most k−1 vertices (edges) from it leaves a connected graph. A directed graph is said to be k-vertex (k-edge) connected if it has at least k + 1 vertices, and the deletion of any set of at most k − 1 vertices (edges) leaves a strongly connected graph. 23.9 (Cheriyan and Thurimella [42]) This exercise develops a 1 + 2/k factor algorithm for the following problem. Problem 23.23 (Minimum k-vertex connected subgraph) Given a nonnegative integer k and an undirected graph G = (V, E) that is k-vertex connected, ﬁnd a minimum cardinality set E ⊂ E such that the subgraph G = (V, E ) is k-vertex connected. Let G = (V, E) be k-vertex connected. We will say that edge e ∈ E is critical if its removal leaves a graph that is not k-vertex connected. A simple cycle C in G is critical if every edge on C is critical. A theorem of Mader,

23.5

Exercises

227

which states that a critical cycle in G must have a vertex of degree exactly k, is central to the algorithm. Algorithm 23.24 (k-vertex connected subgraph) 1. Find a minimum cardinality set M ⊂ E such that ∀v ∈ V : degM (v) ≥ k − 1. 2. Find a minimal set F such that M ∪ F is k-vertex connected. 3. Output G = (V, M ∪ F ).

1. Give a polynomial time algorithm for Step 1 of Algorithm 23.24. Observe that |M | ≤ OPT. Hint: Use a b-matching algorithm on the complement of G. Given an undirected graph G = (V, E) and a function b : V → Z+ specifying an upper bound for each vertex, the b-matching problem asks for a maximum cardinality set M ⊆ E such that ∀v ∈ V, degM (v) ≤ b(v). This problem is in P. 2. Give an eﬃcient implementation for Step 2 of Algorithm 23.24. 3. Use Mader’s theorem to show that F must be acyclic, and hence |F | ≤ |V | −1. Use this to show that Algorithm 23.24 achieves an approximation factor of 1 + 2/k. Hint: Use the fact that k|V |/2 is a lower bound on OPT. 23.10 (Cheriyan and Thurimella [42]) Consider the problem of ﬁnding a minimum k-vertex connected subgraph of a directed graph. Give an algorithm similar to that in Exercise 23.9 for achieving factor 1 + 2/k for this problem. Use the following two facts. 1. In a directed graph, an alternating cycle, C, is an even length sequence of distinct edges (v0 , v1 )(v2 , v1 )(v2 , v3 )(v4 , v3 ) . . . (vm−1 , vm )(v0 , vm ), where vertices are allowed to repeat. Notice that alternate vertices on C have two out-edges (two in-edges). Vertices having two out-edges (two inedges) will be called C-out (C-in) vertices. Mader showed that if G is a k-vertex connected directed graph containing an alternating cycle C, each of whose edges is critical, then C contains either a C-out vertex having out-degree exactly k or a C-in vertex having in-degree exactly k. 2. Given a directed graph G = (V, E), deﬁne its associated bipartite graph H to be the following. Corresponding to each vertex v ∈ V , H has two vertices, v− and v+ , and corresponding to each edge (u, v) ∈ E, H has the edge (u+ , v− ). There is an alternating cycle in G iﬀ its associated bipartite graph contains a cycle.

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23.11 (Khuller and Vishkin [172], using Edmonds [72]) This exercise develops a factor 2 algorithm for the following problem. Problem 23.25 (Minimum k-edge connected subgraph) Given an undirected graph G = (V, E), a function w : E → Q+ , and an integer k, ﬁnd a minimum weight subgraph of G that is k-edge connected. 1. Let r ∈ V be any vertex of G. Consider the problem of ﬁnding a minimum weight subgraph G of G such that for each vertex v ∈ V , there are k edge-disjoint paths from r to v in G . Show that this problem is the same as Problem 23.25, i.e., any solution to one is also a solution to the other. 2. Let G = (V, E) be an edge-weighted directed graph and r ∈ V be one of its vertices. A set E ⊆ E is said to be an r-arborescence if every vertex, other than r, has in-degree 1. In eﬀect, an r-arborescence is a spanning tree directed out of r. Deﬁne the r-connectivity of G to be max{k | ∀v ∈ V ∃k edge-disjoint paths from r to v in G}. Edmonds showed that the maximum number of edge-disjoint r-arborescences in G is equal to the r-connectivity of G. Use this to show that the problem of ﬁnding a minimum weight subgraph of G that has an rconnectivity of k is the same as the problem of ﬁnding a minimum weight subgraph of G that has k edge-disjoint r-arborescences. 3. Edmonds showed that the edges of a directed graph G = (V, E) can be partitioned into k edge-disjoint r-arborescences iﬀ, on ignoring directions, E can be partitioned into k spanning trees, and the in-degree of every vertex, other than r, is exactly k. Use this characterization to show that the problem of ﬁnding a minimum weight subgraph of G that has k edgedisjoint arborescences can be solved in polynomial time. Hint: This problem can be expressed as a matroid intersection problem, the two matroids being a partition matroid and the k-fold union of a graphic matroid (which is also a matroid). 4. Let G = (V, E) be an edge-weighted undirected graph and r ∈ V be one of its vertices. Let OPT(G) denote the weight of an optimal solution to Problem 23.25 on instance G. Obtain graph H by bidirecting G, i.e., by replacing each edge (u, v) ∈ E with the two edges (u → v) and (v → u), each having the same weight as (u, v). Let OPT(H) denote the weight of a minimum weight subgraph of H that can be partitioned into k rarborescences. Show that OPT(G) ≤ OPT(H) ≤ 2 · OPT(G). Use this to obtain a factor 2 approximation algorithm for Problem 23.25. 23.12 (Goemans and Bertsimas [101]) The metric Steiner network problem is the Steiner network problem with the restrictions that G is a complete

23.5

Exercises

229

graph, the cost function on edges satisﬁes the triangle inequality, and ue = ∞ for each edge. It generalizes the metric Steiner tree problem to arbitrary connectivity requirements. For D ⊆ V , deﬁne LPS (D) to be LP-relaxation (23.2), together with a set of equality constraints for vertices in D, as follows. minimize e∈E ce xe xe ≥ f (S), e: e∈δ(S)

(23.4) S⊆V v∈D e∈E

subject to

xe = f ({v}), e: e∈δ({v})

xe ≥ 0,

It turns out that the equality constraints are redundant for the metric Steiner network problem. For any choice of D ⊆ V , an optimal solution to LPS (D) is also an optimal solution to LPS (∅). This is called the parsimonious property. Let us say that a vertex v is Steiner if it has no connectivity requirements, i.e., if ∀v ∈ V, r(u, v) = 0. Use the parsimonious property to prove that there is a fractional optimal solution to the metric Steiner network problem which has no edges incident at Steiner vertices. 23.13 Consider the following integer program for the traveling salesman problem (Problem 3.5). minimize e∈E ce xe xe = 2, e: e

(23.5) v∈V S⊂V e∈E

subject to incident at xe ≥ 2, e: e∈δ(S) v

xe ∈ {0, 1},

Show that optimal solutions to this integer program are optimal TSP tours. The linear relaxation of this program is called the subtour elimination LPrelaxation for TSP. The rest of this exercise deals with the special case of metric TSP and develops a proof that the solution found by Christoﬁdes’ algorithm, Algorithm 3.10, is within a factor of 3/2 of the optimal solution to this LP-relaxation. 1. Give an example that puts a lower bound of (essentially) 4/3 on the integrality gap of this relaxation. Hint: Use the following graph.

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Steiner Network

t

t

t t t

t ✚ ✚ ✚ ✚ t t

t ... t ...

t t t

t t t

t

t ...

✚ ✚ t t t

✚ ✚

t

2. Let v1 be an arbitrary vertex in the given graph G = (V, E). Deﬁne a 1-tree in G to be a spanning tree on the vertices V − {v1 }, together with two distinct edges incident at vertex v1 . Clearly, the cost of an optimal 1-tree is a lower bound on the cost of an optimal TSP tour. LP (12.12) stated in Exercise 12.9 was an exact relaxation for the MST problem. Use it to obtain an exact LP-relaxation for the minimum 1-tree problem. 3. (Held and Karp [123]) Show that the cost of a minimum 1-tree is a lower bound on the cost of an optimal solution to the subtour elimination LP. Hint: Compare the LP obtained above for minimum 1-tree with the following equivalent formulation of the subtour elimination LP. (By e : e ∈ S we mean edges e that have both endpoints in S.) minimize e∈E ce xe xe = 2, e: e

(23.6) v∈V S⊆V e∈E

subject to incident at v xe ≤ |S| − 1, e: e∈S

xe ≥ 0,

4. Use the parsimonious property, introduced in Exercise 23.12, to show that the equality constraints, on vertices, in the subtour elimination LP are redundant. (Observe that the LP obtained on removing these constraints is also an LP-relaxation for the problem of ﬁnding a minimum cost spanning two-edge connected subgraph of G.) 5. For D ⊆ V , let LPT (D) denote the subtour elimination LP for GD , the subgraph of G induced on D. Let OPTf (LPT (D) denote the cost of an optimal solution to LPT (D). Show the following monotonicity property OPTf (LPT (D)) ≤ OPTf (LPT (V )). Hint: Use the relaxation without equality constraints. 6. Let D ⊆ V be of even cardinality. Show that the cost of a minimum cost perfect matching in the subgraph of G induced on D is ≤ 1 OPTf (LPT (D)). 2 Hint: Use LP (12.9), introduced in Exercise 12.8, for matching, and LP (23.6) for TSP.

23.6

Notes

231

7. Show that the metric TSP solution found using Algorithm 3.10, is within a factor of 3/2 of the optimal solution to the subtour elimination LP.

23.6 Notes

The result of this chapter is due to Jain [137]. Cheriyan and Thurimella [42] contains further results on ﬁnding small subgraphs of a given graph with a speciﬁed connectivity, as well as references to Mader’s theorems. The subtour elimination LP-relaxation for TSP was given in Dantzig, Ford, and Fulkerson [59]. The result of Exercise 23.13 was ﬁrst established by Wolsey [260]. The proof developed here is from Shmoys and Williamson [240].

24 Facility Location

The facility location problem has occupied a central place in operations research since the early 1960’s. It models design situations such as deciding placements of factories, warehouses, schools, and hospitals. Modern day applications include placement of proxy servers on the web. In this chapter, we will present a primal–dual schema based factor 3 approximation algorithm for the special case when connection costs satisfy the triangle inequality. The algorithm diﬀers in two respects from previous primal–dual algorithms. First, the primal and dual pair of LPs have negative coeﬃcients and do not form a covering-packing pair. Second, we will relax primal complementary slackness conditions rather than the dual ones. Also, the idea of synchronization, introduced in the primal–dual schema in Chapter 22, is developed further, with an explicit timing of events playing a role. Problem 24.1 (Metric uncapacitated facility location) Let G be a bipartite graph with bipartition (F, C), where F is the set of facilities and C is the set of cities. Let fi be the cost of opening facility i, and cij be the cost of connecting city j to (opened) facility i. The connection costs satisfy the triangle inequality. The problem is to ﬁnd a subset I ⊆ F of facilities that should be opened, and a function φ : C → I assigning cities to open facilities in such a way that the total cost of opening facilities and connecting cities to open facilities is minimized. Consider the following integer program for this problem. In this program, yi is an indicator variable denoting whether facility i is open, and xij is an indicator variable denoting whether city j is connected to the facility i. The ﬁrst set of constraints ensures that each city is connected to at least one facility, and the second ensures that this facility must be open. minimize i∈F, j∈C

cij xij + i∈F fi yi j∈C i ∈ F, j ∈ C i ∈ F, j ∈ C i∈F

(24.1)

subject to i∈F xij ≥ 1, yi − xij ≥ 0, xij ∈ {0, 1}, yi ∈ {0, 1},

24.1

An intuitive understanding of the dual

233

The LP-relaxation of this program is: minimize i∈F, j∈C

cij xij + i∈F fi yi j∈C i ∈ F, j ∈ C i ∈ F, j ∈ C i∈F

(24.2)

subject to i∈F xij ≥ 1, yi − xij ≥ 0, xij ≥ 0, yi ≥ 0,

The dual program is: maximize j∈C αj αj − βij ≤ cij , βij ≤ fi , j∈C (24.3) i ∈ F, j ∈ C i∈F j∈C i ∈ F, j ∈ C

subject to

αj ≥ 0, βij ≥ 0,

24.1 An intuitive understanding of the dual

Let us ﬁrst give the reader some feel for how the dual variables “pay” for a primal solution by considering the following simple setting. Suppose LP (24.2) has an optimal solution that is integral, say I ⊆ F and φ : C → I. Thus, under this solution, yi = 1 iﬀ i ∈ I, and xij = 1 iﬀ i = φ(j). Let (α, β) denote an optimal dual solution. The primal and dual complementary slackness conditions are: (i) (ii) (iii) (iv) ∀i ∈ F, j ∈ C : xij > 0 ⇒ αj − βij = cij ∀i ∈ F : yi > 0 ⇒ βij = fi j∈C ∀j ∈ C : αj > 0 ⇒ i∈F xij = 1

∀i ∈ F, j ∈ C : βij > 0 ⇒ yi = xij

By condition (ii), each open facility must be fully paid for, i.e., if i ∈ I, then βij = fi . j: φ(j)=i

234

24

Facility Location

Consider condition (iv). Now, if facility i is open, but φ(j) = i, then yi = xij , and so βij = 0, i.e., city j does not contribute to opening any facility besides the one it is connected to. By condition (i), if φ(j) = i, then αj − βij = cij . Thus, we can think of αj as the total price paid by city j; of this, cij goes towards the use of edge (i, j), and βij is the contribution of j towards opening facility i.

24.2 Relaxing primal complementary slackness conditions

Suppose the primal complementary slackness conditions were relaxed as follows, while maintaining the dual conditions: ∀j ∈ C : (1/3)cφ(j)j ≤ αj − βφ(j)j ≤ cφ(j)j , and ∀i ∈ I : (1/3)fi ≤ j: φ(j)=i

βij ≤ fi .

Then, the cost of the (integral) solution found would be within thrice the dual found, thus leading to a factor 3 approximation algorithm. However, we would like to obtain the stronger inequality stated in Theorem 24.7. Now, the dual pays at least one-third the connection cost, but must pay completely for opening facilities. This stronger inequality will be needed in order to use this algorithm to solve the k-median problem in Chapter 25. For this reason, we will relax the primal conditions as follows. The cities are partitioned into two sets, directly connected and indirectly connected. Only directly connected cities will pay for opening facilities, i.e., βij can be nonzero only if j is a directly connected city and i = φ(j). For an indirectly connected city j, the primal condition is relaxed as follows: (1/3)cφ(j)j ≤ αj ≤ cφ(j)j . All other primal conditions are maintained, i.e., for a directly connected city j, αj − βφ(j)j = cφ(j)j , and for each open facility i, βij = fi . j: φ(j)=i

24.3

Primal–dual schema based algorithm

235

24.3 Primal–dual schema based algorithm

The algorithm consists of two phases. In Phase 1, the algorithm operates in a primal–dual fashion. It ﬁnds a dual feasible solution and also determines a set of tight edges and temporarily open facilities, Ft . Phase 2 consists of choosing a subset I of Ft to open, and ﬁnding a mapping, φ, from cities to I. Algorithm 24.2 Phase 1 We would like to ﬁnd as large a dual solution as possible. This motivates the following underlying process for dealing with the non-covering-packing pair of LPs. Each city j raises its dual variable, αj , until it gets connected to an open facility. All other primal and dual variables simply respond to this change, trying to maintain feasibility or satisfying complementary slackness conditions. A notion of time is deﬁned in this phase, so that each event can be associated with the time at which it happened; the phase starts at time 0. Initially, each city is deﬁned to be unconnected. Throughout this phase, the algorithm raises the dual variable αj for each unconnected city j uniformly at unit rate, i.e., αj will grow by 1 in unit time. When αj = cij for some edge (i, j), the algorithm will declare this edge to be tight. Henceforth, dual variable βij will be raised uniformly, thus ensuring that the ﬁrst constraint in LP (24.3) is not violated. βij goes towards paying for facility i. Each edge (i, j) such that βij > 0 is declared special. Facility i is said to be paid for if j βij = fi . If so, the algorithm declares this facility temporarily open. Furthermore, all unconnected cities having tight edges to this facility are declared connected and facility i is declared the connecting witness for each of these cities. (Notice that the dual variables αj of these cities are not raised anymore.) In the future, as soon as an unconnected city j gets a tight edge to i, j will also be declared connected and i will be declared the connecting witness for j (notice that βij = 0 and thus edge (i, j) is not special). When all cities are connected, the ﬁrst phase terminates. If several events happen simultaneously, the algorithm executes them in arbitrary order. Remark 24.3 At the end of Phase 1, a city may have paid towards temporarily opening several facilities. However, we want to ensure that a city pays only for the facility that it is eventually connected to. This is ensured in Phase 2, which chooses a subset of temporarily open facilities for opening permanently. Phase 2 Let Ft denote the set of temporarily open facilities and T denote the subgraph of G consisting of all special edges. Let T 2 denote the graph that has edge (u, v) iﬀ there is a path of length at most 2 between u and v in T , and let H

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be the subgraph of T 2 induced on Ft . Find any maximal independent set in H, say I. All facilities in the set I are declared open. For city j, deﬁne F j = {i ∈ Ft | (i, j) is special}. Since I is an independent set, at most one of the facilities in F j is opened. If there is a facility i ∈ F j that is opened, then set φ(j) = i and declare city j directly connected. Otherwise, consider tight edge (i , j) such that i was the connecting witness for j. If i ∈ I, again set φ(j) = i and declare city j directly connected (notice that in this case βi j = 0). In the remaining case that i ∈ I, let i be any / neighbor of i in graph H such that i ∈ I. Set φ(j) = i and declare city j indirectly connected. I and φ deﬁne a primal integral solution: xij = 1 iﬀ φ(j) = i and yi = 1 iﬀ i ∈ I. The values of αj and βij obtained at the end of Phase 1 form a dual feasible solution.

24.4 Analysis

We will show how the dual variables αj ’s pay for the primal costs of opening f e facilities and connecting cities to facilities. Denote by αj and αj the contribuf e tions of city j to these two costs respectively; αj = αj + αj . If j is indirectly f e connected, then αj = 0 and αj = αj . If j is directly connected, then the following must hold: αj = cij + βij , f e where i = φ(j). Now, let αj = βij and αj = cij .

Lemma 24.4 Let i ∈ I. Then, f αj = fi . j: φ(j)=i

Proof: Since i is temporarily open at the end of Phase 1, it is completely paid for, i.e., βij = fi . j: (i,j)

is special

The critical observation is that each city j that has contributed to fi must f be directly connected to i. For each such city, αj = βij . Any other city j f ✷ that is connected to facility i must satisfy αj = 0. The lemma follows. Corollary 24.5 i∈I fi =

j∈C

f αj .

24.4

Analysis

237

f Recall that αj was deﬁned to be 0 for indirectly connected cities. Thus, only the directly connected cities pay for the cost of opening facilities. e Lemma 24.6 For an indirectly connected city j, cij ≤ 3αj , where i = φ(j).

Proof: Let i be the connecting witness for city j. Since j is indirectly connected to i, (i, i ) must be an edge in H. In turn, there must be a city, say j , such that (i, j ) and (i , j ) are both special edges. Let t1 and t2 be the times at which i and i were declared temporarily open during Phase 1. i i s s ❏ ✡❏ ✡ ❏ ✡ ❏ ❏ ❏ ✡ ❏ ✡ ❏ ❏s ❏s ❏✡ ❏ j j Since edge (i , j) is tight, αj ≥ ci j . We will show that αj ≥ cij and αj ≥ ci j . Then, the lemma will follow by using the triangle inequality. Since edges (i , j ) and (i, j ) are tight, αj ≥ cij and αj ≥ ci j . Since both these edges are special, they must both have gone tight before either i or i is declared temporarily open. Consider the time min(t1 , t2 ). Clearly, αj cannot be growing beyond this time. Therefore, αj ≤ min(t1 , t2 ). Finally, since i is the connecting witness for j, αj ≥ t2 . Therefore, αj ≥ αj , and the required inequalities follow. ✷ Theorem 24.7 The primal and dual solutions constructed by the algorithm satisfy: cij xij + 3 i∈F, j∈C i∈F

fi yi ≤ 3 j∈C αj .

e e Proof: For a directly connected city j, cij = αj ≤ 3αj , where φ(j) = i. Combining with Lemma 24.6 we get

cij xij ≤ 3 i∈F,j∈C j∈C

e αj .

Adding to this the equality stated in Corollary 24.5 multiplied by 3 gives the theorem. ✷

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24.4.1

Running time

A special feature of the primal–dual schema is that it yields algorithms with good running times. Since this is especially so for the current algorithm, we will provide some implementation details. We will adopt the following notation: nc = |C| and nf = |F |. The total number of vertices nc + nf = n, and the total number of edges nc × nf = m. Sort all the edges by increasing cost – this gives the order and the times at which edges go tight. For each facility, i, we maintain the number of cities that are currently contributing towards it, and the anticipated time, ti , at which it would be completely paid for if no other event happens on the way. Initially all ti ’s are inﬁnite, and each facility has 0 cities contributing to it. The ti ’s are maintained in a binary heap so we can update each one and ﬁnd the current minimum in O(log nf ) time. Two types of events happen, and they lead to the following updates. • An edge (i, j) goes tight. – If facility i is not temporarily open, then it gets one more city contributing towards its cost. The amount contributed towards its cost at the current time can be easily computed. Therefore, the anticipated time for facility i to be paid for can be recomputed in constant time. The heap can be updated in O(log nf ) time. – If facility i is already temporarily open, city j is declared connected, and αj is not raised anymore. For each facility i that was counting j as a contributor, we need to decrease the number of contributors by 1 and recompute the anticipated time at which it gets paid for. • Facility i is completely paid for. In this event, i will be declared temporarily open, and all cities contributing to i will be declared connected. For each of these cities, we will execute the second case of the previous event, i.e., update facilities that they were contributing towards. The next theorem follows by observing that each edge (i, j) will be considered at most twice. First, when it goes tight. Second, when city j is declared connected. For each consideration of this edge, we will do O(log nf ) work. Theorem 24.8 Algorithm 24.2 achieves an approximation factor of 3 for the facility location problem and has a running time of O(m log m). 24.4.2 Tight example

The following inﬁnite family of examples shows that the analysis of our algorithm is tight: The graph has n cities, c1 , c2 , . . . , cn and two facilities f1 and f2 . Each city is at a distance of 1 from f2 . City c1 is at a distance of 1 from f1 , and c2 , . . . , cn are at a distance of 3 from f1 . The opening cost of f1 and f2 are ε and (n + 1)ε, respectively, for a small number ε.

24.5

Exercises

239

f1

c1 s ✑◗ ✑ 1 ✑ c2 ◗◗ 1 ✑ s ✏ ◗◗ ✑3 ✏✏ 1 ◗ ✑ ✏ c3 ✑✏✏3 ✏ 1 ◗ s ◗ f2 ✑ s s ✏ ❅ ❅ ❅ . . 1 . ❅3 ❅ ❅ ❅ cn ❅s

The optimal solution is to open f2 and connect all cities to it, at a total cost of (n + 1)ε + n. Algorithm 24.2 will however open facility f1 and connect all cities to it, at a total cost of ε + 1 + 3(n − 1).

24.5 Exercises

24.1 Consider the general uncapacitated facility location problem in which the connection costs are not required to satisfy the triangle inequality. Give a reduction from the set cover problem to show that approximating this problem is as hard as approximating set cover and therefore cannot be done ˜ better than O(log n) factor unless NP ⊆ P. Also, give an O(log n) factor algorithm for this problem. 24.2 In Phase 2, instead of picking all special edges in T , pick all tight edges. Show that now Lemma 24.6 does not hold. Give a suitable modiﬁcation to the algorithm that restores Lemma 24.6. Hint: Order facilities in H in the order in which they were temporarily opened, and pick I to be the lexicographically ﬁrst maximal independent set. 24.3 Give a factor 3 tight example for Algorithm 24.2 in which the set of cities and facilities is the same, i.e., C = F . 24.4 Consider the proof of Lemma 24.6. Give an example in which αj > t2 . 24.5 The vector α found by Algorithm 24.2 is maximal in the sense that if we increase any αj in this vector, then there is no way of setting the βij ’s to get a feasible dual solution. Is every maximal solution α within 3 times the optimal solution to dual program for facility location?

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Hint: It is easy to construct a maximal solution that is 2/n times the optimal. Consider n facilities with an opening cost of 1 each and n cities connected to distinct facilities by edges of cost ε each. In addition, there is another city that is connected to each facility with an edge of cost 1. 24.6 Consider the following modiﬁcation to the metric uncapacitated facility location problem. Deﬁne the cost of connecting city j to facility i to be c2 . ij The cij ’s still satisfy the triangle inequality (but the new connection costs, of c2 , do not). Show that Algorithm 24.2 achieves an approximation guarantee ij of factor 9 for this case. 24.7 Consider the following generalization to arbitrary demands. For each city j, a nonnegative demand dj is speciﬁed, and any open facility can serve this demand. The cost of serving this demand via facility i is cij dj . Give an IP and LP-relaxation for this problem, and extend Algorithm 24.2 to get a factor 3 algorithm. Hint: Raise αj at rate dj . 24.8 In the capacitated facility location problem, we are given a number ui for each facility i, and facility i can serve at most ui cities. Show that the modiﬁcation of LP (24.2) to this problem has an unbounded integrality gap. 24.9 Consider the variant of the capacitated metric facility location problem in which each facility can be opened an unbounded number of times. If facility i is opened yi times, it can serve at most ui yi cities. Give an IP and LPrelaxation for this problem, and extend Algorithm 24.2 to obtain a constant factor algorithm. 24.10 (Charikar, Khuller, Mount, and Narshimhan [40]) Consider the prizecollecting variant of the facility location problem, in which there is a speciﬁed penalty for not connecting a city to an open facility. The objective is to minimize the sum of the connection costs, facility opening costs, and penalties. Give a factor 3 approximation algorithm for this problem. 24.11 (Jain and Vazirani [140]) Consider the fault tolerant variant of the facility location problem, in which the additional input is a connection requirement rj for each city j. In the solution, city j needs to be connected to rj distinct open facilities. The objective, as before, is to minimize the sum of the connection costs and the facility opening costs. Decompose the problem into k phases, numbered k down to 1, as in Exercise 23.7. In phase p, all cities having a residual requirement of p are provided one more connection to an open facility. In phase p, the facility location algorithm of this chapter is run on the following modiﬁed graph, Gp . The cost of each facility that is opened in an earlier phase is set to 0. If city j is connected to facility i in an earlier phase, then cij is set to ∞.

24.5

Exercises

241

1. Show that even though Gp violates the triangle inequality at some places, the algorithm gives a solution within factor 3 of the optimal solution for this graph. Hint: Every time short-cutting is needed; the triangle inequality holds. 2. Show that the solution found in phase p is of cost at most 3 · OPT/p, where OPT is the cost of the solution to the entire problem. Hint: Remove ∞ cost edges of Gp from the optimal solution and divide the rest by p. Show that this is a feasible fractional solution for phase p. 3. Show that this algorithm achieves an approximation factor of 3 · Hk for the fault tolerant facility location problem. 24.12 (Mahdian, Markakis, Saberi, and Vazirani [201]) This exercise develops a factor 3 greedy algorithm for the metric uncapacitated facility location problem, together with an analysis using the method of dual ﬁtting. Consider the following modiﬁcation to Algorithm 24.2. As before, dual variables, αj , of all unconnected cities, j, are raised uniformly. If edge (i, j) is tight, βij is raised. As soon as a facility, say i, is paid for, it is declared open. Let S be the set of unconnected cities having tight edges to i. Each city j ∈ S is declared connected and stops raising its αj . So far, the new algorithm is the same as Algorithm 24.2. The main diﬀerence appears at this stage: Each city j ∈ S withdraws its contribution from other facilities, i.e., for each facility i = i, set βi j = 0. When all cities have been declared connected, the algorithm terminates. Observe that each city contributes towards the opening cost of at most one facility – the facility it gets connected to. 1. This algorithm actually has a simpler description as a greedy algorithm. Provide this description. Hint: Use the notion of cost–eﬀectiveness deﬁned for the greedy set cover algorithm. 2. The next 3 parts use the method of dual ﬁtting to analyze this algorithm. First observe that the primal solution found is fully paid for by the dual computed. 3. Let i be an open facility and let {1, . . . , k} be the set of cities that contributed to opening i at some point in the algorithm. Assume w.l.o.g. that α1 ≤ αj for j ≤ k. Show that for j ≤ k, αj − cij ≤ 2α1 . Also, show that k k

αj ≤ 3 j=1 j=1

cij + fi .

Hint: Use the triangle inequality and the following inequality which is a consequence of the fact that at any point, the total amount contributed for opening facility i is at most fi :

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Facility Location

α1 − cij ≤ fi . j: cij ≤α1

4. Hence show that α/3 is a dual feasible solution. 5. How can the analysis be improved – a factor 1.86 analysis is known for this algorithm. 6. Give a time eﬃcient implementation of this algorithm, matching the running time of Algorithm 24.2 7. Do you see room for improving the algorithm? Hint: Suppose city j is connected to open facility i at some point in the algorithm. Later, facility i is opened, and suppose that cij > ci j . Then, connecting j to i will reduce the cost of the solution. 24.13 (Mahdian, Markakis, Saberi, and Vazirani [201]) Consider the following variant of the metric uncapacitated facility location problem. Instead of fi , the opening cost for each facility i ∈ F , we are provided a startup cost si and an incremental cost ti . Deﬁne the new opening cost for connecting k > 0 cities to facility i to be si + kti . Connection costs are speciﬁed by a metric, as before. The object again is to connect each city to an open facility so as to minimize the sum of connection costs and opening costs. Give an approximation factor preserving reduction from this problem to the metric uncapacitated facility location problem. Hint: Modify the metric appropriately.

24.6 Notes

The ﬁrst approximation algorithm for the metric uncapacitated facility location problem, due to Hochbaum [124], achieved an approximation guarantee of O(log n). The ﬁrst constant factor approximation algorithm, achieving a guarantee of 3.16, was due to Shmoys, Tardos, and Aardal [239]. It was based on LP-rounding. The current best algorithm, achieving an approximation guarantee of 1.61, is due to Jain, Mahdian, and Saberi [138]. This algorithm, a small modiﬁcation of the greedy algorithm presented in Exercise 24.12, is analyzed using the method of dual ﬁtting. The primal–dual schema based Algorithm 24.2 is due to Jain and Vazirani [141].

25 k-Median

The k-median problem diﬀers from the facility location problem in two respects – there is no cost for opening facilities and there is an upper bound, k, on the number of facilities that can be opened. It models the problem of ﬁnding a minimum cost clustering, and therefore has numerous applications. The primal–dual schema works by making judicious local improvements and is not suitable for ensuring a global constraint, such as the constraint in the k-median problem that at most k facilities be opened. We will get around this diﬃculty by borrowing the powerful technique of Lagrangian relaxation from combinatorial optimization. Problem 25.1 (Metric k-median) Let G be a bipartite graph with bipartition (F, C), where F is the set of facilities and C is the set of cities, and let k be a positive integer specifying the number of facilities that are allowed to be opened. Let cij be the cost of connecting city j to (opened) facility i. The connection costs satisfy the triangle inequality. The problem is to ﬁnd a subset I ⊆ F, |I| ≤ k, of facilities that should be opened and a function φ : C → I assigning cities to open facilities in such a way that the total connecting cost is minimized.

25.1 LP-relaxation and dual

The following is an integer program for the k-median problem. The indicator variables yi and xij play the same role as in (24.1). minimize i∈F, j∈C

cij xij xij ≥ 1, i∈F (25.1) j∈C i ∈ F, j ∈ C

subject to

yi − xij ≥ 0, −yi ≥ −k i∈F xij ∈ {0, 1}, yi ∈ {0, 1},

i ∈ F, j ∈ C i∈F

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k-Median

The LP-relaxation of this program is: minimize i∈F, j∈C

cij xij xij ≥ 1, i∈F (25.2) j∈C i ∈ F, j ∈ C

subject to

yi − xij ≥ 0, −yi ≥ −k i∈F xij ≥ 0, yi ≥ 0, The dual program is: maximize j∈C i ∈ F, j ∈ C i∈F

αj − zk αj − βij ≤ cij , βij ≤ z, j∈C (25.3) i ∈ F, j ∈ C i∈F j∈C i ∈ F, j ∈ C

subject to

αj ≥ 0, βij ≥ 0, z≥0

25.2 The high-level idea

The similarity between the two problems, facility location and k-median, leads to a similarity in their linear programs, which will be exploited as follows. Take an instance of the k-median problem, assign a cost of z for opening each facility, and ﬁnd optimal solutions to LP (24.2) and LP (24.3), say (x, y) and (α, β), respectively. By the strong duality theorem, cij xij + i∈F, j∈C i∈F

zyi = j∈C αj .

Now, suppose that the primal solution (x, y) happens to open exactly k facilities (fractionally), i.e., i yi = k. Then, we claim that (x, y) and (α, β, z) are optimal solutions to LP (25.2) and LP (25.3), respectively. Feasibility is easy to check. Optimality follows by substituting i yi = k in the above equality and rearranging terms to show that the primal and dual solutions achieve the same objective function value:

25.2

The high-level idea

245

cij xij = i∈F, j∈C j∈C

αj − zk.

Let’s use this idea, together with Algorithm 24.2 and Theorem 24.7, to obtain a “good” integral solution to LP (25.2). Suppose with a cost of z for opening each facility, Algorithm 24.2, happens to ﬁnd solutions (x, y) and (α, β), where the primal solution opens exactly k facilities. By Theorem 24.7, cij xij + 3zk ≤ 3 i∈F, j∈C j∈C

αj .

Now, observe that (x, y) and (α, β, z) are primal (integral) and dual feasible solutions to the k-median problem satisfying cij xij ≤ 3( i∈F, j∈C j∈C

αj − zk).

Therefore, (x, y) is a solution to the k-median problem within thrice the optimal. Notice that the factor 3 proof given above would not work if less than k facilities were opened; if more than k facilities are opened, the solution is infeasible for the k-median problem. The remaining problem is to ﬁnd a value of z so that exactly k facilities are opened. Several ideas are required for this. The ﬁrst is the following principle from economics. Taxation is an eﬀective way of controlling the amount of goods coming across a border – raising tariﬀs will reduce inﬂow and vice versa. In a similar manner, raising z should reduce the number of facilities opened and vice versa. It is natural now to seek a modiﬁcation to Algorithm 24.2 that can ﬁnd a value of z so that exactly k facilities are opened. This would lead to a factor 3 approximation algorithm. Such a modiﬁcation is not known. Instead, we present the following strategy which leads to a factor 6 algorithm. For the rest of the discussion, assume that we never encountered a run of the algorithm which resulted in exactly k facilities being opened. Clearly, when z = 0 the algorithm will open all facilities, and when z is very large it will open only one facility. The latter value of z can be picked to be ncmax , where cmax is the length of the longest edge. We will conduct a binary search on the interval [0, ncmax ] to ﬁnd z2 and z1 for which the algorithm opens k2 > k and k1 < k facilities, respectively, and, furthermore, z1 − z2 ≤ (cmin /12n2 ), where cmin is the length of the shortest nonzero edge. c As before, we will adopt the following notation: nc = |C| and nf = |F |. The total number of vertices nc + nf = n and the total number of edges nc ×nf = m. Let (xs , y s ) and (xl , y l ) be the two primal solutions found, with s l i∈F yi = k1 and i∈F yi = k2 (the superscripts s and l denote “small” and “large,” respectively). Further, let (αs , β s ) and (αl , β l ) be the corresponding dual solutions found.

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k-Median

Let (x, y) = a(xs , y s ) + b(xl , y l ) be a convex combination of these two solutions, with ak1 + bk2 = k. Under these conditions, a = (k2 − k)/(k2 − k1 ) and b = (k − k1 )/(k2 − k1 ). Since (x, y) is a feasible (fractional) solution to the facility location problem that opens exactly k facilities, it is also a feasible (fractional) solution to the k-median problem. In this solution each city is connected to at most two facilities. Lemma 25.2 The cost of (x, y) is within a factor of (3 + 1/nc ) of the cost of an optimal fractional solution to the k-median problem. Proof: By Theorem 24.7 we have cij xs ≤ 3( ij i∈F, j∈C j∈C s αj − z1 k1 ),

and cij xl ≤ 3( ij i∈F, j∈C j∈C l αj − z2 k2 ).

Since z1 > z2 , (αl , β l ) is a feasible dual solution to the facility location problem even if the cost of facilities is z1 . We would like to replace z2 with z1 in the second inequality, at the expense of the increased factor. This is achieved using the upper bound on z1 − z2 and the fact that i∈F, j∈C cij xl ≥ cmin . ij We get cij xl ≤ ij i∈F, j∈C

l αj − z1 k2 .

1 3+ nc

j∈C Adding this inequality multiplied by b with the ﬁrst inequality multiplied by a gives cij xij ≤ i∈F, j∈C

αj − z1 k ,

1 3+ nc

j∈C where α = aαs + bαl . Let β = aβ s + bβ l . Observe that (α, β, z1 ) is a feasible solution to the dual of the k-median problem. The lemma follows. ✷ In Section 25.3 we give a randomized rounding procedure that obtains an integral solution to the k-median problem from (x, y), with a small increase in cost. In Section 25.3.1 we derandomize this procedure.

25.3

Randomized rounding

247

25.3 Randomized rounding

We give a randomized rounding procedure that produces an integral solution to the k-median problem from (x, y). In the process, it increases the cost by a multiplicative factor of 1 + max(a, b). Let A and B be the sets of facilities opened in the two solutions, |A| = k1 and |B| = k2 . For each facility in A, ﬁnd the closest facility in B – these facilities are not required to be distinct. Let B ⊂ B be these facilities. If |B | < k1 , arbitrarily include additional facilities from B − B into B until |B | = k1 . With probability a, open all facilities in A, and with probability b = 1−a, open all facilities in B . In addition, a set of cardinality k − k1 is picked randomly from B − B and facilities in this set are opened. Notice that each facility in B − B has a probability of b of being opened. Let I be the set of facilities opened, |I| = k. The function φ : C → I is deﬁned as follows. Consider city j and suppose that it is connected to i1 ∈ A and i2 ∈ B in the two solutions. If i2 ∈ B , then one of i1 and i2 is opened by the procedure given above, i1 with probability a and i2 with probability b. City j is connected to the open facility. A B k1 k2 i1 ................................. si3 s ❏ ❏ ❏ ❏ ❏s ❏ si2 j B

B−B

If i2 ∈ B − B , let i3 ∈ B be the facility in B that is closest to i1 . City j is connected to i2 if it is open. Otherwise, it is connected to i1 if it is open. If neither i2 or i1 is open, then j is connected to i3 . Denote by cost(j) the connection cost for city j in the fractional solution (x, y); cost(j) = aci1 j + bci2 j . Lemma 25.3 The expected connection cost for city j in the integral solution, E[cφ(j)j ], is ≤ (1 + max(a, b))cost(j). Moreover, E[cφ(j)j ] can be eﬃciently computed. Proof: If i2 ∈ B , E[cφ(j)j ] = aci1 j + bci2 j = cost(j). Consider the second case, that i2 ∈ B . Now, i2 is open with probability b. The probability that / i2 is not open and i1 is open is (1 − b)a = a2 , and the probability that both i2 and i1 are not open is (1 − b)(1 − a) = ab. This gives

248

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k-Median

E[cφ(j)j ] ≤ bci2 j + a2 ci1 j + abci3 j . Since i3 is the facility in B that is closest to i1 , ci1 i3 ≤ ci1 i2 ≤ ci1 j + ci2 j , where the second inequality follows from the triangle inequality. Again, by the triangle inequality, ci3 j ≤ ci1 j + ci1 i3 ≤ 2ci1 j + ci2 j . Therefore, E[cφ(j)j ] ≤ bci2 j + a2 ci1 j + ab(2ci1 j + ci2 j ). Now, a2 ci1 j + abci1 j = aci1 j . Therefore, E[cφ(j)j ] ≤ (aci1 j + bci2 j ) + ab(ci1 j + ci2 j ) ≤ (aci1 j + bci2 j )(1 + max(a, b)). Clearly, in both cases, E[cφ(j)j ] is easy to compute. ✷

Let (xk , y k ) denote the integral solution obtained to the k-median problem by this randomized rounding procedure. Then, Lemma 25.4 E i∈F, j∈C

cij xk ≤ (1 + max(a, b)) ij i∈F, j∈C

cij xij

and, moreover, the expected cost of the solution found can be computed eﬃciently. 25.3.1 Derandomization

Derandomization follows in a straightforward manner using the method of conditional expectation. First, the algorithm opens the set A with probability a and the set B with probability b = 1−a. Pick A, and compute the expected value if k − k1 facilities are randomly chosen from B − B . Next, do the same by picking B instead of A. Choose to open the set that gives the smaller expectation. Second, the algorithm opens a random subset of k − k1 facilities from B − B . For a choice D ⊂ B − B , |D| ≤ k − k1 , denote by E[D, B − (B ∪ D)] the expected cost of the solution if all facilities in D and additionally k − k1 − |D| facilities are randomly opened from B − (B ∪ D). Since each facility of B − (B ∪ D) is equally likely to be opened, we get E[D, B − (B ∪ D)] = 1 |B − (B ∪ D)|

E[D ∪ {i}, B − (B ∪ D ∪ {i})].

i∈B−(B ∪D)

This implies that there is an i such that

25.3

Randomized rounding

249

E[D ∪ {i}, B − (B ∪ D ∪ {i})] ≤ E[B , B − (B ∪ D)]. Choose such an i and replace D with D ∪ {i}. Notice that the computation of E[D ∪ {i}, B − (B ∪ D ∪ {i})] can be done as in Lemma 25.4. 25.3.2 Running time

It is easy to see that a ≤ 1 − 1/nc (this happens for k1 = k − 1 and k2 = nc ) and b ≤ 1 − 1/k (this happens for k1 = 1 and k2 = k + 1). Therefore, 1 + max(a, b) ≤ 2 − 1/nc . Altogether, the approximation guarantee is (2 − 1/nc )(3 + 1/nc ) < 6. This procedure can be derandomized using the method of conditional probabilities, as in Section 25.3.1. The binary search will make O(log2 (n3 cmax /cmin )) = O(L+log n) probes. The running time for each probe is dominated by the time taken to run Algorithm 24.2; randomized rounding takes O(n) time and derandomization takes O(m) time. Hence we get Theorem 25.5 The algorithm given above achieves an approximation factor of 6 for the k-median problem, and has a running time of O(m log m(L + log(n))). 25.3.3 Tight example

A tight example for the factor 6 k-median algorithm is not known. However, below we give an inﬁnite family of instances which show that the analysis of the randomized rounding procedure cannot be improved. The two solutions (xs , y s ) and (xl , y l ) open one facility, f0 , and k + 1 facilities, f1 , . . . , fk+1 , respectively. The distance between f0 and any other fi is 1, and that between two facilities in the second set is 2. All n cities are at a distance of 1 from f0 , and at a distance of ε from fk+1 . The rest of the distances are given by the triangle inequality. The convex combination is constructed with a = 1/k and b = 1 − 1/k. ✏s ✏✏ ✁ ✏✏ 1 f0 ✏ ✁ c2 ✏ s ✏ 1 1 ✁ s ✚ ✚❇ ❙ ✡ ✁ ✡ ✚ ✡ ❇ ❙ 1 ✡ c3 1✚ ✁ s 1 1 ❇1 ❙ ✚ ✡ ε✁ε ✡ ✡ ✚ ❇ ❙ ✁ ✡ε ✚ ✡ ❇ ❙✁ ✡ ✚ ✡ . ❇ ✁✡ ❙ . ✚ . ✡ ✚ ❇ ✁✡ ❙ ✁ ... ✚ s s s s fk+1 ❇✡ f1 ...... f2 ...... f3 .....✡ . . ❍❍ ❙ ..... ..... ...... ..... ..... ..... ...... ..... . ❍ ❙ 2 2 ε ❍ cn ❙ ❍s ❙ ❍ c1 250

25

k-Median

Now, the cost of the convex combination is an + bεn. Suppose the algorithm picks f1 as the closest neighbor of f0 . The expected cost of the solutions produced by the randomized rounding procedure is then n(bε+a2 +ab(2+ε)). Letting ε tend to 0, the cost of the convex combination is essentially na and that of the rounded solution is na(1 + b). 25.3.4 Integrality gap

The algorithm given above places an upper bound of 6 on the integrality gap of relaxation (25.2). The following example places a lower bound of essentially 2. The graph is a star with n + 1 vertices and unit cost edges. F consists of all n + 1 vertices, C consists of all but the center vertex and k = n − 2. An optimal integral solution is to open facilities at n − 2 vertices of C and has a cost of 2. Consider the following fractional solution. Open a facility to the extent of 1/(n − 1) on the center vertex and (n − 2)/(n − 1) on each vertex of C. This has a cost of n/(n − 1), giving a ratio of 2(n − 1)/n.

t ❅ ❅ n−2 n−1

... ❅

t

t

❅

t ❅

1 n−1

t

25.4 A Lagrangian relaxation technique for approximation algorithms

In this section we will abstract away the ideas developed above so they may be more widely applicable. First, let us recall the fundamental technique of Lagrangian relaxation from combinatorial optimization. This technique consists of relaxing a constraint by moving it into the objective function, together with an associated Lagrange multiplier. Let us apply this relaxation to the constraint, in the k-median IP (25.1), that at most k facilities be opened. Let λ be the Lagrangian multiplier. minimize i∈F, j∈C

cij xij + λ i∈F yi − k j∈C

(25.4)

subject to i∈F xij ≥ 1,

25.5

Exercises

251

yi − xij ≥ 0, xij ∈ {0, 1}, yi ∈ {0, 1},

i ∈ F, j ∈ C i ∈ F, j ∈ C i∈F

This is precisely the facility location IP, with the restriction that the cost of each facility is the same, i.e., λ. It contains an additional constant term of −λk in the objective function. We may assume w.l.o.g. that an optimal solution, (x, y), to IP (25.1) opens exactly k facilities. Now, (x, y) is a feasible solution to IP (25.4) as well, with the same objective function value. Hence, for each value of λ, IP (25.4) is a lower bound on IP (25.1). We have shown that a Lagrangian relaxation of the k-median problem is the facility location problem. In doing so, the global constraint that at most k facilities be opened has been replaced with a penalty for opening facilities, the penalty being the Lagrangian multiplier. (See Exercise 25.4 for another application of this idea.) The next important observation was to notice that in the facility location approximation algorithm, Theorem 24.7, the duals pay one-for-one for the cost of opening facilities, i.e., with approximation factor 1. (See Exercise 22.9 for another such algorithm.) The remaining diﬃculty was ﬁnding a value of λ so that the facility location algorithm opened exactly k facilities. The fact that the facility location algorithm works with the linear relaxation of the problem helped. The convex combination of two (integer) solutions was a feasible (fractional) solution. The last step was rounding this (special) fractional solution into an integral one. For the k-median problem we used randomized rounding (see Exercise 25.4 for a diﬀerent rounding procedure).

25.5 Exercises

25.1 (Lin and Vitter [188]) Consider the general k-median problem in which the connection costs are not required to satisfy the triangle inequality. Give a reduction from the set cover problem to show that approximating this problem is as hard as approximating set cover, and therefore cannot be done ˜ with a factor better than O(log n) unless NP ⊆ P. 25.2 Obtain the dual of LP-relaxation to (25.4). (The constant term in the objective function will simply carry over.) How does it relate with the dual of the k-median LP? 25.3 Use the Lagrangian relaxation technique to give a constant factor approximation algorithm for the following common generalization of the facility location and k-median problems. Consider the uncapacitated facility location

252

25

k-Median

problem with the additional constraint that at most k facilities can be opened. This is a common generalization of the two problems solved in this paper: if k is made nf , we get the ﬁrst problem, and if the facility costs are set to zero, we get the second problem. 25.4 (Garg [94] and Chudak, Roughgarden, and Williamson [47]) Consider the following variant of the metric Steiner tree problem. Problem 25.6 (Metric k-MST) We are given a complete undirected graph G = (V, E), a special vertex r ∈ V , a positive integer k, and a function cost : E → Q+ satisfying the triangle inequality. The problem is to ﬁnd a minimum cost tree containing exactly k vertices, including r. We will develop a factor 5 algorithm for this problem. 1. Observe that a Lagrangian relaxation of this problem is the prizecollecting Steiner tree problem, Problem 22.12, stated in Exercise 22.9. 2. Observe that the approximation algorithm for the latter problem, given in Exercise 22.9, pays for the penalties one-for-one with the dual, i.e., with an approximation factor of 1. 3. Use the prize-collecting algorithm as a subroutine to obtain two trees, T1 and T2 , for very close values of the penalty, containing k1 and k2 vertices, with k1 < k < k2 . Obtain a convex combination of these solutions, with multipliers α1 and α2 . 4. We may assume that every vertex in G is at a distance of ≤ OPT from r. (Use the idea behind parametric pruning, introduced in Chapter 5. The parameter t is the length of the longest edge used by the optimal solution, which is clearly a lower bound on OPT. For each value of t, instance G(t) is obtained by restricting G to vertices that are within a distance of t of r. The algorithm is run on each graph of this family, and the best tree is output.) Consider the following procedure for rounding the convex combination. If α2 ≥ 1/2, then cost(T2 ) ≤ 4 · OPT; remove k2 − k vertices from T2 . Otherwise, double every edge of T2 , ﬁnd an Euler tour, and shortcut the tour to a cycle containing only those vertices that are in T2 and not in T1 (i.e., at most k2 − k1 vertices). Pick the cheapest path of length k − k1 − 1 from this cycle, and connect it by means of an edge to vertex r in T1 . The resulting tree has exactly k vertices. Show that the cost of this tree is ≤ 5 · OPT. Hint: Use the fact that α2 = (k − k1 )/(k2 − k1 ). 25.5 Let us apply the Lagrangian relaxation technique to the following linear program. minimize subject to cT x Ax = b (25.5)

25.5

Exercises

253

Then the lower bound is given by max min cT x − y T (Ax − b) = max y y x min (cT − y T A)x + y T b x

If y does not satisfy AT y = c, then by a suitable choice of x, the lower bound given by this expression can be made as small as desired and therefore meaningless. Meaningful lower bounds arise only if we insist that AT y = c. But then we get the following LP: maximize subject to yT b AT y = c (25.6)

Notice that this is the dual of LP (25.5)! Hence, the Lagrangian relaxation of a linear program is simply its dual and is therefore tight. Obtain the Lagrangian relaxation of the following LP: minimize subject to cT x Ax ≥ b x≥0 25.6 (Jain and Vazirani [141]) Consider the l2 clustering problem. Given a 2 set of n points S = {v1 , . . . , vn } in Rd and a positive integer k, the problem is to ﬁnd a minimum cost k-clustering, i.e., to ﬁnd k points, called centers, f1 , . . . , fk ∈ Rd , so as to minimize the sum of squares of distances from each point vi to its closest center. This naturally deﬁnes a partitioning of the n points into k clusters. Give a constant factor approximation algorithm for this problem. Hint: First show that restricting the centers to be a subset S increases the cost of the optimal solution by a factor of at most 2. Apply the solution of Exercise 24.6 to this modiﬁed problem. 25.7 (Korupolu, Plaxton, and Rajaraman [176] and Arya et al. [15]) For a set S of k facilities, deﬁne cost(S) to be the total cost of connecting each city to its closest facility in S. Deﬁne a swap to be the process of replacing one facility in S by a facility from S. A natural algorithm for metric k-median, based on local search, is: Start with an arbitrary set S of k facilities. In each iteration, check if there is a swap that leads to a lower cost solution. If so, execute any such swap and go to the next iteration. If not, halt. The terminating solution is said to be locally optimal. Let G = {o1 , . . . , ok } be an optimal solution and L = {s1 , . . . , sk } be a locally optimal solution. This exercise develops a proof showing cost(L) ≤ 5 · cost(G), as well as a constant factor approximation algorithm. (25.7)

254

25

k-Median

1. For o ∈ G, let NG (o) denote the set of cities connected to facility o in the optimal solution. Similarly, for s ∈ L, let NL (s) denote the set of cities connected to facility s in the locally optimal solution. Say that s ∈ L captures o ∈ G if |NG (o) ∩ NL (s)| > |NG (o)|/2. Clearly, each o ∈ G is captured by at most one facility in L. In this part let us make the simplifying assumption that each facility s ∈ L captures a unique facility in G. Assume that the facilities are numbered so that si captures oi , for 1 ≤ i ≤ k. Use the fact that for 1 ≤ i ≤ k, cost(L + oi − si ) ≥ cost(L) to show that cost(L) ≤ 3 · cost(G). Hint: cost(L + oi − si ) is bounded by the cost of the following solution: The cities in NL (si ) ∪ NG (oi ) are connected as in the locally optimal solution. Those in NG (oi ) are connected to facility oi . Cities in NL (si ) − NG (oi ) are connected to facilities in L − si using “3 hops” in such a way that each connecting edge of G and each connecting edge of L is used at most once in the union of all these hops. 2. Show that without the simplifying assumption of the previous part, cost(L) ≤ 5 · cost(G). Hint: Consider k appropriately chosen swaps so that each facility o ∈ G is swapped in exactly once and each facility s ∈ L is swapped out at most twice. 3. Strengthen the condition for swapping so as to obtain, for any ε > 0 a factor 5 + ε algorithm running in time polynomial in 1/ε and the size of the instance.

25.6 Notes

The ﬁrst approximation algorithm, achieving a factor of O(log n log log n), was given by Bartal [21]. The ﬁrst constant factor approximation algorithm for the k-median problem, achieving a guarantee of 6 2 , was given by Charikar, 3 Guha, Tardos, and Shmoys [39], using ideas from Lin and Vitter [189]. This algorithm used LP-rounding. The results of this chapter are due to Jain and Vazirani [141]. The current best factor is 3 + 2/p, with a running time of O(np ), due to Arya et al. [15]. This is a local search algorithm that swaps p facilities at a time (see Exercise 25.7 for the algorithm for p = 1). The example of Section 25.3.4 is due to Jain, Mahdian, and Saberi [138]. The best upper bound on the integrality gap of relaxation (25.2) is 4, due to Charikar and Guha [38]. For a factor 2 approximation algorithm for the l2 clustering problem (Exercise 25.6), see Drineas, Kannan, Frieze, Vempala, 2 and Vinay [62].

26 Semideﬁnite Programming

In the previous chapters of Part II of this book we have shown how linear programs provide a systematic way of placing a good upper bound on OPT (assuming a minimization problem), for numerous NP-hard problems. As stated earlier, this is a key step in the design of an approximation algorithm for an NP-hard problem. It is natural, then, to ask if there are other widely applicable ways of doing this. In this chapter we provide another class of relaxations, called vector programs. These serve as relaxations for several NP-hard problems, in particular, for problems that can be expressed as strict quadratic programs (see Section 26.1 for a deﬁnition). Vector programs are equivalent to a powerful and well-studied generalization of linear programs, called semideﬁnite programs. Semideﬁnite programs, and consequently vector programs, can be solved within an additive error of ε, for any ε > 0, in time polynomial in n and log(1/ε), using the ellipsoid algorithm (see Section 26.3). We will illustrate the use of vector programs by deriving a 0.87856 factor algorithm for the following problem (see Exercises 2.1 and 16.6 for a factor 1/2 algorithm). Problem 26.1 (Maximum cut (MAX-CUT)) Given an undirected graph G = (V, E), with edge weights w : E → Q+ , ﬁnd a partition (S, S) of V so as to maximize the total weight of edges in this cut, i.e., edges that have one endpoint in S and one endpoint in S.

26.1 Strict quadratic programs and vector programs

A quadratic program is the problem of optimizing (minimizing or maximizing) a quadratic function of integer valued variables, subject to quadratic constraints on these variables. If each monomial in the objective function, as well as in each of the constraints, is of degree 0 (i.e., is a constant) or 2, then we will say that this is a strict quadratic program. Let us give a strict quadratic program for MAX-CUT. Let yi be an indicator variable for vertex vi which will be constrained to be either +1 or −1. The partition (S, S) will be deﬁned as follows. S = {vi | yi = 1} and S = {vi | yi = −1}. If vi and vj are on opposite sides of this partition,

256

26

Semideﬁnite Programming

then yi yj = −1, and edge (vi , vj ) contributes wij to the objective function. On the other hand, if they are on the same side, then yi yj = 1, and edge (vi , vj ) makes no contribution. Hence, an optimal solution to this program is a maximum cut in G. maximize subject to 1 2 wij (1 − yi yj )

1≤i<j≤n

(26.1) vi ∈ V vi ∈ V

2 yi = 1,

yi ∈ Z,

We will relax this program to a vector program. A vector program is deﬁned over n vector variables in Rn , say v 1 , . . . , v n , and is the problem of optimizing (minimizing or maximizing) a linear function of the inner products v i · v j , 1 ≤ i ≤ j ≤ n, subject to linear constraints on these inner products. Thus, a vector program can be thought of as being obtained from a linear program by replacing each variable with an inner product of a pair of these vectors. A strict quadratic program over n integer variables deﬁnes a vector program over n vector variables in Rn as follows. Establish a correspondence between the n integer variables and the n vector variables, and replace each degree 2 term with the corresponding inner product. For instance, the term yi yj in (26.1) is replaced with v i · v j . In this manner, we obtain the following vector program for MAX-CUT. maximize subject to 1 2 wij (1 − v i · v j )

1≤i<j≤n

(26.2) vi ∈ V vi ∈ V

v i · v i = 1, vi ∈ R , n Because of the constraint v i ·v i = 1, the vectors v 1 , . . . , v n are constrained to lie on the n-dimensional sphere, Sn−1 . Any feasible solution to (26.1) yields a solution to (26.2) having the same objective function value, by assigning the vector (yi , 0, . . . , 0) to v i . (Notice that under this assignment, v i · v j is simply yi yj .) Therefore, the vector program (26.2) is a relaxation of the strict quadratic program (26.1). Clearly, this holds in general as well; the vector program corresponding to a strict quadratic program is a relaxation of the quadratic program. Interestingly enough, vector programs are approximable to any desired degree of accuracy in polynomial time, and thus relaxation (26.2) provides an upper bound on OPT for MAX-CUT. To show this, we need to recall some interesting and powerful properties of positive semideﬁnite matrices.

26.2

Properties of positive semideﬁnite matrices

257

Remark 26.2 Vector programs do not always come about as relaxations of strict quadratic programs. Exercise 26.13 gives an NP-hard problem that has vector program relaxation; however, we do not know of a strict quadratic program for it.

26.2 Properties of positive semideﬁnite matrices

Let A be a real, symmetric n×n matrix. Then A has real eigenvalues and has n linearly independent eigenvectors (even if the eigenvalues are not distinct). We will say that A is positive semideﬁnite if ∀x ∈ Rn , xT Ax ≥ 0. We will use the following two equivalent conditions crucially. We provide a proof sketch for completeness. Theorem 26.3 Let A be a real symmetric n × n matrix. Then, the following are equivalent: 1. ∀x ∈ Rn , xT Ax ≥ 0. 2. All eigenvalues of A are nonnegative. 3. There is an n × n real matrix W , such that A = W T W . Proof: (1 ⇒ 2): Let λ be an eigenvalue of A, and let v be a corresponding eigenvector. Therefore, Av = λv. Pre-multiplying by v T we get v T Av = λv T v. Now, by (1), v T Av ≥ 0. Therefore, λv T v ≥ 0. Since v T v > 0, λ ≥ 0. (2 ⇒ 3): Let λ1 , . . . , λn be the n eigenvalues of A, and v 1 , . . . , v n be the corresponding complete collection of orthonormal eigenvectors. Let Q be the matrix whose columns are v 1 , . . . , v n , and Λ be the diagonal matrix with entries λ1 , . . . , λn . Since for each i, Av i = λi v i , we have AQ = QΛ. Since Q is orthogonal, i.e., QQT = I, we get that QT = Q−1 . Therefore, A = QΛQT . Let D be the diagonal matrix whose diagonal entries are the positive square roots of λ1 , . . . , λn (by (2), λ1 , . . . , λn are nonnegative, and thus their square roots are real). Then, Λ = DD T . Substituting, we get A = QDD T QT = (QD)(QD)T . Now, (3) follows by letting W = (QD)T . (3 ⇒ 1): For any x ∈ Rn , xT Ax = xT W T W x = (W x)T (W x) ≥ 0. ✷

258

26

Semideﬁnite Programming

Using Cholesky decomposition (see Section 26.7), a real symmetric matrix can be decomposed, in polynomial time, as A = U ΛU T , where Λ is a diagonal matrix whose diagonal entries are the eigenvalues of A. Now A is positive semideﬁnite iﬀ all the entries of Λ are nonnegative, thus giving a polynomial time test for positive semideﬁniteness. The decomposition W W T is not polynomial time computable because in general it may contain irrational entries. However, it can be approximated to any desired degree by approximating the square roots of the entries of Λ. In the rest of this chapter we will assume that we have an exact decomposition, since the inaccuracy resulting from an approximate decomposition can be absorbed into the approximation factor (see Exercise 26.6). It is easy to see that the sum of two n × n positive semideﬁnite matrices is also positive semideﬁnite (e.g., using characterization (1) of Theorem 26.3). This is also true of any convex combination of such matrices.

26.3 The semideﬁnite programming problem

Let Y be an n × n matrix of real valued variables whose (i, j)th entry is yij . The problem of maximizing a linear function of the yij ’s, subject to linear constraints on them, and the additional constraint that Y be symmetric and positive semideﬁnite, is called the semideﬁnite programming problem. Let us introduce some notation to state this formally. Denote by Rn×n the space of n × n real matrices. Recall that the trace of a matrix A ∈ Rn×n is the sum of its diagonal entries and is denoted by tr(A). The Frobenius inner product of matrices A, B ∈ Rn×n , denoted A • B, is deﬁned to be A • B = tr(AT B) = n n

aij bij , i=1 j=1

where aij and bij are the (i, j)th entries of A and B, respectively. Let Mn denote the cone of symmetric n×n real matrices. For A ∈ Mn , A 0 denotes the fact that matrix A is positive semideﬁnite. Let C, D 1 , . . . , D k ∈ Mn and d1 , . . . dk ∈ R. Following is a statement of the general semideﬁnite programming problem. Let us denote it by S. maximize subject to C •Y D i • Y = di , Y 0, Y ∈ Mn . Observe that if C, D 1 , . . . , D k are all diagonal matrices, this is simply a linear programming problem. As in the case of linear programs, it is easy to 1≤i≤k (26.3)

26.3

The semideﬁnite programming problem

259

see that allowing linear inequalities, in addition to equalities, does not make the problem more general. Let us call a matrix in Rn×n satisfying all the constraints of S a feasible solution. Since a convex combination of positive semideﬁnite matrices is positive semideﬁnite, it is easy to see that the set of feasible solutions is convex, i.e., if A ∈ Rn×n and B ∈ Rn×n are feasible solutions then so is any convex combination of these solutions. Let A ∈ Rn×n be an infeasible point. Let C ∈ Rn×n . A hyperplane C • Y ≤ b is called a separating hyperplane for A if all feasible points satisfy it and point A does not satisfy it. In the next theorem we show how to ﬁnd a separating hyperplane in polynomial time. As a consequence, for any ε > 0, semideﬁnite programs can be solved within an additive error of ε, in time polynomial in n and log(1/ε), using the ellipsoid algorithm (see Section 26.7 for more eﬃcient methods). Theorem 26.4 Let S be a semideﬁnite programming problem, and A be a point in Rn×n . We can determine, in polynomial time, whether A is feasible for S and, if it is not, ﬁnd a separating hyperplane. Proof: Testing for feasibility involves ensuring that A is symmetric and positive semideﬁnite and that it satisﬁes all the linear constraints. By remarks made in Section 26.2, this can be done in polynomial time. If A is infeasible, a separating hyperplane is obtained as follows. • If A is not symmetric, aij > aji for some i, j. Then yij ≤ yji is a separating hyperplane. • If A is not positive semideﬁnite, then it has a negative eigenvalue, say λ. Let v be the corresponding eigenvector. Now (vv T ) • Y = v T Y v ≥ 0 is a separating hyperplane. • If any of the linear constraints is violated, it directly yields a separating hyperplane. ✷ Next, let us show that vector programs are equivalent to semideﬁnite programs, thereby showing that the former can be solved eﬃciently to any desired degree of accuracy. Let V be a vector program on n n-dimensional vector variables v 1 , . . . , v n . Deﬁne the corresponding semideﬁnite program, S, over n2 variables yij , 1 ≤ i, j ≤ n, as follows. Replace each inner product v i ·v j occurring in V by the variable yij . The objective function and constraints are now linear in the yij ’s. Additionally, require that matrix Y , whose (i, j)th entry is yij , be symmetric and positive semideﬁnite. Lemma 26.5 Vector program V is equivalent to semideﬁnite program S. Proof: We will show that corresponding to each feasible solution to V, there is a feasible solution to S of the same objective function value, and vice

260

26

Semideﬁnite Programming

versa. Let a1 , . . . , an be a feasible solution to V. Let W be the matrix whose columns are a1 , . . . , an . Then, it is easy to see that A = W T W is a feasible solution to S having the same objective function value. For the other direction, let A be a feasible solution to S. By Theorem 26.3, there is an n × n matrix W such that A = W T W . Let a1 , . . . , an be the columns of W . Then, it is easy to see that a1 , . . . , an is a feasible solution to V having the same objective function value. ✷ Finally, we give the semideﬁnite programming relaxation to MAX-CUT that is equivalent to vector program 26.2. maximize subject to 1 2 wij (1 − yi yj )

1≤i<j≤n

(26.4) vi ∈ V

2 yi = 1,

Y

0,

Y ∈ Mn .

26.4 Randomized rounding algorithm

We now present the algorithm for MAX-CUT. For convenience, let us assume that we have an optimal solution to the vector program (26.2). The slight inaccuracy in solving it can be absorbed into the approximation factor (see Exercise 26.6). Let a1 , . . . , an be an optimal solution, and let OPTv denote its objective function value. These vectors lie on the n-dimensional unit sphere Sn−1 . We need to obtain a cut (S, S) whose weight is a large fraction of OPTv . Let θij denote the angle between vectors ai and aj . The contribution of this pair of vectors to OPTv is wij (1 − cos θij ). 2 Clearly, the closer θij is to π, the larger this contribution will be. In turn, we would like vertices vi and vj to be separated if θij is large. The following method accomplishes precisely this. Pick r to be a uniformly distributed vector on the unit sphere Sn−1 , and let S = {vi | ai · r ≥ 0}. Lemma 26.6 Pr[vi and vj are separated ] = θij . π

Proof: Project r onto the plane containing v i and v j . Now, vertices vi and vj will be separated iﬀ the projection lies in one of the two arcs of angle θij shown below.

26.4

Randomized rounding algorithm

261

vi vj θ ij θ ij

Since r has been picked from a spherically symmetric distribution, its projection will be a random direction on this plane. The lemma follows. ✷ The next lemma shows how to generate vectors that are uniformly distributed on the unit sphere Sn−1 . Lemma 26.7 Let x1 , . . . , xn be picked independently from the normal distribution with mean 0 and unit standard deviation. Let d = (x2 + . . . + x2 )1/2 . n 1 Then, (x1 /d, . . . , xn /d) is a random vector on the unit sphere Sn−1 . Proof: Consider the vector r = (x1 , . . . , xn ). The distribution function for r has density f (y1 , . . . , yn ) =

1 2 1 1 √ e−yi /2 = e− 2 n/2 (2π) 2π i=1

n

i

2 yi

.

Notice that the density function depends only on the distance of the point from the origin. Therefore, the distribution of r is spherically symmetric. Hence, dividing by the length of r, i.e., d, we get a random vector on Sn−1 . ✷ The algorithm is summarized below. Algorithm 26.8 (MAX-CUT) 1. Solve vector program (26.2). Let a1 , . . . , an be an optimal solution. 2. Pick r to be a uniformly distributed vector on the unit sphere Sn−1 . 3. Let S = {vi | ai · r ≥ 0}. Let W be the random variable denoting the weight of edges in the cut picked by Algorithm 26.8, and let α= θ 2 min . π 0≤θ≤π 1 − cos θ

262

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Semideﬁnite Programming

One can show that α > 0.87856 (see Exercise 26.3). Lemma 26.9 E[W ] ≥ α · OPTv . Proof: By the deﬁnition of α we have that for any θ, 0 ≤ θ ≤ π, θ ≥α π 1 − cos θ 2 . (26.5)

Using this and Lemma 26.6, we get E[W ] =

1≤i<j≤n

wij Pr[v i and v j are separated] θij ≥α· π 1 wij (1 − cos θij ) = α · OPTv . 2 ✷

=

1≤i<j≤n

wij

1≤i<j≤n

Let us deﬁne the integrality gap for relaxation (26.2) to be inf

I

OPT(I) , OPTv (I)

where the inﬁmum is over all instances I of MAX-CUT. Corollary 26.10 The integrality gap for relaxation (26.2) is at least α > 0.87856. Theorem 26.11 There is a randomized approximation algorithm for MAXCUT achieving an approximation factor of 0.87856. Proof: Let us ﬁrst obtain a “high probability” statement using the bound on expectation established in Lemma 26.9. Let T denote the sum of weights of all edges in G, and deﬁne a so that E[W ] = aT . Let p = Pr[W < (1 − ε)aT ], where ε > 0 is a constant. Since the random variable W is always bounded by T , we get aT ≤ p(1 − ε)aT + (1 − p)T. Therefore, p≤ 1−a . 1 − a + aε

26.5

Improving the guarantee for MAX-2SAT

263

Now, T ≥ E[W ] = aT ≥ α · OPTv ≥ α · OPT ≥ αT , 2

where the last inequality follows from the fact that OPT ≥ T /2 (see Exercise 2.1). Therefore, 1 ≥ a ≥ α/2. Using this upper and lower bound on a, we get p≤1− where c= εα/2 . 1 + ε − α/2 εα/2 ≤ 1 − c, 1 + ε − α/2

Run Algorithm 26.8 1/c times, and output the heaviest cut found in these runs. Let W be the weight of this cut. Then, 1 Pr[W ≥ (1 − ε)aT ] ≥ 1 − (1 − c)1/c ≥ 1 − . e Since aT ≥ α · OPT > 0.87856 OPT, we can pick a value of ε > 0 so that (1 − ε)aT ≥ 0.87856 OPT. ✷ Example 26.12 The following example shows that the bound on the integrality gap of relaxation (26.2) given in Corollary 26.10 is almost tight. Consider a graph which is a 5-cycle v1 , v2 , v3 , v4 , v5 , v1 . Then, an optimal solution to relaxation (26.2) is to place the ﬁve vectors in a 2-dimensional subspace within which they are given by v i = (cos( 4iπ ), sin( 4iπ )), for 1 ≤ i ≤ 5 (see 5 5 √ Exercise 26.5). The cost of this solution is OPTv = 5 (1 + cos π ) = 25+5 5 . 2 5 8 Since OPT = 4 for this graph, the integrality gap for this example is 32√ = 0.88445.... ✷ 25+5 5

26.5 Improving the guarantee for MAX-2SAT

MAX-2SAT is the restriction of MAX-SAT (Problem 16.1) to formulae in which each clause contains at most two literals. In Chapter 16 we obtained a factor 3/4 algorithm for this problem using randomization, followed by the method of conditional expectation. We will give an improved algorithm using semideﬁnite programming. The key new idea needed is a way of converting the obvious quadratic program (see Exercise 26.8) for this problem into a strict quadratic program. We will accomplish this as follows. Corresponding to each Boolean variable

264

26

Semideﬁnite Programming

xi , introduce variable yi which is constrained to be either +1 or −1, for 1 ≤ i ≤ n. In addition, introduce another variable, say y0 , which is also constrained to be +1 or −1. Let us impose the convention that Boolean variable xi is true if yi = y0 and false otherwise. Under this convention we can write the value of a clause in terms of the yi ’s, where the value, v(C), of clause C is deﬁned to be 1 if C is satisﬁed and 0 otherwise. Thus, for clauses containing only one literal, v(xi ) = 1 + y0 yi 1 − y0 yi and v(xi ) = . 2 2

Consider a clause containing 2 literals, e.g., (xi ∨ xj ). Its value is v(xi ∨ xj ) = 1 − v(xi )v(xj ) = 1 − = 1 − y 0 yi 1 − y0 yj 2 2

1 2 3 + y0 yi + y0 yj − y0 yi yj 4 1 + y0 yj 1 − yi yj 1 + y0 y i + + . = 4 4 4

2 Observe that in this derivation we have used the fact that y0 = 1. In all the remaining cases as well, it is easy to check that the value of a 2 literal clause consists of a linear combination of terms of the form (1 + yi yj ) or (1 − yi yj ). Therefore, a MAX-2SAT instance can be written as the following strict quadratic program, where the aij ’s and bij ’s are computed by collecting terms appropriately.

maximize

0≤i<j≤n

aij (1 + yi yj ) + bij (1 − yi yj )

2 yi = 1,

(26.6) 0≤i≤n 0≤i≤n

subject to

yi ∈ Z,

Following is the vector program relaxation for (26.6), where vector variable v i corresponds to yi . maximize

0≤i<j≤n

aij (1 + v i · v j ) + bij (1 − v i · v j ) v i · v i = 1, vi ∈ R n+1 (26.7) 0≤i≤n 0≤i≤n

subject to

,

The algorithm is similar to that for MAX-CUT. We solve vector program (26.7). Let a0 , . . . , an be an optimal solution. Pick a vector r uniformly distributed on the unit sphere in (n + 1) dimensions, Sn , and let yi = 1 iﬀ

26.6

Exercises

265

r · ai ≥ 0, for 0 ≤ i ≤ n. This gives a truth assignment for the Boolean variables. Let W be the random variable denoting the weight of this truth assignment. Lemma 26.13 E[W ] ≥ α · OPTv . Proof: E[W ] = 2

0≤i<j≤n

aij Pr[yi = yj ] + bij Pr[yi = yj ].

Let θij denote the angle between ai and aj . By inequality (26.5), Pr[yi = yj ] = By Exercise 26.4, Pr[yi = yj ] = 1 − Therefore, E[W ] ≥ α ·

0≤i<j≤n

θij α ≥ (1 − cos θij ). π 2

α θij ≥ (1 + cos θij ). π 2

aij (1 + cos θij ) + bij (1 − cos θij ) = α · OPTv . ✷

26.6 Exercises

26.1 Is matrix W in Theorem 26.3 unique (up to multiplication by −1)? Hint: Consider the matrix QDQT . 26.2 Let B be obtained from matrix A by throwing away a set of columns and the corresponding set of rows. We will say that B is a principal submatrix of A. Show that the following is another equivalent condition for a real symmetric matrix to be positive semideﬁnite: that all of its principal submatrices have nonnegative determinants. (See Theorem 26.3 for other conditions.) 26.3 Show, using elementary calculus, that α > 0.87856. 26.4 Show that for any φ, 0 ≤ φ ≤ π, 1− φ α ≥ (1 + cos φ). π 2

266

26

Semideﬁnite Programming

Hint: Substitute θ = π − φ in inequality (26.5). 26.5 Show that for a 5-cycle, the solution given in Example 26.12 is indeed an optimal solution to the vector program relaxation for MAX-CUT. 26.6 Show that the inaccuracies resulting from the fact we do not have an optimal solution to the vector program (26.2) and that matrix A is not exactly decomposed as W W T (see end of Section 26.2) can be absorbed into the approximation factor for MAX-CUT. Hint: Use the idea behind the proof of Theorem 26.11 and the fact that the solution to program (26.2) lies in the range [T /2, T ], where T is the sum of weights of all edges in G. 26.7 Theorem 26.11 shows how to obtain a “high probability” statement from Lemma 26.9. Obtain a similar statement for MAX-2SAT, using Lemma 26.13, thereby obtaining a 0.87856 factor algorithm for MAX-2SAT. 26.8 Give a quadratic program for MAX-2SAT. 26.9 (Linial, London, and Rabinovich [190]) Let G be the complete undirected graph on n vertices, V , and let w be a function assigning nonnegative weights to the edges of G. The object is to ﬁnd an optimal distortion 2 2 embedding of the vertices of G. Let vertex i be mapped to v i ∈ Rn by such an embedding. The embedding should satisfy: 1. no edge is overstretched, i.e., for 1 ≤ i < j ≤ n, ||v i − v j ||2 ≤ w( ij), and 2. the maximum shrinkage is minimized, i.e., maximize

(i,j):wij =0

min

(||v i − v j ||2 /wij ).

Give a vector program for ﬁnding such an optimal embedding and give the equivalent semideﬁnite program. Hint: The vector program is: minimize subject to c v i · v i + v j · v j − 2v i · v j ≤ wij , v i · v i + v j · v j − 2v i · v j ≥ cwij , vi ∈ R , n (26.8) 1≤i<j≤n 1≤i<j≤n 1≤i≤n

26.10 (Knuth [174]) Give an eﬃcient algorithm for sampling from the normal distribution with mean 0 and unit standard deviation, given a source of unbiased random bits.

26.6

Exercises

267

26.11 Give a strict quadratic program for the MAX k-CUT and maximum directed cut problems, Problems 2.14 and 2.15 stated in Exercises 2.3 and 2.4. Give a vector program relaxation and an equivalent semideﬁnite program as well. 26.12 (Goemans and Williamson [106]) Consider MAX-CUT with the additional constraint that speciﬁed pairs of vertices be on the same/opposite sides of the cut. Formally, we are speciﬁed two sets of pairs of vertices, S1 and S2 . The pairs in S1 need to be separated, and those in S2 need to be on the same side of the cut sought. Under these constraints, the problem is to ﬁnd a maximum weight cut. Assume that the constraints provided by S1 and S2 are not inconsistent. Give a strict quadratic program and vector program relaxation for this problem. Show how Algorithm 26.8 can be adapted to this problem so as to maintain the same approximation factor. 26.13 (Karger, Motwani, and Sudan [158]) Let G = (V, E) be an undirected graph. Consider a vector program with n n-dimensional vectors corresponding to the vertices of G, and constraints that the vectors lie on the unit sphere, Sn−1 , and that for each edge (i, j) ∈ G, vi · vj ≤ − 1 . k−1

Show that this vector program is a relaxation of the k-coloring problem, i.e., if G is k-colorable, then this vector program has a feasible solution. Hint: Consider the following k vectors in Rn . Each vector has 0 in the last n − k positions. Vector i has − the remaining positions. k−1 k

in the ith position and 1/

k(k − 1) in

26.14 (Chor and Sudan [43]) Consider the following problem: Problem 26.14 (Betweenness) We are given a set S = {x1 , x2 , . . . , xn } of n items and a set T of m triplets T ⊆ S × S × S. Each triplet consists of three distinct items. A total ordering (permutation) of S, xπ1 < xπ2 < . . . < xπn satisﬁes a triplet (xi , xj , xk ) ∈ T if xj occurs between xi and xk in the ordering, i.e., if either xi < xj < xk holds or xk < xj < xi holds. The problem is to ﬁnd a total ordering that maximizes the number of satisﬁed triplets. 1. Show that a random ordering (i.e., a permutation chosen uniformly at random among all possible permutations) will satisfy in expectation one third of all triplets in T . 2. Use the method of conditional expectation to derandomize the above algorithm, thereby obtaining a factor 1/3 approximation algorithm. What upper bound on OPT is this algorithm using? Give an example showing that with this upper bound a better algorithm is not possible.

268

26

Semideﬁnite Programming

3. The rest of the exercise develops an algorithm based on semideﬁnite programming. The ideas can be illustrated more simply by assuming that the instance is satisﬁable, i.e., that all m triplets can be satisﬁed simultaneously. Note that checking for this condition is NP-hard, so the restriction of the betweenness problem to such instances is not an NP-optimization problem (see Exercise 1.9). Show that an instance is satisﬁable iﬀ the following strict quadratic program in variables pi ∈ R, i = 1, . . . , n, has a solution: (pi − pj )2 ≥ 1 (pi − pj )(pk − pj ) ≤ 0 for all i, j, for all (xi , xj , xk ) ∈ T.

4. Obtain the vector programming relaxation of this strict quadratic program as well as the equivalent semideﬁnite program. 5. Give an instance where the above semideﬁnite program is satisﬁable but the instance itself is not satisﬁable. 6. Let us assume that n × n matrix Y is a feasible solution to the above semideﬁnite program, and let vi ∈ Rn for i = 1, · · · , n be vectors such that T Yij = vi vj . Now select r uniformly at random on the unit sphere Sn−1 . Consider the random ordering obtained by sorting rT vi . Show that, in expectation, this random ordering satisﬁes at least half of the constraints in T . Hint: What is the probability that a single triplet is satisﬁed? What is the angle between vi − vj and vk − vj ?

26.7 Notes

The results of this chapter are based on the seminal work of Goemans and Williamson [106] that introduced the use of semideﬁnite programs in approximation algorithms. Experimental results reported in their paper show that Algorithm 26.8 performs much better on typical instances than the worst case guarantee. Mahajan and Ramesh [200] give a derandomization of Algorithm 26.8, as well as the MAX-2SAT algorithm, using the method of conditional expectation. Karloﬀ [161] provides a family of tight examples for Algorithm 26.8, for which the expected weight of the cut produced is arbitrarily close to α · OPTv . Feige and Schechtman [84] strengthen this to showing that there are graphs such that even the best hyperplane (rather than a random one, as prescribed in Algorithm 26.8) gives a cut of weight only α · OPTv . They also show that the integrality gap of the semideﬁnite relaxation (26.2) for MAX-CUT is α. For eﬃcient algorithms, using interior point methods, for approximating semideﬁnite programs, see Alizadeh [4], Nesterov and Nemirovskii [214] and Overton [215]. For a duality theory for semideﬁnite programs, see Wolkowitz [260] and Vandeberghe and Boyd [250].

26.7

Notes

269

Lov´sz and Schrijver [196] use semideﬁnite programming to provide an a automatic way of strengthening any convex relaxation (having a convex feasible region) of a 0/1 integer program. They also show that if the original relaxation can be optimized in polynomial time, then so can the strengthened relaxation (however, in order to guarantee polynomial running time, this process can be applied only a constant number of times). Feige and Goemans [81] improve the approximation factor for MAX2SAT to 0.931. They also give a 0.859 factor for the maximum directed cut problem (see Exercise 26.11). For semideﬁnite-programming-based algorithms for the MAX k-CUT problem see Frieze and Jerrum [90]. Karger, Motwani, and Sudan [158] use the relaxation in Exercise 26.13 to obtain an O(n1−3/(k+1) log1/2 n) coloring for k-colorable graphs.

Part III Other Topics

27 Shortest Vector

The shortest vector problem is a central computational problem in the classical area of geometry of numbers. The approximation algorithm presented below has many applications in computational number theory and cryptography. Two of its most prominent applications are the derivation of polynomial time algorithms for factoring polynomials over the rationals and for simultaneous diophantine approximation. Problem 27.1 (Shortest vector) Given n linearly independent vectors a1 , . . . , an ∈ Qn , ﬁnd the shortest vector, in Euclidean norm, in the lattice generated by these vectors. The lattice L generated by a1 , . . . , an is the set of all integer linear combinations of these vectors, i.e., L = {λ1 a1 + . . . + λn an | λi ∈ Z}. Remark 27.2 We will consider only full rank lattices, i.e., lattices that span the entire space on which they are deﬁned. Most of the results apply to the general case as well, although the details can get more involved. We will present an exponential (in n) factor algorithm for this problem. The problem of ﬁnding a polynomial factor algorithm for shortest vector has remained open for over two decades. It is worth pointing out that the exponential factor algorithm does perform well in practice and is widely used. The shortest vector in the 1-dimensional lattice generated by two integers is simply the greatest common divisor (gcd) of these integers, which is polynomial time computable by Euclid’s algorithm. When restricted to two dimensions, the shortest vector problem is polynomial time solvable. This follows from an algorithm of Gauss that was originally formulated in the language of quadratic forms. It will be instructive to ﬁrst study these two algorithms, since Gauss’ algorithm can be viewed as a generalization of Euclid’s algorithm, and the n-dimensional algorithm can be seen as a generalization of Gauss’ algorithm. Several peculiarities set the shortest vector problem apart from other problems studied in this book. Unlike other NP-hard problems, we do not know if there is an instance of shortest vector that has exponentially many solutions, i.e., a lattice that has exponentially many shortest vectors. Indeed, this problem is not known to be NP-hard in the usual sense; it is only known to be NP-hard under randomized reductions. Another point of diﬀerence is that the lower bounding scheme used for obtaining the exponential factor

274

27

Shortest Vector

algorithm can, in principle, be used to obtain a polynomial factor algorithm – by an existential argument we will prove that this scheme gives rise to a factor n approximate No certiﬁcate for the shortest vector problem. In contrast, for all other problems studied in this book, the best known lower bound is actually polynomial time computable.

27.1 Bases, determinants, and orthogonality defect

In this chapter all vectors will be assumed to be row vectors, unless otherwise stated. Let A denote the n × n matrix whose rows are the given vectors, a1 , . . . , an . Let vectors b1 , . . . , bn ∈ L, and let B denote the n × n matrix whose rows are b1 , . . . , bn . Since a1 , . . . , an can generate b1 , . . . , bn , B = ΛA, where Λ is an n × n integer matrix. Therefore det(B) is an integer multiple of det(A). We will say that b1 , . . . , bn form a basis for lattice L if the lattice generated by these vectors is precisely L. A square matrix with integer entries whose determinant is ±1 will be called unimodular. Observe that its inverse is also unimodular. Theorem 27.3 Let vectors b1 , . . . , bn ∈ L. The following conditions are equivalent: 1. b1 , . . . , bn form a basis for lattice L 2. |det(B)| = |det(A)| 3. there is an n × n unimodular matrix U such that B = U A Proof: Since a1 , . . . , an can generate b1 , . . . , bn , B = ΛA, where Λ is an n × n integer matrix. 1 ⇒ 2: If b1 , . . . , bn form a basis for L, they can generate a1 , . . . , an . Therefore, A = Λ B, where Λ is an n × n integer matrix. Hence, det(Λ)det(Λ ) = 1. Since both Λ and Λ have integer determinants, det(Λ) = ±1, and so |det(B)| = |det(A)|. 2 ⇒ 3: Since |det(B)| = |det(A)|, det(Λ) = ±1, and hence Λ is unimodular. 3 ⇒ 1: Since U is unimodular, U −1 is also unimodular. Now, A = U −1 B, and so each of a1 , . . . , an can be written as an integer linear combination of b1 , . . . , bn . Therefore, b1 , . . . , bn form a basis for L. ✷ By Theorem 27.3, the determinant of a basis for L is invariant, up to sign. We will call |det(A)| the determinant of lattice L and will denote it as det L. Observe that det L is the volume of the parallelohedron deﬁned by the basis vectors. Theorem 27.3 also tells us that we can move from basis to basis by applying unimodular transforms. We will use this fact in our algorithms. The most desirable basis to obtain is an orthogonal basis, since it must contain a shortest vector of L (see Exercise 27.1). However, not every lattice

27.1

Bases, determinants, and orthogonality defect

275

admits such a basis. For instance, the following 2-dimensional lattice has no orthogonal basis. (The two shaded parallelohedra have volume det L each.)

11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 111111111111 000000000000 11111111 00000000 111111111111 000000000000 11111111 00000000 111111111111 000000000000 11111111 00000000 111111111111 000000000000 11111111 00000000 111111111111 000000000000 11111111 00000000 111111111111 000000000000

We will denote the Euclidean norm of vector a by a . Recall Hadamard’s inequality, which states that for an n × n real matrix A, |det(A)| ≤ a1 . . . an . This inequality holds with equality iﬀ one of the rows of A is the zero vector or else the rows are mutually orthogonal. Applying this inequality we get that any basis b1 , . . . , bn satisﬁes det L ≤ b1 . . . bn . Since none of the basis vectors is the zero vector, the inequality holds with equality iﬀ the basis is orthogonal. Let us deﬁne the orthogonality defect of basis b1 , . . . , bn to be b1 . . . bn . det L Since det L is invariant, the smaller the orthogonality defect of a basis, the shorter its vectors must be. Let us say that linearly independent vectors b1 , . . . , bk ∈ L are primitive if they can be extended to a basis of L. For a single vector, primitivity is easy to characterize. Let us say that a ∈ L is shortest in its direction if xa is not a vector in L for 0 < |x| < 1. Theorem 27.4 Vector a ∈ L is primitive iﬀ a is shortest in its direction. Proof: Suppose there is a basis B of L containing a. Any vectors of the form xa, where x is a scalar, that can be generated using B must have x ∈ Z. Therefore, a is shortest in its direction.

276

27

Shortest Vector

Suppose a is shortest its direction. Since a ∈ L, we can write a = λ1 a1 + . . . + λn an , where λi ∈ Z, and a1 , . . . , an form a basis for L. Since a is shortest in its direction, gcd(λ1 , . . . , λn ) is 1. Therefore, there exists a unimodular n × n matrix, say Λ, whose ﬁrst row is λ1 , . . . , λn (see Exercise 27.2). Let B = ΛA. By Theorem 27.3, B is a basis for L. Since the ﬁrst row of B is a, the theorem follows. ✷

27.2 The algorithms of Euclid and Gauss

In one dimension the lattice generated by basis vector a is simply all integer multiples of a. Hence, the shortest vector problem is trivial in the 1-dimensional case. Instead, consider the following problem. Given integers a and b, consider all integer linear combinations of a and b, and ﬁnd the smallest number in this lattice. This will be the gcd of a and b, denoted by (a, b). Assume w.l.o.g. that a ≥ b ≥ 0. The idea behind Euclid’s algorithm is that since (a, b) = (a − b, b), we can replace the original problem by a smaller one. Continuing in this manner we are left with ﬁnding the smallest number, in absolute value, in the sequence {|a − mb|, for m ∈ Z}. Let this be c. If c = 0, then (a, b) = b, and we are done. Otherwise, (a, b) = (b, c), and we proceed with the pair b, c. Observe that c ≤ b/2. Hence this process terminates in log2 b iterations. Next, let us present Gauss’ algorithm in 2 dimensions. For this case, a weaker condition than orthogonality is suﬃcient for ensuring that a basis contains a shortest vector. Let θ denote the angle between basis vectors b1 and b2 , 0◦ < θ < 180◦ . Thus, det L = b1 · b2 sin θ. Number the vectors so that b1 ≤ b2 . Theorem 27.5 If 60◦ ≤ θ ≤ 120◦ , then b1 is a shortest vector in lattice L. Proof: Assume for the purpose of contradiction that there is a vector b ∈ L that is shorter than b1 . Since b1 and b2 are both primitive, b cannot be linearly dependent on b1 or b2 . By a simple case analysis it is easy to see that b must form an angle of at most 60◦ with one of b1 , b2 , −b1 , or −b2 .

27.2

The algorithms of Euclid and Gauss

277

b2

<120ο -b 1

>60ο b1

-b 2

Let D be the 2 × 2 matrix whose rows consist of this vector and b. Observe that |det(D)| is nonzero and is strictly smaller than det L = b1 · b2 sin θ. This contradicts the fact that det(D) is an integer multiple of det L. Hence b1 is a shortest vector in L. ✷ Deﬁne µ21 = b2 · b1 , b1 2

where · represents inner product. Observe that µ21 b1 is the component of b2 in the direction of b1 . The following proposition suggests an algorithm for ﬁnding a basis satisfying the condition of Theorem 27.5. Proposition 27.6 If basis (b1 , b2 ) satisﬁes • b1 ≤ b2 and • |µ21 | ≤ 1/2, then 60◦ ≤ θ ≤ 120◦ . Proof: Note that, cos θ = b1 · b2 µ21 b1 = . b1 · b2 b2 ✷

Therefore, by the two conditions, | cos θ| ≤ 1/2. Hence, 60◦ ≤ θ ≤ 120◦ .

278

27

Shortest Vector

Gauss gave the following algorithm for transforming an arbitrary basis into one satisfying the above conditions. Algorithm 27.7 (Shortest vector for 2 dimensions) 1. Initialization: assume b1 ≤ b2 . 2. While the conditions of Proposition 27.6 are not satisﬁed, do: (a) If |µ21 | > 1/2, let b2 ← b2 − mb1 , where m is the integer closest to µ21 . (b) If b1 > b2 , interchange b1 and b2 . 3. Output b1 . Observe that the operations involve interchanging the rows of B and subtracting a multiple of one row from another. These are clearly unimodular operations (the absolute value of the determinant remains unchanged). Therefore, we always have a basis for L. Clearly, after step 2(a) is performed, |µ21 | ≤ 1/2. Observe that the operation in this step is quite similar to that in Euclid’s gcd algorithm. It minimizes the component of b2 in the direction of b1 by subtracting a suitable multiple of b1 . Since b1 · b2 decreases after each iteration, the algorithm must terminate (there are only a bounded number of vectors of L within a given radius). For polynomial time termination of Algorithm 27.7, see Exercises 27.3, 27.4, and 27.5, and the comments in Section 27.7.

27.3 Lower bounding OPT using Gram–Schmidt orthogonalization

In this section we will present the Gram–Schmidt lower bound on OPT, the length of a shortest vector in lattice L. Intuitively, the Gram–Schmidt orthogonalization of basis b1 , . . . , bn gives the n “altitudes” of the parallelohedron deﬁned by this basis. Formally, it is a set of mutually orthogonal vectors b∗ , . . . , b∗ , where b∗ = b1 and b∗ is the 1 n 1 i component of bi orthogonal to b∗ , . . . , b∗ , for 2 ≤ i ≤ n. b∗ can be obtained 1 i−1 i from bi by subtracting from it components in the directions of b∗ , . . . , b∗ , 1 i−1 as given by the following recurrence: b∗ = b1 1 b∗ = bi − i i−1 j=1

bi · b∗ ∗ j b , i = 2, . . . , n. b∗ 2 j j

(27.1)

27.3

Lower bounding OPT using Gram–Schmidt orthogonalization

279

b3*

b3 b2* b2

b1=b1*

For 1 ≤ j < i ≤ n, deﬁne µij = bi · b∗ j , b∗ 2 j

and deﬁne µii = 1. Then i bi = j=1 µij b∗ , i = 1, . . . , n. j

(27.2)

For j ≤ i, deﬁne bi (j) to be the component of bi orthogonal to b1 , . . . , bj−1 , i.e., bi (j) = µij b∗ + µi,j+1 b∗ + · · · + b∗ . j j+1 i It is easy to see that det L = b∗ . . . b∗ . 1 n Therefore, the orthogonality defect of this basis can be written as b1 . . . bn 1 , ∗ ∗ = b1 . . . bn sin θ2 . . . sin θn where θi is the angle between bi and the vector space spanned by b1 , . . . , bi−1 , for 2 ≤ i ≤ n. This angle is deﬁned as follows. Let bi be the projection of bi onto the space spanned by b1 , . . . , bi−1 . Then

280

27

Shortest Vector

θi = cos−1

bi bi

.

Notice that the Gram–Schmidt orthogonalization depends not only on the basis chosen, but also on the ordering chosen among the basis vectors. It provides a lower bound on OPT as follows. Lemma 27.8 Let b1 , . . . , bn be a basis for lattice L, and b∗ , . . . , b∗ be the n 1 Gram–Schmidt orthogonalization for it. Then OPT ≥ min{ b∗ , . . . , b∗ }. n 1 Proof: Let v be a shortest vector in L. Suppose k is the largest indexed basis vector used in generating v, i.e., k v= i=1 λi bi ,

where λk = 0. Using (27.2), v can also be expressed as a linear combination of the vectors b∗ , . . . , b∗ . In this combination, the coeﬃcient of b∗ is λk , since 1 k k µk,k = 1. Since the vectors b∗ , . . . , b∗ are orthogonal, 1 n v Hence OPT = v ≥ b∗ ≥ min{ b∗ , . . . , b∗ }. n k 1 ✷

2

≥ λ2 b∗ k k

2

≥ b∗ k

2

.

27.4 Extension to n dimensions

In this section we will extend Gauss’ 2 dimensions algorithm to n dimensions. As usual, our main eﬀort will be on improving the lower bound on OPT. Recall that for any basis, b1 , . . . , bn , b∗ . . . b∗ = det L 1 n is an invariant, and min{ b∗ , . . . , b∗ } is a lower bound on OPT. In a 1 n typical basis, the minimum will be attained towards the end of the sequence b∗ , . . . , b∗ . We will try to make this sequence lexicographically small, 1 n thereby ensuring that the last entries, and hence the minimum, are not too

27.4

Extension to n dimensions

281

small. The conditions in Proposition 27.6 suggest a way of accomplishing this. By the ﬁrst condition we have b1

2

≤ b2 2 , i.e.,

b∗ 1

2

≤ µ2 b∗ 21 1

2

+ b∗ 2 . 2

Substituting for µ21 from the second condition, we get b∗ 1 Therefore, 2 b∗ ≤ √ b∗ . 2 1 3 √ Thus b∗ cannot be smaller than b∗ by more than a factor of 2/ 3. Let 2 1 us impose similar conditions on each consecutive pair of basis vectors, so √ that for each i, b∗ is not smaller than b∗ by more than a factor of 2/ 3. i i+1 Then the ﬁrst vector, b1 , will be within an exponential factor of the lower bound. Algorithm 27.7 extends in a natural way to iteratively ensuring all these conditions. However, we do not know how to establish termination for this extended algorithm (see Exercise 27.7 for details). To ensure termination in a polynomial number of iterations, we will use an idea similar to that in Exercise 27.3. Let us say that basis b1 , . . . , bn is Gauss reduced if, for 1 ≤ i ≤ n − 1,

2 • bi (i) ≤ √3 bi+1 (i) • |µi+1,i | ≤ 1/2. 2

≤

1 ∗ b 4 1

2

+ b∗ 2

2

The algorithm for obtaining a Gauss reduced basis is a simple generalization of Algorithm 27.7. Algorithm 27.9 (Shortest vector algorithm) 1. While basis b1 , . . . , bn is not Gauss reduced, do: (a) For each i, 1 ≤ i ≤ n − 1, ensure that |µi+1,i | ≤ 1/2. If |µi+1,i | > 1/2 then bi+1 ← bi+1 − mbi , where m is the integer closest to µi+1,i . 2 (b) Pick any i such that bi > √3 bi+1 , and interchange bi and bi+1 . 2. Output b1 .

Theorem 27.10 Algorithm 27.9 terminates in a polynomial number of iterations and achieves an approximation guarantee of 2(n−1)/2 .

282

27

Shortest Vector

Proof: For 1 ≤ i ≤ n − 1 we have bi (i)

2

= b∗ i ≤

2

≤

4 1 ∗ ( b 3 4 i

4 bi+1 (i) 3

2

2

=

4 2 (µ b∗ 3 i+1,i i

2

+ b∗ 2 ) i+1

+ b∗ 2 ). i+1

Therefore, b∗ i and so b1 ≤ 2(n−1)/2 min{ b∗ } ≤ 2(n−1)/2 OPT. i i 2

≤ 2 b∗ 2 . i+1

(27.3)

In order to show termination in a polynomial number of iterations, we will use the following potential function. n Φ= i=1 b∗ i

(n−i)

.

Notice that step 1(a) in Algorithm 27.9 does not change the Gram–Schmidt orthogonalization of the basis and can be executed in O(n) arithmetic operations (each addition, subtraction, multiplication, and division counts as one operation). In an iteration, at most one interchange of vectors is√ performed in Step 1(b). If this happens, Φ drops by a factor√ at least 2/ 3. Indeed of this was the reason to introduce the factor of 2/ 3 in the deﬁnition of a Gauss reduced basis. Let us assume w.l.o.g. that the initial basis has integer components. Then, the initial value of Φ is at most (max ai )n(n−1)/2 . i We will show that Φ ≥ 1 throughout the execution of the algorithm. Consequently, if the number of iterations is m, then 2 √ 3 Therefore, m≤ n(n − 1) maxi log ai √ . 2 log(2/ 3) m ≤ (max ai )n(n−1)/2 . i Express basis vectors b1 , . . . bn in the orthogonal basis b∗ , . . . , b∗ . For 1 ≤ k ≤ 1 n n − 1, let B k be the k × k lower-triangular matrix whose rows are b1 , . . . , bk . Clearly,

27.4

Extension to n dimensions

283

det(B k ) = b∗ . . . b∗ . 1 k Now, the (i, j)th entry of B k B T is bi · bj , which is an integer. Therefore, k det(B k B T ) = b∗ 1 k Hence,

(n−1) 2

. . . b∗ k

2

≥ 1.

Φ= k=1 det(B k B T ) ≥ 1. k ✷

We have shown that Algorithm 27.9 terminates in a polynomial number of arithmetic operations. Strictly speaking, we need to upper bound the number of bit operations, i.e., we need to show that the numbers being handled can be written using a polynomial number of bits. Below we deﬁne the stronger notion of a Lov´sz reduced basis. One can show that a simple extension of a Algorithm 27.9 that ﬁnds a Lov´sz reduced basis executes only a polynomial a number of bit operations. However, the proof is tedious and is omitted. Another motivation for deﬁning the stronger notion of a Lov´sz reduced a basis is that its orthogonality defect can also be bounded. Let us say that basis b1 , . . . , bn is weakly reduced if for 1 ≤ j ≤ i ≤ n, |µij | ≤ 1/2. The basis is said to be Lov´sz reduced if it is Gauss reduced and weakly reduced. One a can obtain a weakly reduced basis from a given basis without changing the Gram–Schmidt orthogonalization, using at most n(n − 1)/2 operations (see Exercise 27.9). Substituting step 1(a) in Algorithm 27.9 by this procedure, we obtain an algorithm for ﬁnding a Lov´sz reduced basis. The bound on the a number of iterations established in Theorem 27.10 applies to this algorithm as well. Theorem 27.11 The orthogonality defect of a Lov´sz reduced basis is bounda ed by 2n(n−1)/4 . Proof: We will use inequality (27.3) established in Theorem 27.10. bi

2 i

= j=1 µ2 b∗ ij j

2

i−1

≤ j=1 1 ∗ b 4 j

2

2

+ b∗ i

2

1 ≤ (1 + (2 + · · · + 2i−1 )) b∗ i 4 Therefore,

≤ 2i−1 b∗ 2 . i

284

27 n Shortest Vector

bi i=1 2

≤ 2n(n−1)/2

n i=1

b∗ i

2

= 2n(n−1)/2 (det L)2 . ✷

The theorem follows.

Example 27.12 Consider the following basis, together with its Gram– Schmidt orthogonalization. 1 0 0 ... 1/2 ρ 0 ... 1/2 ρ/2 ρ2 . . . . . . 0 0 0 0 0 0

1/2 ρ/2 ρ2 /2 . . . ρn−2 /2 ρn−1

√ Here ρ = 3/2. It is easy to verify that this basis is Lov´sz reduced. Each a basis vector is of unit length. The Gram–Schmidt lower bound is ρ(n−1) , i.e., exponentially smaller than any of the basis vectors. Subtracting the last two rows we obtain the vector 0, . . . , 0, ρ(n−2) , −ρ(n−1) . 2 ✷

This vector is exponentially smaller than any of the basis vectors.

27.5 The dual lattice and its algorithmic use

As in the case of linear programs, one can deﬁne a structure that can be viewed as the dual of a given lattice. The power of duality in linear programs derives from the potent min–max relation connecting a linear program and its dual. In contrast, there is no such relation connecting a lattice with its dual. Yet, the dual lattice does seem to have algorithmic signiﬁcance. In this section we will use it to show that the Gram–Schmidt lower bound is good by producing a basis for which this lower bound is at least OPT/n. The dual lattice, L∗ , is deﬁned by L∗ = {v ∈ Rn | ∀b ∈ L, b · v ∈ Z}. For an n × n matrix B, we will denote (B −1 )T by B −T . Theorem 27.13 Let b1 , . . . , bn be any basis for L. Then, the rows of B −T form a basis for the dual lattice L∗ . Furthermore, det L∗ = 1/det L. Proof: Let us name the rows of B −T as v 1 , . . . , v n . Then, bi · v j = 1 if i = j 0 otherwise.

27.5

The dual lattice and its algorithmic use

285

Therefore, any integer linear combination of v 1 , . . . , v n has an integer inner product with each of b1 , . . . , bn , and hence is in L∗ . Conversely, let v ∈ L∗ . Let v · bi = ai ∈ Z, for 1 ≤ i ≤ n. Let a = (a1 , . . . , an ). Thus, we have Bv T = aT . Therefore, v = aB −T , i.e., v is an integer linear combination of the vectors v 1 , . . . , v n . Hence v 1 , . . . , v n form a basis for L∗ . Finally, det L∗ = det(B −T ) = 1 . det L ✷ Let v ∈ R be a nonzero vector. Then, v will denote the (n − 1)dimensional vector space {b ∈ Rn | b · v = 0}. A set L ⊂ L that is a lattice in its own right will be called a sublattice of L. Its dimension is the dimension of the vector space spanned by L . Lemma 27.14 and Exercise 27.11 will establish that there is a one-to-one correspondence between (n−1)dimensional sublattices of L and primitive vectors of L∗ . Lemma 27.14 Let v ∈ L∗ be primitive. Then • L ∩ (v ⊥ ) is an (n − 1)-dimensional sublattice of L. • There is a vector b ∈ L such that v · b = 1. Proof: Since v is primitive, there is a basis of L∗ containing v. Let it be v, v 2 , . . . , v n , and let V be the n × n matrix whose rows are these basis vectors. Let b1 , . . . , bn be the rows of V −T . Since v · bi = 0, for 2 ≤ i ≤ n, the (n − 1)-dimensional sublattice of L generated by b2 , . . . , bn lies in v ⊥ . The second assertion follows from the fact that v · b1 = 1. ✷ Next we will present a key primitive for constructing a basis with a good Gram–Schmidt orthogonalization. Let v ∈ L∗ be primitive, L = L ∩ (v ⊥ ), and w ∈ L with v · w = 1. Let b1 , . . . , bn−1 be a basis for L with Gram– Schmidt orthogonalization b∗ , . . . , b∗ . 1 n−1 Lemma 27.15 b1 , . . . , bn−1 , w is a basis for L, with Gram–Schmidt orthogonalization b∗ , . . . , b∗ , 1 n−1 v . v 2 n ⊥

Proof: Let b ∈ L be an arbitrary vector. Let us ﬁrst observe that b − (b · v)w ∈ L . This is so because (b − (b · v)w) · v = 0. Thus, we can write n−1 b − (b · v)w = i=1 λi bi ,

where λ1 , . . . , λn−1 ∈ Z. Therefore,

286

27

Shortest Vector n−1 b= i=1 λi bi + (b · v)w.

Hence, every vector of L can be written as an integer linear combination of b1 , . . . , bn−1 , w, thus establishing the ﬁrst part. Let µi b∗ be the component of w in the direction of b∗ , for 1 ≤ i ≤ n − 1. i i Let w =w− i=1 ∗ ∗ n−1

µi b∗ . i

Since w is orthogonal to each of b1 , . . . , bn−1 , it is linearly dependent on v. Since v · w = 1, v · w∗ = 1. Therefore, w∗ = Hence, w∗ = v . v 2 ✷ 1 . v

Let us use the primitive given above as follows. Let L1 = L, and let L∗ be 1 its dual. Pick a primitive vector v 1 ∈ L∗ , and let sublattice L2 = L1 ∩ (v ⊥ ). 1 1 Let L∗ denote the dual of L2 (it lies in the vector space spanned by L2 ). 2 Now we proceed by picking another primitive vector v 2 ∈ L∗ , and deﬁne 2 L3 = L2 ∩(v ⊥ ), and so on. In general, we pick a primitive vector v i ∈ L∗ , and i 2 deﬁne sublattice Li+1 = Li ∩(v ⊥ ). Observe that although Li is a sublattice of i L, L∗ is not necessarily a sublattice of L∗ , for 1 ≤ i ≤ n. Next, use the second i part of Lemma 27.14 to pick wi ∈ Li such that v i · wi = 1, for 1 ≤ i ≤ n. We summarize this procedure below. (We call it a “procedure” rather than an “algorithm” because we do not know how to execute some of its steps in polynomial time.) Procedure 27.16 1. Initialization: L1 ← L, and L∗ ← dual of L. 1 2. For i = 1 to n do: Pick a primitive vector v i ∈ L∗ . i Pick wi ∈ Li such that v i · wi = 1. Li+1 ← Li ∩ (v ⊥ ). i L∗ ← dual of Li+1 . i+1 3. Output wn , wn−1 , . . . , w1 and vn v1 ,..., 2 vn v1

2

.

27.5

The dual lattice and its algorithmic use

287

Using Lemma 27.15 and applying induction, we get Lemma 27.17 The output of Procedure 27.16 satisﬁes that wn , wn−1 , . . . , w1 is a basis for L with Gram–Schmidt orthogonalization vn v1 ,..., 2 vn v1

2

.

We will need the following fundamental theorem of Minkowski. √ √ Theorem 27.18 There is a vector b ∈ L such that b ≤ n n det L. √ Proof: Let a ∞ denote the ∞ norm of vector a. Now, a ≤ n a ∞ . Therefore,√ is suﬃcient to show that there is a vector b ∈ L such that it b ∞ ≤ n det L. Further, w.l.o.g. assume that det L = 1 (since we can scale). Now, let C = {v ∈ Rn | v ∞ ≤ 1/2} be the unit cube with center at the origin. Place a cube at each lattice point of L, i.e., consider the cubes C + b, for each lattice point b. Deﬁne the density of a lattice to be the number of lattice points per unit volume. Clearly, the density of a lattice L is 1/det L . Since det L = 1, the density of lattice L is 1. Now, we claim that two of these cubes must intersect. Suppose the cubes are disjoint. Then, each cube contains exactly one lattice point. Since the cubes are of unit volume, this implies that the density of lattice points is strictly smaller than 1, leading to a contradiction. Let the intersecting cubes be centered at lattice points b1 and b2 . Then, b1 − b2 ∞ ≤ 1. Since b = b1 − b2 is also a lattice point, the theorem follows. ✷ Theorem 27.19 There is a basis for L whose Gram–Schmidt lower bound is at least OPT/n. Proof: By Theorem 27.18, there are vectors b ∈ L and v ∈ L∗ such that b v ≤ n. Reﬁne Procedure 27.16 as follows. Let v i ∈ L∗ be a primitive i vector, and bi ∈ Li such that v i bi ≤ n+1−i ≤ n. Vectors wi are deﬁned as above. By Lemma 27.17, wn , . . . , w1 is a basis for L with Gram–Schmidt orthogonalization v1 vn ,..., vn 2 v1

2

.

Let b denote the shortest vector among b1 , . . . , bn . Then, b = min{ b1 , . . . , bn } ≤ n min 1 1 ,..., v1 vn .

288

27

Shortest Vector

Hence, the Gram–Schmidt lower bound, min 1 1 ,..., v1 vn ≥ b OPT ≥ . n n ✷ As a result of Theorem 27.19, there is a factor n approximate No certiﬁcate for the shortest vector problem (see Section 1.2 for deﬁnition). Thus, if the answer to the question “Is the shortest vector in lattice L of length at most α?” is “no”, and in fact α < OPT/n, then there is a polynomial sized guess that enables us to verify in polynomial time that the answer is indeed “no”. The guess is a basis whose existence is demonstrated in Theorem 27.19, and the veriﬁcation simply involves conﬁrming that α is less than the lower bound established by this basis.

27.6 Exercises

27.1 Show that if lattice L has a basis whose vectors are mutually orthogonal, then the shortest vector in the basis is a shortest vector in L. Hint: Write the length of a shortest vector v ∈ L in terms of the basis vectors. 27.2 Recall that if (a, b) = 1, then Euclid’s algorithm gives integers x, y such that ax + by = 1. Generalize this fact to the following: If gcd(a1 , . . . , an ) = 1, then there is an n × n unimodular matrix U whose ﬁrst row is a1 , . . . , an . Hint: Prove by induction on n. 27.3 For the purpose of showing polynomial time termination, let us relax the ﬁrst condition of Gauss’ algorithm in 2 dimensions to ensuring that vector b1 is not much bigger than b2 , i.e., b1 ≤ (1 + ε) b2 . The second condition is as before. Show that the algorithm terminates after a polynomial – in the length of the initial vectors and 1/ε – number of iterations. What is the guarantee on the length of b1 at termination? Hint: Use the Gram–Schmidt lower bound to establish the guarantee. 27.4 Polynomial time termination for Algorithm 27.7 is diﬃcult to establish because in any one iteration, the progress may be minimal. Give an example to show this. √ Hint: Consider for instance b1 = (1, 0) and b2 = ( 1 + ε, 23 ). 2 27.5 (Kaib and Schnorr [152]) To get around the diﬃculty mentioned in Exercise 27.4, consider the following small modiﬁcation to Algorithm 27.7. Say that basis (b1 , b2 ) is well ordered if

27.6

Exercises

289

b1 ≤ b1 − b2 < b2 . 1. Assume b1 ≤ b2 . Show that one of the three bases (b1 , b2 ), (b1 , b1 − b2 ), and (b1 −b2 , b1 ) is well ordered. Hence, modify the initialization step to start with a well ordered basis. 2. The main algorithm is as follows: (a) If |µ21 | > 1/2, let b2 ← b2 − mb1 , where m is the integer closest to µ21 . (b) If b1 − b2 > b1 + b2 , then let b2 ← −b2 . (c) If b1 ≤ b2 , output (b1 , b2 ) and halt. Otherwise, interchange b1 and b2 and go to step (a). Show that the basis obtained after step (b) is well ordered. 3. Suppose the modiﬁed algorithm, starting with well ordered basis (b1 , b2 ), executes k iterations and ends with a reduced basis. Renumber the vectors encountered in reverse order as (a1 , a2 , . . . , ak , ak+1 ), where b1 = ak , b2 = ak+1 , and the terminating basis is (a1 , a2 ). Prove that for i > 1, ai ≤ (1/2) ai+1 , thereby establishing a polynomial bound on the running time. 27.6 Give an example of a 3-dimensional lattice L and vectors b1 , b2 ∈ L such that b1 and b2 are individually primitive, but b1 , b2 is not. 27.7 Suppose basis b1 , . . . , bn satisﬁes the following conditions for 1 ≤ i ≤ n − 1: • bi (i) ≤ bi+1 (i) • |µi+1,i | ≤ 1/2, Show that b1 ≤ 2 √ 3 i b∗ , i+1

for 1 ≤ i ≤ n − 1. Hence, b1 ≤ 2 √ 3 n−1 min{ b∗ } ≤ i i 2 √ 3

n−1

OPT.

Hint: By a calculation similar to the 2-dimensional case show that for 1 ≤ i ≤ n − 1, 2 b∗ ≤ √ b∗ . i i+1 3

290

27

Shortest Vector

27.8 By modifying the deﬁnition of a Gauss reduced basis appropriately, √ obtain a factor c(n−1)/2 algorithm, for any constant c > 2/ 3, for the shortest vector problem. √ Hint: Replace the factor of 2/ 3 in the deﬁnition of Gauss reduced basis by 1 + ε, for a suitable ε > 0. 27.9 Show that a basis can be made weakly reduced using at most n(n−1)/2 arithmetic operations. Hint: In Algorithm 27.9, the operations of step 1(a) can be carried out in any order. In contrast, for obtaining a weakly reduced basis, order is important. Pick a pair (i, j) with |µij | > 1/2 such that j is the largest possible, and carry out the operation of step 1(a) in Algorithm 27.9. 27.10 Prove that the dual of lattice L∗ is L. 27.11 Prove the converse of Lemma 27.14. • If L is an (n − 1)-dimensional sublattice of L, then there is a vector v ∈ L∗ such that L = L ∩ (v ⊥ ). • If v ∈ L∗ and ∃b ∈ L such that v · b = 1, then v is primitive. 27.12 Let v ∈ L∗ be primitive and L be an (n − 1)-dimensional sublattice of L such that L = L ∩ (v ⊥ ). Prove that det L = v · det L. 27.13 This exercise develops a partial converse for Lemma 27.15. Let v and b1 , . . . , bn−1 be as deﬁned in Lemma 27.15. Show that if b1 , . . . , bn−1 , w form a basis for L, then v · w = 1. Hint: By Lemma 27.14, there is a vector w ∈ L such that v · w = 1. Express w in the basis b1 , . . . , bn−1 , w, and use this to argue that v · w = 1. 27.14 Show that the basis whose existence is established in Theorem 27.19 can be found in polynomial time, given an oracle for the shortest vector problem. Hint: A shortest vector satisﬁes the condition of Theorem 27.18. 27.15 (Korkhine and Zolotarav [175]) A basis b1 , . . . , bn for lattice L is said to be KZ reduced if: • b1 is a shortest vector in L, b2 is such that b∗ is shortest possible while 2 ensuring the primitiveness of b1 , b2 , etc. In general, bi is such that b∗ is i shortest possible while ensuring the primitiveness of b1 , . . . , bi , i.e., the component of bi orthogonal to the vector space spanned by b1 , . . . , bi−1 is being minimized, while ensuring primitiveness.

27.6

Exercises

291

• For 1 ≤ j < i ≤ n, |µij | ≤ 1/2. Give a polynomial time algorithm for ﬁnding a KZ reduced basis in a lattice, given an oracle for the shortest vector problem. 27.16 (Lov´sz [193]) Let p(n) be a polynomial function. Show that if there a is a polynomial time algorithm that ﬁnds a vector v ∈ L such that √ n v ≤ p(n) det L, then there is a factor p2 (n) approximation algorithm for the shortest vector problem. Hint: The main idea is present in Theorem 27.19. 27.17 (Designed by K. Jain and R. Venkatesan, based on Frieze [88]) Given a set S of n nonnegative integers S = {a1 , . . . , an } and an integer b, the subset sum problem asks if there is a subset of S that adds up to b. This NP-hard problem lends itself naturally to the design of a one–way function, a basic cryptographic primitive, as follows. An n bit message c = c1 , . . . , cn can be encoded as n fS (c) = i=1 ai ci = b, say.

Clearly, fS is easy to compute. The NP-hardness of the subset sum problem implies that fS is hard to invert in general, i.e., given a1 , . . . , an and b, it is hard to compute c. This exercise shows how, using Algorithm 27.9, fS can be inverted with high probability if a1 , . . . , an are picked uniformly and independently in the 2 (large) range [1, B], where B ≥ 2n . In practice, it is desirable to pick B to be a large number so that fS turns out to be one-to-one. √ Let m = 2n/2 n + 1 and p = m+1. Pick the following basis vectors for an (n + 1)-dimensional lattice L. b0 = (pb, 0, ...0), b1 = (−pa1, 1, 0...0), . . . , bn = (−pan , 0, 0, ..., 1). 1. Let u1 , . . . , un be ﬁxed integers. Show that if the ai ’s are picked uniformly and independently in the range [1, B], then n Pr[ i=1 ui ai = 0] ≤

1 , B

where the probability is over the choice of √ ’s. ai 2. Observe that L has a vector of length ≤ n. Show that if there is no other vector in L of length ≤ m, then Algorithm 27.9 can be used to ﬁnd c such that fS (c) = b.

292

27

Shortest Vector

3. Suppose there is another vector in L of length ≤ m, say v. Let v = n λb0 + i=1 vi bi , say, with λ, vi ∈ Z. Clearly each vi ∈ [−m, m]. Show that λ ∈ [−2m, 2m]. n Hint: We can assume w.l.o.g. that b ≥ ( 1 ) i=1 ai (if not, then decode 2 n i=1 ai − b). The ﬁrst coordinate of v must be zero, which gives n n

|λ|b ≤ i=1 ai |vi | ≤ ( i=1 ai ) v ≤ 2bm.

4. Show that if there is another vector in L of length ≤ m then there are n integers u1 , . . . , un , with each ui ∈ [−3m, 3m], such that i=1 ui ai = 0. n Hint: i=1 ai (λci − vi ) = 0. 5. Show that if the ai ’s are picked uniformly and independently in the range n [1, B], then the probability that i=1 ui ai = 0 for integers u1 , . . . , un , with each ui ∈ [−3m, 3m], is very small. Hint: Use the fact that the number of choices for u1 , . . . , un is (6m+1)n . 6. Hence prove that fS can be inverted with high probability. 27.18 (Goldreich, Micciancio, Safra, and Seifert [108]) Consider the closest vector problem. Given basis vectors b1 , . . . , bn for lattice L and a target vector b ∈ Rn , ﬁnd a lattice vector that is closest to b in Euclidean distance. Show that for any function f (n), an f (n) factor approximation algorithm for the closest vector problem can be used as a subroutine to give an f (n) factor approximation algorithm for the shortest vector problem. Hint: Given basis vectors b1 , . . . , bn for a lattice L, suppose b = i λi bi is a shortest vector in L. Clearly all λi cannot be even. Suppose λi is odd. Now consider lattice L whose basis vectors are b1 , . . . , bi−1 , 2bi , bi+1 , . . . , bn . Observe that if the closest vector in lattice L to target vector bi is v, then b = v − bi . Since i is not known, construct n such questions and pick the best answer.

27.7 Notes

Algorithm 27.9 and Theorem 27.11 are due to Lenstra, Lenstra, and Lov´sz a [183]. Schnorr [235] modiﬁed this algorithm to achieve an approximation factor of (1 + ε)n , for any ε > 0. More precisely, Schnorr’s algorithm runs in polynomial time for a suitable choice of ε = o(1), which results in a slightly subexponential factor algorithm, i.e., an algorithm with an approximation factor of 2o(n) . This is currently the best approximation guarantee known for shortest vector. Algorithm 27.7 appears in Gauss [99], though it is stated there in terms of quadratic forms. Using a complex potential function argument, Lagarias [180] showed that this algorithm has polynomial running time (in the number

27.7

Notes

293

of bits needed to write the input basis). For a polynomial time algorithm for shortest vector in d-dimensions, for ﬁxed constant d, see Kannan [154]. Example 27.12 is due to Kannan [153]. Ajtai [2] showed that the shortest vector problem is NP-hard under randomized reductions. Theorem 27.19 is due to Lagarias, Lenstra, and Schnorr [181].

28 Counting Problems

The techniques for approximately counting the number of solutions to #Pcomplete problems are quite diﬀerent from those for obtaining approximation algorithms for NP-hard optimization problems. Much of the former theory is built around the Markov chain Monte Carlo method, see Section 28.4 for references. In this chapter, we will present combinatorial algorithms (not using Markov chains) for two fundamental problems, counting the number of satisfying truth assignments for a DNF formula, and estimating the failure probability of an undirected network. Intuitively, the class #P captures the problems of counting the number of solutions to NP problems. Let us formalize this notion. Let L be a language in NP, M be its associated veriﬁer, and polynomial p be the bound on the length of its Yes certiﬁcates (see Section A.1). For string x ∈ Σ ∗ , deﬁne f (x) to be the number of strings y such that |y| ≤ p(|x|) and M (x, y) accepts. Functions f : Σ ∗ → Z+ that arise in this manner constitute the class #P. Function f ∈ #P is said to be #P-complete if every function g ∈ #P can be reduced to f in the following sense. There is a polynomial time transducer R : Σ ∗ → Σ ∗ , that, given an instance, x, of g, produces an instance, R(x), of f . Furthermore, there is a polynomial time computable function S : Σ ∗ × Z+ → Z+ that given x and f (R(x)) computes g(x), i.e., ∀x ∈ Σ ∗ , g(x) = S(x, f (R(x))). In other words, an oracle for f can be used to compute g in polynomial time. The solution counting versions of all known NP-complete problems are #P-complete.1 Interestingly enough, other than a handful of exceptions, this is true of problems in P as well. This raises the question of designing polynomial time algorithms for approximately counting the number of solutions to these latter problems (see Exercise 28.3 regarding the question of approximately counting the number of solutions to NP-complete problems). These problems admit only two interesting possibilities: they either allow approximability to any required degree, or essentially not at all (see Section 28.4).

1

In fact, typically a polynomial time reduction from one NP-complete problem to another maps solutions of the given instance to solutions of the transformed instance, and so preserves the number of solutions; hence, the proof of #Pcompleteness follows directly from the proof of NP-completeness.

28.1

Counting DNF solutions

295

The former possibility is captured in the deﬁnition of a fully polynomial randomized approximation scheme, abbreviated FPRAS. Consider a problem in P whose counting version, f , is #P-complete. An algorithm A is an FPRAS for this problem if for each instance x ∈ Σ ∗ , and error parameter ε > 0, Pr[|A(x) − f (x)| ≤ εf (x)] ≥ 3 , 4

and the running time of A is polynomial in |x| and 1/ε. (See Exercise 28.1 for a method for reducing the error probability of an FPRAS.)

28.1 Counting DNF solutions

Problem 28.1 (Counting DNF solutions) Let f = C1 ∨ C2 ∨ . . . ∨ Cm be a formula in disjunctive normal form on n Boolean variables x1 , . . . , xn . Each clause Ci is of the form Ci = l1 ∧ l2 ∧ . . . ∧ lri , where each lj is a literal, i.e., it is either a Boolean variable or its negation. We may assume w.l.o.g. that each clause is satisﬁable, i.e., does not contain a variable and its negation. The problem is to compute #f, the number of satisfying truth assignments of f . The main idea is to deﬁne an eﬃciently samplable random variable X which is an unbiased estimator for #f, i.e., E[X] = #f. If in addition, the standard deviation of X is within a polynomial factor of E[X], then an FPRAS for #f can be obtained in a straightforward manner by sampling X a polynomial number of times (in n and 1/ε) and outputting the mean. Constructing an unbiased estimator for #f is easy. Let random variable Y have uniform distribution on all 2n truth assignments, and let Y (τ ) be 2n if τ satisﬁes f , and 0 otherwise (see Exercise 28.4). However, this random variable can have a very large standard deviation, and does not yield an FPRAS. For instance, suppose f has only polynomially many satisfying truth assignments. Then, with high probability, a polynomial number of randomly picked truth assignments will all have Y = 0, giving a poor estimate for #f. We will rectify this by deﬁning a random variable that assigns nonzero probability to only the satisfying truth assignments of f . Let Si denote the set of truth assignments to x1 , . . . , xn that satisfy clause Ci . Clearly, |Si | = 2n−ri , where ri is the number of literals in clause Ci . Also, #f = | ∪m Si |. Let i=1 c(τ ) denote the number of clauses that truth assignment τ satisﬁes. Let M denote the multiset union of the sets Si , i.e., it contains each satisfying truth assignment, τ , c(τ ) number of times. Notice that |M | = i |Si | is easy to compute. Pick a satisfying truth assignment, τ , for f with probability c(τ )/|M |, and deﬁne X(τ ) = |M |/c(τ ). We will ﬁrst show that X can be eﬃciently sampled, i.e., using a randomized polynomial time algorithm.

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Counting Problems

Lemma 28.2 Random variable X can be eﬃciently sampled. Proof: Picking a random element from the multiset M ensures that each truth assignment is picked with the desired probability. The following twostep process will accomplish this. First pick a clause so that the probability of picking clause Ci is |Si |/|M |. Next, among the truth assignments satisfying the picked clause, pick one at random. Now, the probability with which truth assignment τ is picked is 1 |Si | c(τ ) × . = |M | |Si | |M | ✷ Lemma 28.3 X is an unbiased estimator for #f. Proof: E[X] = τ i:τ

satisﬁes

Ci

Pr[τ is picked] · X(τ ) = τ satisﬁes

f

|M | c(τ ) × = #f. |M | c(τ ) ✷

X takes values only in a “polynomial range”, thereby ensuring that its standard deviation is not large compared to its expectation. This fact is proved in the next lemma, and leads to the FPRAS construction. Lemma 28.4 If m denotes the number of clauses in f , then σ(X) ≤ m − 1. E[X] Proof: Denote |M |/m by α. Clearly, E[X] ≥ α. For each satisfying truth assignment τ of f , 1 ≤ c(τ ) ≤ m. Therefore, X(τ ) lies in the range [α, mα], and so the random variable deviates from its mean by at most (m − 1)α, i.e., |X(τ ) − E[X]| ≤ (m − 1)α. Therefore, the standard deviation of X is bounded by (m − 1)α. Using the lower bound on E[X] stated above, we get the lemma. ✷ Finally, we will show that sampling X polynomially many times (in n and 1/ε) and simply outputting the mean leads to an FPRAS for #f. Let Xk denote the mean of k samples of X. Lemma 28.5 For any ε > 0, Pr[|Xk − #f | ≤ ε#f ] ≥ 3/4,

28.2

Network reliability

297

where k = 4(m − 1)2 /ε2 . Proof: We will use Chebyshev’s inequality (see Section B.2), with a = εE[Xk ]. Using the value of k stated above we get Pr[|Xk − E[Xk ]| ≥ εE[Xk ]] ≤ σ(Xk ) εE[Xk ]

2

=

σ(X) √ ε kE[X]

2

≤

1 , 4

where √ equality follows by noting that E[Xk ] = E[X] and σ(Xk ) = the σ(X)/ k, and the last inequality follows by applying Lemma 28.4. The lemma follows. ✷ Theorem 28.6 There is an FPRAS for the problem of counting DNF solutions.

28.2 Network reliability

Problem 28.7 (Network reliability) Given a connected, undirected graph G = (V, E), with failure probability pe speciﬁed for each edge e, compute the probability that the graph becomes disconnected. Graph G will become disconnected if all edges in some cut (C, C), C ⊂ V fail. We will present an FPRAS for this problem. Let us ﬁrst handle the case that each edge has the same failure probability, denoted by p. However, we will allow G to have parallel edges between any two vertices. Denote by FAIL(p) the probability that G gets disconnected. If FAIL(p) is at least inverse polynomial, then it can be eﬃciently estimated by Monte Carlo sampling (see proof of Theorem 28.11 for details). Let us handle the diﬃcult case that FAIL(p) is small. Assume that FAIL(p) ≤ n−4 . The reason for this choice will become clear below. The probability that cut (C, C) gets disconnected is simply pc where c is the number of edges crossing this cut. Since the failure probability of a cut decreases exponentially with capacity, the most important cuts for the purpose of estimating FAIL(p) are cuts with “small” capacity. The algorithm is built around two ideas: 1. For any ε > 0, we will show that only polynomially many “small” cuts (in n and 1/ε) are responsible for 1−ε fraction of the total failure probability FAIL(p). Moreover, these cuts, say E1 , . . . Ek , Ei ⊆ E, can be enumerated in polynomial time. 2. We will construct a polynomial sized DNF formula f whose probability of being satisﬁed is precisely the probability that at least one of these cuts fails.

298

28

Counting Problems

As a result of the ﬁrst idea, it is suﬃcient to estimate the probability that one of the cuts E1 , . . . Ek fails. However, because of correlations, this is nontrivial. The second idea reduces this problem to counting DNF solutions, for which we have an FPRAS. Formula f has a Boolean variable xe for each edge e. xe is set to true with probability p, the failure probability of edge e. Suppose cut Ei = {e1 , . . . , ej }. Construct the clause Di = xe1 ∧ · · · ∧ xej , i.e., the conjunct of all variables corresponding to edges in this cut. The probability that this clause is satisﬁed is precisely the failure probability of cut Ei . Finally, f = D1 ∨ · · · ∨ Dk , i.e., the disjunct of clauses corresponding to cuts. 28.2.1 Upperbounding the number of near-minimum cuts

The ﬁrst idea has its roots in the fact that one can place upper bounds on the number of minimum and near-minimum capacity cuts in an undirected graph. Let c be the capacity of a minimum cut in G. Recall that all edges in G are assumed to be of unit capacity, and that G is allowed to have parallel edges between any two vertices. Lemma 28.8 The number of minimum cuts in G = (V, E) is bounded by n(n − 1)/2. Proof: By contracting an edge (u, v) in a graph we mean merging the vertices u and v into a single vertex. All edges running between u and v are discarded. Those running between u or v and some other vertex w will now run between the merged vertex and w, their number being conserved. Now consider the following random contraction process. Iteratively, pick a random edge (u, v) in the current graph and contract it. Terminate when exactly two vertices are left. Suppose these two vertices correspond to sets S and V − S, S ⊂ V , of vertices of the starting graph G. Then, the algorithm outputs the cut (S, S). We will say that this cut survives. Clearly, a cut survives iﬀ none of its edges is contracted during the algorithm. Let (C, C) be any minimum cut in G. We will show Pr[(C, C) survives] ≥ 1 n 2

.

This statement yields the lemma via an interesting argument. Let M be the number of minimum cuts in G. The survival of each of these cuts is a mutually exclusive event, and the total probability of these events adds up to at most 1. Hence M/(n(n − 1)/2) ≤ 1, thereby giving the desired bound. Consider an arbitrary iteration in the random contraction process, and let H be a graph at the beginning of this iteration. Since the process of contraction cannot decrease the capacity of the minimum cut, the capacity of each cut in H is at least c. This holds for cuts separating one vertex of H

28.2

Network reliability

299

from the rest. Therefore, the degree of each vertex in H must be at least c. Hence, H must have at least cm/2 edges, where m is the number of vertices in H. Now, the conditional probability that cut (C, C) survives the current itc eration, given that it has survived so far, is at least (1 − cm/2 ) = (1 − 2/m) (this is simply the probability that the randomly chosen edge in this iteration is not picked from the cut (C, C)). The probability that (C, C) survives the whole algorithm is simply the product of these conditional probabilities. This gives Pr[(C, C) survives] ≥ 1− 2 n 1− 2 n−1 ... 1 − 2 3 = 1 n 2

. ✷

αc.

For α ≥ 1, we will say that a cut is an α-min cut if its capacity is at most

Lemma 28.9 For any α ≥ 1, the number of α-min cuts in G is at most n2α . Proof: We will prove the lemma for the case that α is a half-integer. The proof for arbitrary α follows by applying the same ideas to generalized binomial coeﬃcients. Let 2α = k. Consider the following two-phase process: First, run the random contraction algorithm until there are k vertices remaining in the graph. Next, pick a random cut among all 2k−1 cuts in this graph. This will deﬁne a cut in the original graph. Let (C, C) be any α-min cut in G. We will show that the probability that it survives the two phase process is at least 1/n2α , thereby proving the desired bound. Let H be the graph at the beginning of an arbitrary iteration in the ﬁrst phase. As argued in Lemma 28.8, if H has m vertices, it has at least mc/2 edges. Therefore, the conditional probability that (C, C) survives the current αc iteration, given that it has survived so far, is at least 1 − mc/2 = 1 − 2α/m. The probability that (C, C) survives the ﬁrst phase is at least 1− 2α n 1− 2α n−1 ... 1 − 2α 3 = 1 n k

.

The conditional probability that (C, C) survives the second phase, given that it has survived the ﬁrst, is 1/2k−1 . Therefore, Pr[(C, C) survives both phases] ≥ n k

1 1 1 ≥ k = 2α . k−1 n n 2 ✷

300

28

Counting Problems

28.2.2

Analysis

Recall that we are considering the case that FAIL(p) ≤ n−4 . We can now justify this choice. The failure probability of a minimum cut is pc ≤ FAIL(p) ≤ n−4 . Let pc = n−(2+δ) , where δ ≥ 2. Now, by Lemma 28.9, for any α ≥ 1, the total failure probability of all cuts of capacity αc is at most pαc n2α = n−αδ . This rapid decrease in the total failure probability of all cuts of capacity αc will enable us to bound the total failure probability of “large” capacity cuts. Lemma 28.10 For any α ≥ 1, Pr[some cut of capacity > αc fails] ≤ n−αδ 1 + 2 δ .

Proof: Number all cuts in G by increasing capacity. Let ck and pk denote the capacity and failure probability of the kth cut in this numbering. Let a be the number of the ﬁrst cut of capacity greater than αc. It suﬃces to show that pk ≤ n−αδ 1 + k≥a 2 δ

.

We will evaluate this sum in two steps. First, we will consider the ﬁrst n2α terms. Each of these terms is at most pαc = n−α(2+δ) . Therefore, their sum is at most n−αδ . Next, let us bound the sum of the remaining terms. Clearly, this sum is bounded by k>n2α pk . By Lemma 28.9, there are at most n2α cuts of capacity bounded by αc. Therefore, cn2β ≥ βc. Writing k = n2β we get ck ≥ c ln k/2 ln n, and pk ≤ (pc ) 2 ln n = k −(1+δ/2) . Therefore, pk ≤ k>n2α ∞ n2α ln k

k −(1+δ/2) dk ≤

2 −αδ . n δ ✷

This proves the lemma. Theorem 28.11 There is an FPRAS for estimating network reliability.

Proof: We will ﬁrst consider the case that each edge in graph G has the same failure probability, p.

28.2

Network reliability

301

If FAIL(p) > n−4 , then we will resort to Monte Carlo sampling. Flip a coin with bias p for failure of each edge, and check if G is disconnected. Repeat this experiment O(log n/(ε2 FAIL(p))) times, and output the mean number of times G is disconnected. A straightforward application of Chernoﬀ bounds shows that the mean lies in [(1−ε)FAIL(p), (1+ε)FAIL(p)] with high probability. Next, assume that FAIL(p) ≤ n−4 . Now, for any ε > 0, we want to determine α such that the total failure probability of all cuts of capacity > αc is bounded by εFAIL(p). By Lemma 28.10, it suﬃces to ﬁnd α such that n−αδ 1 + Solving, we get α=1+ ln(ε/2) 2 ln(ε/2) ≤2− . − δ ln n 2 ln n δ 2 δ ≤ εFAIL(p) ≤ εn−(2+δ) .

By Lemma 28.9, cn2α > αc. For the value of α computed, Pr[one of the ﬁrst n2α fails] ≥ (1 − ε)FAIL(p). The ﬁrst n2α = O(n4 /ε) cuts can be enumerated in polynomial time (see Exercises). We will use these to construct the corresponding DNF formula, and estimate the probability that it is satisﬁable, as described above. Finally, we show how to “reduce” the case of arbitrary edge failure probabilities to the simpler case analyzed above. Suppose edge e has failure probability pe . Choose a small parameter θ. Replace edge e with ke = −(ln pe )/θ parallel edges each with failure probability 1 − θ. Then, the probability that all ke edges fail is (1 − θ)−(ln pe )/θ . As θ → 0, this failure probability converges to pe . Let H be the graph obtained by doing this transformation on each edge of G. In the limit as θ → 0, each cut in H has the same failure probability as that in G. Let us give an eﬃcient implementation of this idea. All we really want is a listing of the “small” capacity cuts in G. Once this is done, we can apply the more general DNF counting algorithm developed in Exercise 28.5, where each variable is set to true with its own probability pe . Observe that changing θ scales the capacities of cuts in H without changing their relative values. Thus, it suﬃces to assign a weight of − ln pe to each edge e of G, and ﬁnd “small” capacity cuts in this graph. This completes the proof. ✷

302

28

Counting Problems

28.3 Exercises

28.1 Given an FPRAS for a problem, show that its success probability can be improved to 1−δ, for any δ > 0, by a multiplicative increase in the running time of only O(log(1/δ)). Hint: Run the FPRAS O(log(1/δ)) times and output the median value. 28.2 Suppose we make the deﬁnition of an FPRAS more stringent by requiring it to have a ﬁxed additive error α with high probability, i.e., Pr[f (x) − α ≤ A(x) ≤ f (x) + α] ≥ 3 . 4

Show that if there were such an algorithm for a #P-complete problem, then P= NP. 28.3 Show that if there were an FPRAS for counting the number of satisfying truth assignments to SAT then every problem in NP could be solved in random polynomial time. How weak an approximate counting algorithm for SAT suﬃces to give this consequence? What does this say for the question of approximately counting the number of solutions to other NP-complete problems? Hint: Use solution ampliﬁcation. Given SAT formula f , deﬁne formula f over k new Boolean variables which is a tautology. Then the number of solutions of φ = f ∧ f is #f · 2k . 28.4 Given a DNF formula f , let Y be a random variable that on a random truth assignment τ is 2n if τ satisﬁes f and 0 otherwise. Show that Y is an unbiased estimator for #f. How large can the ratio σ(Y )/E[Y ] be? 28.5 (Karp and Luby [165]) You are given a DNF formula f on n Boolean variables, x1 , . . . , xn , and probabilities p1 , . . . , pn with which these variables are (independently) set to true. Let D denote the resulting probability distribution over the 2n truth assignments to the Boolean variables, and p denote the probability that f is satisﬁed by a truth assignment picked from D. Construct an FPRAS for estimating p. Hint: Let qi denote the probability that clause Ci is satisﬁed by a truth assignment picked from D, and Q = i qi . Now, consider random variable X that assigns to each satisfying truth assignment τ a probability of PrD [τ ]c(τ )/Q, and deﬁne X(τ ) = Q/c(τ ). 28.6 A uniform generator for an NP problem Π is a randomized polynomial time algorithm A that given an instance I of a problem, outputs either a solution to I, or else the special symbol “⊥”, such that

28.3

Exercises

303

• each solution to I is output with the same probability, i.e., there is a number α ∈ (0, 1] such that Pr[A outputs s] = α, for each solution s of I, and • the probability of outputting ⊥, i.e., failing to output a solution, is < 1/2. Give a uniform generator for picking a random satisfying truth assignment to a given DNF formula. Hint: The essential idea behind the construction of random variable X works. 28.7 (Jerrum, Valiant, and Vazirani [147]) Let Π be an NP problem that is self-reducible (see Section A.5 and Exercise 1.15). Show that there is an FPRAS for Π iﬀ there is an almost uniform generator for it. An almost uniform generator for Π is a randomized polynomial time algorithm A such that for any µ > 0 and instance I of Π, there is a number α ∈ (0, 1] such that • for each solution s of I, Pr[A outputs s] ∈ [(1 − µ)α, (1 + µ)α], • Pr[A fails to output a solution] < 1/2, and • the running time of A is polynomial in |I| and log(1/µ). Observe that unlike an FPRAS, which can only achieve inverse polynomial error, a uniform generator can achieve inverse exponential error (µ), in polynomial time. Hint: For the forward direction, ﬁrst construct a uniform generator, assuming that the FPRAS makes no error. (Traverse down the self-reducibility tree for I, with biases determined by estimates on the number of solutions. Accept leaf with appropriate probability, to achieve uniform generation.) Use the fact that the error probability of the FPRAS can be made exponentially small to obtain an almost uniform generator. For the reverse direction, obtain instance Iα , with |Iα | < |I|, and a good estimate of the ratio of the number of solutions to I and Iα . 28.8 (Jerrum, Valiant, and Vazirani [147]) This exercise leads to strong evidence that the problem of estimating the number of simple cycles in a directed graph is essentially not approximable. Show that if there is an almost uniform generator for this problem, then there is a randomized polynomial time algorithm for deciding if a given directed graph has a Hamiltonian cycle. Hint: Obtain a graph G from G that ampliﬁes the number of cycles of each length. However, it ampliﬁes bigger cycles more than it ampliﬁes smaller cycles, so that most cycles in G are of maximum length and correspond to the largest cycles in G.

304

28

Counting Problems

28.9 Show that the random contraction algorithm of Lemma 28.8 can be used to obtain a randomized algorithm for ﬁnding a minimum cut in an undirected graph. 28.10 (Karger and Stein [159]) Obtain a randomized algorithm for enumerating all α-min cuts in G using the random contraction algorithm and Lemma 28.9. In the next three exercises (from Vazirani and Yannakakis [252]), we will develop a deterministic algorithm for enumerating in an undirected graph by increasing weight, with polynomial delay, i.e., the algorithm spends polynomial time between successive outputs. Assume that graph G = (V, E) has n vertices besides s and t, numbered 1 to n. Every s–t cut in G can be represented as an n bit 0/1 vector. A partially speciﬁed cut, in which the sides of only vertices numbered 1 to k are decided, is represented as a k bit 0/1 vector. Consider a binary tree T of height n. Its leaves represent s–t cuts and internal nodes represent partially speciﬁed cuts. All cuts consistent with a partially speciﬁed cut lie in the subtree rooted at it. Clearly, a minimum weight cut in this subtree can be computed with one max-ﬂow computation. 28.11 Let a be an n − k bit 0/1 vector representing a partially speciﬁed cut, as well as an internal node in T . The subtree, T , rooted at this node is of height k and contains 2k leaves (s–t cuts). Among these 2k cuts, let a be a minimum weight cut. Show how the remaining 2k − 1 cuts of T can be partitioned into k subtrees which are of height 0, 1, . . . , k − 1. 28.12 Using a heap, give an algorithm for enumerating s–t cuts in G by increasing weight. Hint: The heap is initialized with a minimum cut in G. At an arbitrary point, the cuts not enumerated so far can be partitioned into subtrees (see Exercise 28.11). The heap contains a minimum cut from each subtree. 28.13 Give an algorithm for enumerating all cuts in an undirected graph by increasing weight with polynomial delay. Hint: Assume that the graph has a special vertex s, which always goes on side 0 of the cut, and n other vertices, numbered 1 to n. A cut is speciﬁed by an n bit vector specifying the sides of vertices numbered 1 to n. The main diﬀerence arises in ﬁnding a minimum cut in the subtree rooted at the internal node 0k , k < n. This is done by ﬁnding a minimum cut separating the vertices s, 1, . . . , i from vertex i + 1 for k ≤ i < n, and picking the lightest of these cuts. 28.14 (Karger [156]) Consider the generalization of network reliability to estimating the probability that G disconnects into r or more components, where r is a ﬁxed constant. Obtain an FPRAS for this problem.

28.4

Notes

305

28.4 Notes

The counting class #P was deﬁned by Valiant [249]. The FPRAS for counting DNF solutions is due to Karp and Luby [165], who also gave the deﬁnition of FPRAS (see also Karp, Luby, and Madras [166]). The FPRAS for estimating network reliability is due to Karger [156]. Most algorithms for approximate counting work by constructing an almost uniform generator for the problem and appealing to the equivalence established in Exercise 28.7. Broder [33] introduced the use of rapidly mixing Markov chains for almost uniform generation (see also Mihail [205]). Jerrum and Sinclair [145] gave the ﬁrst FPRAS using this approach, for counting the number of perfect matchings in dense bipartite graphs (each vertex should have a degree ≥ n/2; see also Section 30.3). They also showed that a crude approximate counter, with polynomial error, can be transformed into an FPRAS (with inverse polynomial error), by deﬁning an appropriate Markov chain on the self-reducibility tree of an instance. As a result, #P-complete problems either admit an FPRAS or are essentially not approximable at all (see Exercise 28.8) For Markov–chain based approximate counting algorithms, see Jerrum and Sinclair [142], Sinclair [242], and the references in Section 30.3.

29 Hardness of Approximation

A remarkable achievement of the theory of exact algorithms is that it has provided a fairly complete characterization1 of the intrinsic complexity of natural computational problems, modulo some strongly believed conjectures. Recent impressive developments raise hopes that we will some day have a comprehensive understanding of the approximability of NP-hard optimization problems as well. In this chapter we will give a brief overview of these developments. Current hardness results fall into three important classes. For minimization problems, the hardness factors for these classes are constant (> 1), Ω(log n), and nε for a ﬁxed constant ε > 0, where n is the size of the instance. For maximization problems, the factors are constant (< 1), O(1/ log n), and 1/nε for a ﬁxed ε > 0. In this chapter we will present hardness results for MAX-3SAT, vertex cover, and Steiner tree in the ﬁrst class, set cover in the second class, and clique in the third class. For all these problems, we will establish hardness for their cardinality versions, i.e., the unit cost case.

29.1 Reductions, gaps, and hardness factors

Let us start by recalling the methodology for establishing hardness results for exact optimization problems. The main technical core is the Cook–Levin theorem which establishes the hardness, assuming P = NP, of distinguishing between instances of SAT that are satisﬁable and those that are not. To show hardness of computing an optimal solution to, say the cardinality vertex cover problem, one shows, via a polynomial time reduction from SAT, that it is hard to distinguish between graphs that have covers of size at most k from graphs that don’t, where k is provided as part of the input. Since an exact algorithm can make this distinction, this reduction establishes the non-existence of an eﬃcient exact algorithm. The main technical core of hardness of approximation results is the PCP theorem, which is stated in Section 29.2. For establishing a hardness of approximation result for, say, the vertex cover problem, this theorem is used to

1

A few (important) exceptions, such as the graph isomorphism problem, remain uncharacterized.

29.1

Reductions, gaps, and hardness factors

307

show the following polynomial time reduction. It maps an instance φ of SAT to a graph G = (V, E) such that • if φ is satisﬁable, G has a vertex cover of size ≤ 2 |V |, and 3 • if φ is not satisﬁable, the smallest vertex cover in G is of size > α · 2 |V |, 3 where α > 1 is a ﬁxed constant. Claim 29.1 As a consequence of the reduction stated above, there is no polynomial time algorithm for vertex cover that achieves an approximation guarantee of α, assuming P = NP. Proof: Essentially, this reduction establishes the hardness, assuming P = NP, of distinguishing graphs having a cover of size ≤ 2 |V | from those having 3 a cover of size > α· 2 |V |. An approximation algorithm for vertex cover, having 3 a guarantee of α or better, will ﬁnd a cover of size ≤ α · 2 |V | when given 3 a graph G from the ﬁrst class. Thus, it will be able to distinguish the two classes of graphs, leading to a contradiction. ✷ The reduction stated above introduces a gap, of factor α, in the optimal objective function value achieved by the two classes of graphs (if α = 1 then this is an ordinary polynomial time reduction from SAT to vertex cover). Let us formally state the central notion of a gap-introducing reduction. The deﬁnition is slightly diﬀerent for minimization and maximization problems. For simplicity, let us assume that we are always reducing from SAT. Let Π be a minimization problem. A gap-introducing reduction from SAT to Π comes with two parameters, functions f and α. Given an instance φ of SAT, it outputs, in polynomial time, an instance x of Π, such that • if φ is satisﬁable, OPT(x) ≤ f (x), and • if φ is not satisﬁable, OPT(x) > α(|x|) · f (x). Notice that f is a function of the instance (such as 2 |V | in the example 3 given above), and α is a function of the size of the instance. Since Π is a minimization problem, the function α satisﬁes α(|x|) ≥ 1. If Π is a maximization problem, we want the reduction to satisfy • if φ is satisﬁable, OPT(x) ≥ f (x), and • if φ is not satisﬁable, OPT(x) < α(|x|) · f (x). In this case, α(|x|) ≤ 1. The gap, α(|x|), is precisely the hardness factor established by the gap-introducing reduction for the NP-hard optimization problem. Once we have obtained a gap-introducing reduction from SAT (or any other NP-hard problem) to an optimization problem, say Π1 , we can prove a hardness result for another optimization problem, say Π2 , by giving a special reduction, called a gap-preserving reduction, from Π1 to Π2 . Now there are four possibilities, depending on whether Π1 and Π2 are minimization or maximization problems. We give the deﬁnition below assuming that Π1 is

308

29

Hardness of Approximation

a minimization problem and Π2 is a maximization problem. The remaining cases are similar. A gap-preserving reduction, Γ , from Π1 to Π2 comes with four parameters (functions), f1 , α, f2 , and β. Given an instance x of Π1 , it computes, in polynomial time, an instance y of Π2 such that • • OPT(x) ≤ f1 (x) ⇒ OPT(y) ≥ f2 (y), OPT(x) > α(|x|)f1 (x) ⇒ OPT(y) < β(|y|)f2 (y).

Observe that x and y are instances of two diﬀerent problems, and so it would be more appropriate to write OPTΠ1 (x) and OPTΠ2 (y) instead of OPT(x) and OPT(y), respectively. However, we will avoid this extra notation, since the context clariﬁes the problems being talked about. In keeping with the fact that Π1 is a minimization problem and Π2 is a maximization problem, α(|x|) ≥ 1 and β(|y|) ≤ 1. Composing a gap-introducing reduction with a gap-preserving reduction gives a gap-introducing reduction, provided all the parameters match up. For example, suppose that in addition to the reduction Γ deﬁned above, we have obtained a gap-introducing reduction, Γ , from SAT to Π1 , with parameters f1 and α. Then, composing Γ with Γ , we get a gap-introducing reduction from SAT to Π2 , with parameters f2 and β. This composed reduction shows that there is no β(|y|) factor approximation algorithm for Π2 , assuming P = NP. In each gap-preserving reduction stated below, we will take special care to ensure that the parameters match up. Remark 29.2 • The “gap” β can, in general, be bigger or smaller than α. In this sense, “gap-preserving” is a slight misnomer. • We do not require any guarantee from reduction Γ if instance x of Π1 falls in the ﬁrst gap, i.e., satisﬁes f1 (x) < OPT(x) ≤ α(|x|)f1 (x). • An approximation algorithm for Π2 together with a gap-preserving reduction Γ from Π1 to Π2 does not necessarily yield an approximation algorithm for Π1 . Observe the contrast with an approximation factor preserving reduction (see Section A.3.1 for deﬁnition). The latter reduction additionally requires a means of transforming a near-optimal solution to the transformed instance y of Π2 into a near-optimal solution to the given instance x of Π1 . On the other hand, Γ together with an appropriate gap-introducing reduction from SAT to Π1 does suﬃce for proving a hardness of approximation result for Π2 . Obviously the less stringent requirement on gap-preserving reductions makes them easier to design. • We have already presented some gap-introducing reductions, e.g., Theorems 3.6 and 5.7. The reader may wonder why these do not suﬃce as the starting point for further hardness results and why the PCP theorem was needed. The reason is that these reductions simply exploit the freedom to choose edge costs and not the deep combinatorial structure of the problem.

29.2

The PCP theorem

309

The following ﬁgure shows the gap-preserving reductions presented in this chapter: PCP theorem ❅ ❅

✠ MAX-3SAT ❄ MAX-3SAT(5) ❄ Vertex cover ❄ Steiner tree

❅

❄ Set cover

❅ ❅ ❘ ❅ Clique

29.2 The PCP theorem

Probabilistic characterizations of the class NP yield a general technique for obtaining gap-introducing reductions. The most useful of these characterizations is captured in the PCP theorem. PCP stands for probabilistically checkable proof systems. Recall the usual deﬁnition of NP (see Appendix A) as the class of languages whose yes instances support short (polynomial in the length of the input) witnesses that can be veriﬁed quickly (in polynomial time). Informally, a probabilistically checkable proof for an NP language encodes the witness in a special way so that it can be veriﬁed probabilistically by examining very few of its bits. A probabilistically checkable proof system comes with two parameters, the number of random bits required by the veriﬁer, and the number of bits of the witness that the veriﬁer is allowed to examine. In keeping with established terminology, let us call a witness string the proof. The most useful setting for these parameters is O(log n) and O(1), respectively. This deﬁnes the class PCP(log n, 1). The veriﬁer is a polynomial time Turing machine which, besides its input tape and work tape, has a special tape that provides it with a string of random bits and another special tape on which it is provided with the proof. The machine can read any bit of the proof by simply specifying its location. Of course, the particular locations it examines are a function of the input

310

29

Hardness of Approximation

string and the random string. At the end of its computation, the machine goes into either an accept state or a reject state. ❄ Veriﬁer V ❄ y Input x Proof

❄ Random bits r Work tape

❄

A language L ∈ PCP(log n, 1) if there is a veriﬁer V , and constants c and q, such that on input x, V obtains a random string, r, of length c log |x| and queries q bits of the proof. Furthermore, • if x ∈ L, then there is a proof y that makes V accept with probability 1, • if x ∈ L, then for every proof y, V accepts with probability < 1/2, / where the probability is over the random string r. The probability of accepting in case x ∈ L is called the error probability. / In general, for two functions r(n) and q(n), we can deﬁne the class PCP(r(n), q(n)), under which the veriﬁer obtains O(r(n)) random bits and queries O(q(n)) bits of the proof. The acceptance criteria for input strings are the same as above. In this terminology, NP = PCP(0, poly(n)), where poly(n) = k≥0 {nk }. In this case, the veriﬁer is not allowed any random bits. It must deterministically accept strings in the language and reject strings not in the language, as in the deﬁnition of NP. The PCP theorem gives another characterization of NP. Theorem 29.3 NP = PCP(log n, 1). One half of this theorem, that PCP(log n, 1) ⊆ NP, is easy to prove (see Exercise 29.1). The other half, that NP ⊆ PCP(log n, 1), is a diﬃcult result, and gives a useful tool for establishing hardness of approximation results. The currently known proof of this half is too complicated for exposition in this book. Fortunately, the statement of the theorem is suﬃcient to derive the hardness results. In order to provide the reader with some feel for the PCP theorem, let us make an observation. It is easy to construct a veriﬁer for 3SAT whose error

29.3

Hardness of MAX-3SAT

311

probability (i.e., probability of accepting unsatisﬁable formulae) is ≤ 1−1/m, where m is the number of clauses in the input 3SAT formula, say φ. The veriﬁer expects a satisfying truth assignment to φ as the proof. It uses the O(log n) random bits to pick a random clause of φ. It then reads the truth assignments for the three variables occurring in this clause. Notice that this is only a constant number of bits. It accepts iﬀ the truth setting for these three variables satisﬁes the clause. Clearly, if φ is satisﬁable, there is a proof that makes the veriﬁer accept with probability 1, and if φ is not satisﬁable, on every proof, the veriﬁer accepts with probability ≤ 1 − 1/m. The interesting and diﬃcult part of the PCP theorem is decreasing the error probability to < 1/2, even though the veriﬁer is allowed to read only a constant number of bits of the proof. It involves a complex algebraic construction that ensures that small parts of the proof depend on every bit of the input. The PCP theorem directly gives an optimization problem – in particular, a maximization problem – for which there is no factor 1/2 approximation algorithm, assuming P = NP. Problem 29.4 (Maximize accept probability) Let V be a PCP(log n, 1) veriﬁer for SAT. On input φ, a SAT formula, ﬁnd a proof that maximizes the probability of acceptance of V . Claim 29.5 Assuming P = NP, there is no factor 1/2 approximation algorithm for Problem 29.4. Proof: If φ is satisﬁable, then there is a proof that makes V accept with probability 1, and if φ is not satisﬁable, then on every proof, V accepts with probability < 1/2. Suppose there is a factor 1/2 approximation algorithm for Problem 29.4. If φ is satisﬁable, then this algorithm must provide a proof on which V ’s acceptance probability is ≥ 1/2. The acceptance probability can be computed in polynomial time, by simply simulating V for all random strings of length O(log n). Thus, this approximation algorithm can be used for deciding SAT in polynomial time, contradicting the assumption P = NP. ✷ Claim 29.5 directly gives the following corollary. In subsequent sections, we will use the PCP theorem to obtain hardness results for natural computational problems. A similar corollary follows in each case. Corollary 29.6 Assuming P = NP, there is no PTAS for Problem 29.4.

29.3 Hardness of MAX-3SAT

MAX-3SAT is the restriction of MAX-SAT (see Problem 16.1) to instances in which each clause has at most three literals. This problem plays a similar role in hardness of approximation as 3SAT plays in the theory of NP-hardness,

312

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Hardness of Approximation

as a “seed” problem from which reductions to numerous other problems have been found. The main result of this section is: Theorem 29.7 There is a constant εM > 0 for which there is a gapintroducing reduction from SAT to MAX-3SAT that transforms a Boolean formula φ to ψ such that • if φ is satisﬁable, OPT(ψ) = m, and • if φ is not satisﬁable, OPT(ψ) < (1 − εM )m, where m is the number of clauses in ψ. Corollary 29.8 There is no approximation algorithm for MAX-3SAT with an approximation guarantee of 1 − εM , assuming P = NP, where εM > 0 is the constant deﬁned in Theorem 29.7. The exact solution of MAX-3SAT is shown hard under the assumption P = NP. It is interesting to note that hardness of approximate solution of MAX-3SAT is also being established under the same assumption. For clarity, let us break the proof into two parts. We will ﬁrst prove hardness for the following problem. Problem 29.9 (MAX k-FUNCTION SAT) Given n Boolean variables x1 , . . . , xn and m functions f1 , . . . , fm , each of which is a function of k of the Boolean variables, ﬁnd a truth assignment to x1 , . . . , xn that maximizes the number of functions satisﬁed. Here k is assumed to be a ﬁxed constant. Thus, we have a class of problems, one for each value of k. Lemma 29.10 There is a constant k for which there is a gap-introducing reduction from SAT to MAX k-FUNCTION SAT that transforms a Boolean formula φ to an instance I of MAX k-FUNCTION SAT such that • if φ is satisﬁable, OPT(I) = m, and • if φ is not satisﬁable, OPT(I) < 1 m, 2 where m is the number of formulae in I. Proof: Let V be a PCP(log n, 1) veriﬁer for SAT, with associated parameters c and q. Let φ be an instance of SAT of length n. Corresponding to each string, r, of length c log n (the “random” string), V reads q bits of the proof. Thus, V reads a total of at most qnc bits of the proof. We will have a Boolean variable corresponding to each of these bits. Let B be the set of Boolean variables. Thus, the relevant part of each proof corresponds to a truth assignment to the variables in B. We will establish the lemma for k = q. Corresponding to each string r, we will deﬁne a Boolean function, fr . This will be a function of q variables from B. The acceptance or rejection of V is of course a function of φ, r, and the q bits of the proof read by V . For ﬁxed φ and r, consider the restriction of this function to the q bits of the proof. This is the function fr .

29.4

Hardness of MAX-3SAT with bounded occurrence of variables

313

Clearly, there is a polynomial time algorithm which, given input φ, outputs the m = nc functions fr . If φ is satisﬁable, there is a proof that makes V accept with probability 1. The corresponding truth assignment to B satisﬁes all nc functions fr . On the other hand, if φ is not satisﬁable, then on every proof, V accepts with probability < 1/2. Thus, in this case every truth assignment satisﬁes < 1 nc of these functions. The lemma follows. ✷ 2 Proof of Theorem 29.7: Using Lemma 29.10 we transform a SAT formula φ to an instance of MAX k-FUNCTION SAT. We now show how to obtain a 3SAT formula from the nc functions. Each Boolean function fr constructed in Lemma 29.10 can be written as a SAT formula, say ψr , containing at most 2q clauses. Each clause of ψr contains at most q literals. Let ψ be the SAT formula obtained by taking the conjunct of all these formulae, i.e., ψ = r ψr . If a truth assignment satisﬁes formula fr , then it satisﬁes all clauses of ψr . On the other hand, if it does not satisfy fr , then it must leave at least one clause of ψr unsatisﬁed. Therefore, if φ is not satisﬁable, any truth assignment must leave > 1 nc clauses of ψ unsatisﬁed. 2 Finally, let us transform ψ into a 3SAT formula. This is done using the standard trick of introducing new variables to obtain small clauses from a big clause. Consider clause C = (x1 ∨ x2 ∨ . . . ∨ xk ), with k > 3. Introduce k − 2 new Boolean variables, y1 , . . . , yk−2 , and consider the formula f = (x1 ∨ x2 ∨ y1 ) ∧ (y 1 ∨ x3 ∨ y2 ) ∧ . . . ∧ (y k−2 ∨ xk−1 ∨ xk ). Let τ be any truth assignment to x1 , . . . , xk . If τ satisﬁes C, then it can be extended to a truth assignment satisfying all clauses of f . On the other hand, if τ does not satisfy C, then for every way of setting y1 , . . . , yk−2 , at least one of the clauses of f remains unsatisﬁed. We apply this construction to every clause of ψ containing more than 3 literals. Let ψ be the resulting 3SAT formula. It contains at most nc 2q (q − 2) clauses. If φ is satisﬁable, then there is a truth assignment satisfying all clauses of ψ . If φ is not satisﬁable, > 1 nc of the clauses remain unsatis2 ﬁed, under every truth assignment. Setting εM = 1/(2q+1 (q − 2)) gives the theorem. ✷

29.4 Hardness of MAX-3SAT with bounded occurrence of variables

For each ﬁxed k, deﬁne MAX-3SAT(k) to be the restriction of MAX-3SAT to Boolean formulae in which each variable occurs at most k times. This problem leads to reductions to some key optimization problems.

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Hardness of Approximation

Theorem 29.11 There is a gap preserving reduction from MAX-3SAT to MAX-3SAT(29) that transforms a Boolean formula φ to ψ such that • if OPT(φ) = m, then OPT(ψ) = m , and • if OPT(φ) < (1 − εM )m, then OPT(ψ) < (1 − εb )m , where m and m are the number of clauses in φ and ψ, εM is the constant determined in Theorem 29.7, and εb = εM /43. Proof: The proof critically uses expander graphs. Recall, from Section 20.3, that graph G = (V, E) is an expander if every vertex has the same degree, and for any nonempty subset S ⊂ V , |E(S, S)| > min(|S|, |S|), where E(S, S) denotes the set of edges in the cut (S, S), i.e., edges that have one endpoint in S and the other in S. Let us assume that such graphs are eﬃciently constructible in the following sense. There is an algorithm A and a constant N0 such that for each N ≥ N0 , A constructs a degree 14 expander graph on N vertices in time polynomial in N (Remark 29.12 clariﬁes this point). Expanders enable us to construct the following device whose purpose is to ensure that in any optimal truth assignment, a given set of Boolean variables must have consistent assignment, i.e., all true or all false. Let k ≥ N0 , and let Gx be a degree 14 expander graph on k vertices. Label the vertices with distinct Boolean variables x1 , . . . , xk . We will construct a CNF formula ψx on these Boolean variables. Corresponding to each edge (xi , xj ) of Gx , we will include the clauses (xi ∨ xj ) and (xj ∨ xi ) in ψx . A truth assignment to x1 , . . . , xk is said to be consistent if either all the variables are set to true or all are set to false. An inconsistent truth assignment partitions the vertices of Gx into two sets, say S and S. Assume w.l.o.g. that S is the smaller set. Now, corresponding to each edge in the cut (S, S), ψx will have an unsatisﬁed clause. Therefore, the number of unsatisﬁed clauses, |E(S, S)|, is at least |S| + 1. We will use this fact critically. Next, we describe the reduction. We may assume w.l.o.g. that every variable occurs in φ at least N0 times. If not, we can replicate each clause N0 times without changing the approximability properties of the formula in any essential way. Let B denote the set of Boolean variables occurring in φ. For each variable x ∈ B, we will do the following. Suppose x occurs k ≥ N0 times in φ. Let Vx = {x1 , . . . , xk } be a set of completely new Boolean variables. Let Gx be a degree 14 expander graph on k vertices. Label its vertices with variables from Vx and obtain formula ψx as described above. Finally, replace each occurrence of x in φ by a distinct variable from Vx . After this process is carried out for each variable x ∈ B, every occurrence of a variable in φ is replaced by a distinct variable from the set of new variables

29.4

Hardness of MAX-3SAT with bounded occurrence of variables

315

V = x∈B Vx .

Let φ be the resulting formula. In addition, corresponding to each variable x ∈ B, a formula ψx has been constructed. Finally, let ψ =φ ∧( x∈B ψx ).

Observe that for each x ∈ B, each variable of Vx occurs exactly 29 times in ψ – once in φ , and 28 times in ψx . Therefore, ψ is an instance of MAX3SAT(29). We will say that the clauses of φ are Type I clauses, and the remaining clauses of ψ are Type II clauses. Now, the important claim is that an optimal truth assignment for ψ must satisfy all Type II clauses, and therefore must be consistent for each set Vx , x ∈ B. Suppose that this is not the case. Let τ be an optimal truth assignment that is not consistent for Vx , for some x ∈ B. τ partitions the vertices of Gx into two sets, say S and S, with S being the smaller set. Now, ﬂip the truth assignment to variables in S, keeping the rest of the assignment the same as τ . As a result, some Type I clauses that were satisﬁed under τ may now be unsatisﬁed. Each of these must contain a variable of S, and so their number is at most |S|. On the other hand we get at least |S| + 1 new satisﬁed clauses corresponding to the edges in the cut (S, S). Thus, the ﬂipped assignment satisﬁes more clauses than τ , contradicting the optimality of τ . Let m and m be the number of clauses in φ and ψ. The total number of occurrences of all variables in φ is at most 3m. Each occurrence participates in 28 Type II two-literal clauses, giving a total of at most 42m Type II clauses. In addition, ψ has m Type I clauses. Therefore, m ≤ 43m. If φ is satisﬁable, then so is ψ. Next, consider the case that OPT(φ) < (1 − εM )m, i.e., > εM m clauses of φ remain unsatisﬁed under any truth assignment. If so, by the above claim, > εM m ≥ εM m /43 of the clauses of ψ must remain unsatisﬁed. The theorem follows. ✷ Remark 29.12 The assumption about the eﬃcient construction of expander graphs is slightly untrue. It is known that for each N ≥ N0 , an expander of size ≤ N (1+o(1)) can be constructed eﬃciently (see Section 29.9). The reader can verify that this does not change the status of Theorem 29.11. Exercise 29.4 extends Theorem 29.11 to establishing hardness for MAX3SAT(5).

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Hardness of Approximation

29.5 Hardness of vertex cover and Steiner tree

In this section, we will apply the machinery developed above to some graph theoretic problems. For integer d ≥ 1, let VC(d) denote the restriction of the cardinality vertex cover problem to instances in which each vertex has degree at most d. Theorem 29.13 There is a gap preserving reduction from MAX-3SAT(29) to VC(30) that transforms a Boolean formula φ to a graph G = (V, E) such that • if OPT(φ) = m, then OPT(G) ≤ 2 |V |, and 3 • if OPT(φ) < (1 − εb )m, then OPT(G) > (1 + εv ) 2 |V |, 3 where m is the number of clauses in φ, εb is the constant determined in Theorem 29.11, and εv = εb /2. Proof: Assume w.l.o.g. that each clause of φ has exactly 3 literals (this can be easily accomplished by repeating the literals within a clause, if necessary). We will use the standard transformation. Corresponding to each clause of φ, G has 3 vertices. Each of these vertices is labeled with one literal of the clause. Thus, |V | = 3m. G has two types of edges (see the illustration below): • for each clause, G has 3 edges connecting its 3 vertices, and • for each u, v ∈ V , if the literals labeling u and v are negations of each other, then (u, v) is an edge in G. Each vertex of G has two edges of the ﬁrst type and at most 28 edges of the second type. Hence, G has degree at most 30. We claim that the size of a maximum independent set in G is precisely OPT(φ). Consider an optimal truth assignment and pick one vertex, corresponding to a satisﬁed literal, from each satisﬁed clause. Clearly, the picked vertices form an independent set. Conversely, consider an independent set I in G, and set the literals corresponding to its vertices to be true. Any extension of this truth setting to all variables must satisfy at least |I| clauses. The complement of a maximum independent set in G is a minimum vertex cover. Therefore, if OPT(φ) = m then OPT(G) = 2m. If OPT(φ) < (1−εb )m, then OPT(G) > (2 + εb )m. The theorem follows. ✷ As an illustration, consider the formula (x1 ∨ x2 ∨ x3 ) ∧ (x1 ∨ x2 ∨ x3 ). The graph produced by the reduction given in Theorem 29.13 is given below:

29.5

Hardness of vertex cover and Steiner tree

317

✔ t x2 ✔ ✍

✔ ✔

✎ x1 t ✔❚ ✔ ❚ ❚ ✔ ❚ ✔

x1

❚

❚

❚

❚t x3

✔ ✔ x2 t ✌

✔

✔

✔

t ✔❚ ✔ ❚ ❚ ✔

❚

❚

❚

❚

❚t x3

Theorem 29.14 There is a gap preserving reduction from VC(30) to the Steiner tree problem. It transforms an instance G = (V, E) of VC(30) to an instance H = (R, S, cost) of Steiner tree, where R and S are the required and Steiner vertices of H, and cost is a metric on R ∪ S. It satisﬁes: • if OPT(G) ≤ 2 |V |, then OPT(H) ≤ |R| + 2 |S| − 1, and 3 3 • if OPT(G) > (1 + εv ) 2 |V |, then OPT(H) > (1 + εs )(|R| + 2 |S| − 1), 3 3 where εs = 4εv /97, and εv is the constant determined in Theorem 29.13. Proof: Graph H = (R, S, cost) will be such that G has a vertex cover of size c iﬀ H has a Steiner tree of cost |R| + c − 1. H will have a required vertex re corresponding to each edge e ∈ E and a Steiner vertex sv corresponding to each vertex v ∈ V . The edge costs are as follows. An edge between a pair of Steiner vertices is of cost 1, and an edge between a pair of required vertices is of cost 2. An edge (re , sv ) is of cost 1 if edge e is incident at vertex v in G, and it is of cost 2 otherwise. Let us show that G has a vertex cover of size c iﬀ H has a Steiner tree of cost |R| + c − 1. For the forward direction, let Sc be the set of Steiner vertices in H corresponding to the c vertices in the cover. Observe that there is a tree in H covering R ∪ Sc using cost 1 edges only (since every edge e ∈ E must be incident at a vertex in the cover). This Steiner tree has cost |R| + c − 1. For the reverse direction, let T be a Steiner tree in H of cost |R| + c − 1. We will show below that T can be transformed into a Steiner tree of the same cost that uses edges of cost 1 only. If so, the latter tree must contain exactly c Steiner vertices. Moreover, every required vertex of H must have a unit cost edge to one of these Steiner vertices. Therefore, the corresponding c vertices of G form a cover. Let (u, v) be an edge of cost 2 in T . We may assume w.l.o.g. that u and v are both required. (If u is Steiner, remove (u, v) from T , getting two components. Throw in an edge from v to a required vertex to connect the two sides, and get a Steiner tree of the same cost as T .) Let eu and ev be the edges, in G, corresponding to these vertices. Since G is connected, there is a path, p, from one of the endpoints of eu to one of the endpoints of ev in G. Now, removing (u, v) from T gives two connected components. Let the

318

29

Hardness of Approximation

set of required vertices in these two sets be R1 and R2 . Clearly, u and v lie in diﬀerent sets, so path p must have two adjacent edges, say (a, b) and (b, c) such that their corresponding vertices, say w and w , lie in R1 and R2 , respectively. Let the Steiner vertex, in H, corresponding to b be sb . Now, throwing in the edges (sb , w) and (sb , w ) must connect the two components. Observe that these two edges are of unit cost. Now, if OPT(G) ≤ 2 |V |, then OPT(H) > |R|+ 2 |S|−1, and if OPT(G) > 3 3 (1 + εv ) 2 |V |, then OPT(H) > |R| + (1 + εv ) 2 |S| − 1. The theorem follows. ✷ 3 3 The reduction is illustrated below. Graph G is an instance of the vertex cover problem. The highlighted vertices form a cover. Graph H shows the Steiner tree corresponding to this cover in the reduced graph. Required vertices have been marked with squares, and the three Steiner vertices corresponding to the cover have been marked with circles (the remaining Steiner vertices have been omitted for clarity). The edge between two Steiner vertices in the tree is dotted to distinguish it from the remaining edges, which connect required and Steiner vertices. ✉ .✉ .. .. .. .. .. .. ✉ .. ✉ .. ✚ ✚ .. ✚ ✚ .. ✚ ✚ .. ✚ ✚ .. .. ✚ ✚ .✉ ✉ ✚ ✚ G H

29.6 Hardness of clique

The best approximation algorithms known for some problems, including clique, are extremely weak – to the extent that the solution produced by the best known algorithm is only very slightly better than picking a trivial feasible solution. Recent hardness results have been invaluable in explaining why this is so: these problems are inherently inapproximable (essentially). In this section, we will establish this for clique: Problem 29.15 (Clique) Given an undirected graph G = (V, E) with nonnegative weights on vertices, ﬁnd a clique of maximum weight. A clique in G is a subset of vertices, S ⊆ V , such that for each pair u, v ∈ S, (u, v) ∈ E. Its weight is the sum of weights of its vertices. Consider the cardinality version of this problem, i.e., when all vertex weights are unit. In this section we will show that there is a constant εq > 0, such that there is no 1/(nεq ) factor approximation algorithm for this problem, assuming P = NP. Let us ﬁrst prove the following weaker result.

29.6

Hardness of clique

319

Lemma 29.16 For ﬁxed constants b and q, there is a gap-introducing reduction from SAT to clique that transforms a Boolean formula φ of size n to a graph G = (V, E), where |V | = 2q nb , such that • if φ is satisﬁable, OPT(G) ≥ nb , and • if φ is not satisﬁable, OPT(G) < 1 nb . 2 Proof: Let F be a PCP(log n, 1) veriﬁer for SAT that requires b log n random bits and queries q bits of the proof. We will transform a SAT instance, φ, of size n to a graph G = (V, E) as follows. For each choice of a binary string, r, of b log n bits, and each truth assignment, τ , to q Boolean variables, there is a vertex vr,τ in G. Thus, |V | = 2q nb . Let Q(r) represent the q positions in the proof that F queries when it is given string r as the “random” string. We will say that vertex vr,τ is accepting if F accepts when it is given random string r and when it reads τ in the Q(r) positions of the proof; it is rejecting otherwise. Vertices vr1 ,τ1 and vr2 ,τ2 are consistent if τ1 and τ2 agree at each position at which Q(r1 ) and Q(r2 ) overlap. Clearly, a necessary condition for consistency is that r1 = r2 . Two distinct vertices vr1 ,τ1 and vr2 ,τ2 are connected by an edge in G iﬀ they are consistent and they are both accepting. Vertex vr,τ is consistent with proof p if positions Q(r) of p contain τ . If φ is satisﬁable, there is a proof, p, on which F accepts for each choice, r, of the random string. For each r, let p(r) be the truth setting assigned by proof p to positions Q(r). Now, the vertices {vr,p(r) | |r| = b log n} form a clique in G of size nb . Next, suppose that φ is not satisﬁable, and let C be a clique in G. Since the vertices of C are pairwise consistent, there is a proof, p, that is consistent with all vertices of C. Therefore, the probability of acceptance of F on proof p is at least |C|/nb (notice that the vertices of C must correspond to distinct random strings). Since the probability of acceptance of any proof is < 1/2 the largest clique in G must be of size < 1 nb . ✷ 2 As a consequence of Lemma 29.16, there is no factor 1/2 approximation algorithm for clique assuming P = NP. Observe that the hardness factor established is precisely the bound on the error probability of the probabilistically checkable proof for SAT. By the usual method of simulating the veriﬁer a constant number of times, this can be made 1/k for any constant k, leading to a similar hardness result for clique. In order to achieve the claimed hardness, the error probability needs to be made inverse polynomial. This motivates generalizing the deﬁnition of PCP as follows. Let us deﬁne two additional parameters, c and s, called completeness and soundness, respectively. A language L ∈ PCPc,s [r(n), q(n)] if there is a veriﬁer V , which on input x of length n, obtains a random string of length O(r(n)), queries O(q(n)) bits of the proof, and satisﬁes: • if x ∈ L, there is a proof y that makes V accept with probability ≥ c,

320

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Hardness of Approximation

• if x ∈ L, then for every proof y, V accepts with probability < s. / Thus, the previously deﬁned class PCP[r(n), q(n)] = PCP1, 1 [r(n), q(n)]. In 2 general, c and s may be functions of n. We would like to obtain a PCP characterization of NP which has inverse polynomial soundness. An obvious way of reducing soundness is to simulate a PCP[log n, 1] veriﬁer multiple number of times and accept iﬀ the veriﬁer accepts each time. Simulating k times will reduce soundness to 1/2k ; however, this will increase the number of random bits needed to O(k log n) and the number of query bits to O(k). Observe that the number of vertices in the graph constructed in Lemma 29.16 is 2O(r(n)+q(n)) . To achieve inverse polynomial soundness, k needs to be Ω(log n). For this value of k, the number of bits queried is O(log n), which is not a problem. However, the number of random bits needed is O(log2 n), which leads to a superpolynomial sized graph. The following clever idea overcomes this diﬃculty. We will use a constant degree expander graph to generate O(log n) strings of b log n bits each, using only O(log n) truly random bits. The veriﬁer will be simulated using these O(log n) strings as the “random” strings. Clearly, these are not truly random strings. Properties of expanders help show that they are “almost random” – the probability of error still drops exponentially in the number of times the veriﬁer is simulated. Let H be a constant degree expander on nb vertices, each vertex having a unique b log n bit label. A random walk on H of length O(log n) can be constructed using only O(log n) bits, b log n bits to pick the starting vertex at random and a constant number of bits to pick each successive vertex. (Observe that the random walk is started in the stationary distribution, which is uniform since the graph is regular.) The precise property of expanders we will need is the following. Theorem 29.17 Let S be any set of vertices of H of size < (nb )/2. There is a constant k such that Pr[ all vertices of a k log n length random walk lie in S ] < 1 . n

For intuitive justiﬁcation for Theorem 29.17, observe that a constant fraction of the edges incident at vertices of S have their other end points in S – these help the walk escape from S. The following ﬁgure shows a walk on H that does not lie in S:

29.6

Hardness of clique

321

S

Theorem 29.18

1 NP = PCP1, n [log n, log n]

Proof: We will prove the diﬃcult half,

1 PCP1, 1 [log n, 1] ⊆ PCP1, n [log n, log n], 2

and leave the rest as Exercise 29.5. Let L ∈ PCP1, 1 [log n, 1]. Let F be a 2 veriﬁer for L which requires b log n random bits and queries q bits of the proof, where b and q are constants. 1 Next, we give a PCP1, n [log n, log n] veriﬁer for L, F , which constructs the expander graph H deﬁned above. It then constructs a random walk of length k log n on H, using O(log n) random bits. Both constructions can be accomplished in polynomial time. The label of each vertex on this path speciﬁes a b log n bit string. It uses these k log n + 1 strings as the “random” strings on which it simulates veriﬁer F . F accepts iﬀ F accepts on all k log n+ 1 runs. Consider string x ∈ L, and let p be a proof that makes veriﬁer F accept x with probability 1. Clearly, F , given proof p, also accepts x with probability 1. Hence the completeness of the new proof system is 1. Next, consider string x ∈ L, and let p be an arbitrary proof supplied to F . / When given proof p, veriﬁer F accepts on < (nb )/2 random strings of length b log n. Let S denote the corresponding set of vertices of H, |S| < (nb )/2. Now, F accepts x iﬀ the random walk remains entirely in S. Since the probability of this event is < 1/n, the soundness of F is 1/n. Finally observe that F requires only O(log n) random bits and queries O(log n) bits of the proof. ✷

322

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Theorem 29.19 For ﬁxed constants b and q, there is a gap-introducing reduction from SAT to clique that transforms a Boolean formula φ of size n to a graph G = (V, E), where |V | = nb+q , such that • if φ is satisﬁable, OPT(G) ≥ nb , and • if φ is not satisﬁable, OPT(G) < nb−1 .

1 Proof: Let F be a PCP1, n [log n, log n] veriﬁer for SAT that requires b log n random bits and queries q log n bits of the proof. The transformation of SAT instance φ to graph G is exactly as in Lemma 29.16. The only diﬀerence is that the increased number of bits queried results in a larger number of vertices. The correctness of the construction also along the lines of Lemma 29.16. If φ is satisﬁable, let p be a good proof, and pick the nb vertices of G that are consistent with p, one for each choice of the random string. These vertices will form a clique in G. Furthermore, any clique C in G gives rise to a proof that is accepted by F with probability ≥ |C|/nb . Since the soundness of F is ✷ 1/n, if φ is not satisﬁable, the largest clique in G is of size < nb−1 .

Corollary 29.20 There is no 1/(nεq ) factor approximation algorithm for the cardinality clique problem, assuming P = NP, where εq = 1/(b + q), for constants b and q deﬁned in Theorem 29.19.

29.7 Hardness of set cover

As stated in Chapter 2, the simple greedy algorithm for the set cover problem, which is perhaps the ﬁrst algorithmic idea one would attempt, has remained essentially the best algorithm. Since set cover is perhaps the single most important problem in the theory of approximation algorithms, a lot of eﬀort was expended on obtaining an improved algorithm. In this section, we will present the remarkable result that the approximation factor of this algorithm is tight up to a constant multiplicative factor. Improved hardness results show that it is tight up to lower order terms as well (see Section 29.9). This should put to rest nagging doubts about the true approximability of this central problem. 29.7.1 The two-prover one-round characterization of NP

Observe that for the purpose of showing hardness of MAX-3SAT and clique (Theorems 29.7 and 29.19), we did not require a detailed description of the kinds of queries made by the veriﬁer – we only required a bound on the number of queries made. In contrast, this time we do need a description, and moreover, we want to ﬁrst establish that a particularly simple veriﬁer

29.7

Hardness of set cover

323

suﬃces. For this purpose, we will introduce a new model for probabilistically checkable proofs, the two-prover one-round proof system. This model is best understood by thinking of the proof system as a game between the prover and the veriﬁer. The prover is trying to cheat – it is trying to convince the veriﬁer that a “no” instance for language L is actually in L. Is there a veriﬁer that can ensure that the probability of getting cheated is < 1/2 for every “no” instance? In the two-prover model, the veriﬁer is allowed to query two noncommunicating provers, denoted P1 and P2 . Since the veriﬁer can cross-check the provers’ answers, the provers’ ability to cheat gets restricted in this model. In turn, we will impose restrictions on the veriﬁer as well, and thereby obtain a new characterization of NP. Under a one-round proof system, the veriﬁer is allowed only one round of communication with each prover. The simplest way of formalizing this is as follows. We will assume that the two proofs are written in two alphabets, say Σ1 and Σ2 . In general, the sizes of these alphabets may be unbounded and may depend on the size of the input. The veriﬁer is allowed to query one position in each of the two proofs. The two-prover one-round model comes with three parameters: completeness, soundness and the number of random bits provided to the veriﬁer, denoted by c, s and r(n), respectively. This deﬁnes the class 2P1Rc,s (r(n)). A language L is in 2P1Rc,s (r(n)) if there is a polynomial time bounded veriﬁer V that receives O(r(n)) truly random bits and satisﬁes:

∗ ∗ • for every input x ∈ L, there is a pair of proofs y1 ∈ Σ1 and y2 ∈ Σ2 that makes V accept with probability ≥ c, ∗ ∗ • for every input x ∈ L and every pair of proofs y1 ∈ Σ1 and y2 ∈ Σ2 , V / accepts with probability < s.

The PCP theorem implies, and in fact is equivalent to, the fact that there is a gap-introducing reduction from SAT to MAX-3SAT(5) (see Theorem 29.7 and Exercises 29.3 and 29.4). We will use this to show: Theorem 29.21 There is a constant εP > 0 such that NP = 2P1R1,1−εP (log(n)). Proof: We will establish the diﬃcult half, i.e., NP ⊆ 2P1R1,1−εP (log(n)), and leave the rest as Exercise 29.7. Clearly, it is suﬃcient to show that SAT ∈ 2P1R1,1−εP (log(n)). As a result of Theorem 29.7 and Exercise 29.4, there is gap-introducing reduction from SAT to MAX-3SAT(5)2 . More precisely, there is a constant ε5 > 0 for which there is a reduction Γ from SAT to MAX-3SAT(5) that transforms a Boolean formula φ to ψ such that • if φ is satisﬁable, OPT(ψ) = m, and

2

The bounded occurrence version of MAX-3SAT is not essential for this theorem; however, we will require it in the main reduction.

324

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Hardness of Approximation

• if φ is not satisﬁable, OPT(ψ) < (1 − ε5 )m, where m is the number of clauses in ψ. The two-prover one-round veriﬁer, V , for SAT works as follows. Given a SAT formula φ, it uses the above stated reduction to obtain a MAX-3SAT(5) instance ψ. It assumes that P1 contains an optimal truth assignment, τ , for ψ and P2 contains, for each clause, the assignment to its three Boolean variables under τ (hence, |Σ1 | = 2 and |Σ2 | = 23 ). It uses the O(log n) random bits to pick a random clause, C, from ψ, and further, a random Boolean variable, x, occurring in C. V obtains the truth assignments to x and the three variables in C by querying P1 and P2 , respectively. It accepts iﬀ C is satisﬁed and the two proofs agree on their assignment for x. If φ is satisﬁable, then so is ψ. Clearly, there are proofs y1 and y2 such that V accepts with probability 1. Next assume that φ is not satisﬁable. Any truth assignment to ψ must leave strictly more than ε5 fraction of the clauses unsatisﬁed. Consider any pair of proofs (y1 , y2 ). Interpret y1 as a truth assignment, say τ . The random clause, C, picked by V is not satisﬁed by τ with probability > ε5 . If so, and if the assignment for C contained in y2 is satisfying, then y1 and y2 must be inconsistent. In the latter case, the veriﬁer catches this with probability ≥ 1/3. Hence overall, V must reject with probability > ε5 /3. ✷ Remark 29.22 Using standard techniques (see Exercise 29.8), Γ can be modiﬁed to ensure that the instance of MAX-3SAT(5) produced satisﬁes the following uniformity conditions: each Boolean variable occurs in exactly 5 clauses and each clause contains 3 distinct variables (negated or unnegated). This modiﬁcation changes the constant ε5 to some other constant, say ε5 > 0. These uniformity conditions will be needed in the main reduction. Remark 29.23 As a result of the uniformity conditions, if ψ has n variables, then it has 5n/3 clauses. Therefore, the two proofs are of length n and 5n/3, respectively. For carrying out the main reduction, it will be important to ensure that the two proofs are of equal length. This can be easily achieved by repeating the ﬁrst proof 5 times and the second proof 3 times. The veriﬁer will query a random copy of each proof. It is easy to verify that Theorem 29.21 still holds (even though the “copies” may be diﬀerent). 29.7.2 The gadget

The following set system will be a basic gadget in the main reduction: (U, C1 , . . . , Cm , C 1 , . . . , C m ), where U is the universal set and C1 , . . . , Cm are subsets of U . Clearly, U can be covered by picking a set Ci and its complement C i . Such a cover will be called a good cover. A cover that does not include a set and its complement will be called a bad cover. The following theorem, which can be proven using the probabilistic method (see Exercise 29.9), shows the existence of such set systems for which the sizes of good and

29.7

Hardness of set cover

325

bad covers are widely diﬀerent. Moreover, they can be constructed eﬃciently, with high probability. Theorem 29.24 There exists a polynomial p(., .) such that there is a randomized algorithm which generates, for each m and l, a set system (U, C1 , . . . , Cm , C 1 , . . . , C m ), with |U | = p(m, 2l ). With probability > 1/2 the gadget produced satisﬁes that every bad cover is of size > l. Moreover, the running time of the algorithm is polynomial in |U |. A good cover is well coordinated – it involves picking a set Ci and its complement. Acceptance in the two-prover one-round proof system also involves coordination – on random string r, the veriﬁer queries the two proofs and accepts iﬀ the answers are coordinated. The choice of this proof system, for establishing hardness of set cover, should be more convincing in light of this observation. 29.7.3 Reducing error probability by parallel repetition

Before presenting the reduction, we would like improve the soundness of the two-prover one-round proof system for SAT. The usual way of accomplishing this is parallel repetition: The veriﬁer picks k clauses randomly and independently, and a random Boolean variable from each of the clauses. It queries P1 on the k variables and P2 on the k clauses, and accepts iﬀ all answers are accepting. One would expect that probability that the provers manage to cheat drops to < (1 − εP )k . Surprisingly enough, this is not true. Since each prover is allowed to look at all k questions before providing its k answers, it may be able to coordinate its answers and thereby cheat with a higher probability. Example 29.25 illustrates this in a simple setting. If the provers are required to answer each question before being given the next question, the probability of error drops in the usual fashion; however, this requires k rounds of communication and falls outside the two-prover one-round model. Example 29.25 Consider the following setting in which the two noncommunicating provers are attempting to agree on a random bit. The veriﬁer gives random, independent bits r1 and r2 to P1 and P2 , respectively. The protocol succeeds if the two provers manage to commit to one of the two bits, i.e., either both provers output (1, r1 ) or both provers output (2, r2 ); the ﬁrst element of a pair says whose bit the provers are outputting and the second element is the bit itself. Since P1 does not know r2 and P2 does not know r1 , the probability of their succeeding is 1/2. Now consider parallel repetitions of this protocol. The veriﬁer gives two bits, r1 and s1 , to P1 and two bits, r2 and s2 , to P2 . The four bits are random

326

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Hardness of Approximation

and independent. The provers succeed iﬀ they can commit to one of the r’s and one of the s’s. One would expect the probability of success to be 1/4. However, by cleverly coordinating answers, the provers can make it 1/2 as follows. The answers of P1 are (1, r1 ) and (2, r1 ), and those of P2 are (1, s2 ) and (2, s2 ). The provers succeed iﬀ r1 = s2 , which happens with probability 1/2. ✷ Despite this diﬃculty, one can still prove that the probability of error does drop exponentially with k. However, the proof of this useful fact is not easy. Theorem 29.26 Let the error probability of a two-prover one-round proof system be δ < 1. Then the error probability on k parallel repetitions is at most δ dk , where d is a constant that depends only on the length of the answers of the original proof system. 29.7.4 The reduction

We will prove the following. Theorem 29.27 There is a constant c > 0 for which there is a randomized gap-introducing reduction Γ , requiring time nO(log log n) , from SAT to the cardinality set cover problem that transforms a Boolean formula φ to a set system S over a universal set of size nO(log log n) such that • if φ is satisﬁable, OPT(S) = 2nk , and • if φ is not satisﬁable, Pr[OPT(G) > cnk k log n] > 1/2, where n is the length of each of the two proofs for SAT under the two-prover one-round model (see Remark 29.23); n is polynomial in the size of φ. The parameter k is O(log log n). Remark 29.28 This is slight abuse of notation, since gap-introducing reductions were deﬁned to run in polynomial time. Proof: Let V be the two-prover one-round veriﬁer for SAT, described in Theorem 29.21. Assume further that the MAX-3SAT(5) formula produced by V satisﬁes the uniformity conditions stated in Remark 29.22 and that the two proofs queried by V are of equal length, say n, as stated in Remark 29.23. Denote by ψ the MAX-3SAT(5) formula produced by V when given SAT formula φ. Let V be a two-prover one-round veriﬁer that executes k parallel repetitions of V , as described in Section 29.7.3. Now, each of the proofs is of length nk . Each position of P1 contains a truth assignment to k Boolean variables (not necessarily distinct) and each position of P2 contains a truth assignment to the 3k Boolean variables occurring in k clauses. Thus, proofs P1 and P2 are written in alphabets Σ1 and Σ2 whose sizes are 2k and 23k , respectively. k will be ﬁxed to be O(log log n) for reasons clariﬁed below.

29.7

Hardness of set cover

327

Veriﬁer V uses random bits provided to it to pick k random clauses of ψ, and a random Boolean variable from each of these k clauses, thereby specifying a position in P1 and a position in P2 . These involve picking from one of nk and 3k choices, respectively. Therefore, the total number of random strings is (3n)k . Denote by Q1 (r) and Q2 (r) the positions in P1 and P2 , respectively, speciﬁed by random string r. Suppose the answers in positions Q1 (r) and Q2 (r) are a and b, respectively. Recall that V accepts on random string r iﬀ b satisﬁes all k clauses picked, and a and b assign the same truth values to the k chosen variables. Given r and the answer in Q2 (r), say b, the “acceptable” answer in Q1 (r) is uniquely speciﬁed. Let projection function π(r, b) denote this answer. ✛ nk positions

✲ ✻

✛ ✻ k bits

nk positions

✲

b

3k bits

a

❄

i = Q1 (r) P1 P2

j = Q2 (r)

❄

The parameters m and l for the gadget are ﬁxed as follows. We will set m = |Σ1 | = 2k , and l = O(k log n) = O(log n log log n). Let (U, C1 , . . . , C2k , C 1 , . . . , C 2k ) be the gadget with these parameters. Thus, corresponding to each answer a ∈ Σ1 , we have a unique set Ca . As stated in Theorem 29.24, |U | = p(2k , 2l ) = nO(log log n) , and the gadget can be constructed probabilistically in time polynomial in |U |. The gadget will be constructed once, and as stated in Theorem 29.24, will satisfy the chosen parameters with probability > 1/2. For the rest of the proof, assume that it does. We will make (3n)k copies of the gadget over disjoint universal sets. Each copy corresponds to a random string. Denote the r r r r copy corresponding to random string r to be (U r , C1 , . . . , C2k , C 1 , . . . , C 2k ). The reduction Γ transforms φ to a set cover instance S as follows. The universal set U= r U r,

where the union is over all (3n)k random strings. Clearly, |U| = |U |(3n)k = nO(log log n) . The subsets of U speciﬁed by S are of two kinds. First, corresponding to each position i in P1 and answer a ∈ Σ1 , there is a set Si,a = r:Q1 (r)=i r Ca ,

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Hardness of Approximation

where the union is over all random strings r such that Q1 (r) = i. Second, corresponding to each position j in P2 and answer b ∈ Σ2 , there is a set Sj,b . If b does not satisfy all k clauses of ψ, speciﬁed by position Q2 (r), then Sj,b = ∅. Otherwise, Sj,b = r:Q2 (r)=j

C π(r,b) ,

r

where the union is over all random strings r such that Q2 (r) = j. Let r be a random string, and let Q1 (r) = i and Q2 (r) = j. Then, the only sets in S that contain elements of U r are: • Si,a , for a ∈ Σ1 , and • Sj,b , for b ∈ Σ2 such that b satisﬁes the k clauses speciﬁed by position j in P2 . r r Moreover, each set of the ﬁrst type contains exactly one set from C1 , . . . , C2k r r and each set of the second type contains exactly one set from C 1 , . . . , C 2k . Let r be a random string, and let Q1 (r) = i and Q2 (r) = j. Observe that Si,a ∪ Sj,b covers U r iﬀ π(r, b) = a and b satisﬁes the k clauses speciﬁed by position j in P2 . Let C be a cover for U. If C contains such a pair of sets then we will say that C contains a good cover for U r . If C does not contain a good cover for U r , then it must contain > l sets of the form Si,a , Sj,b , a ∈ Σ1 , b ∈ Σ2 in order to cover U r . In this case, we will say that C contains a bad cover for U r. Suppose φ is satisﬁable. Then there is a pair of proofs (y1 , y2 ) on which the veriﬁer accepts with probability 1. Let us pick a cover C as follows. Corresponding to each position i in P1 and j in P2 pick sets Si,a and Sj,b , where a and b are the answers for these queries in y1 and y2 , respectively. Hence, |C| = 2nk . It is easy to see that C contains a good cover for each set U r . Next suppose that φ is not satisﬁable. Now, V will reject any pair of proofs with high probability. We have assumed that the gadget found satisﬁes the chosen parameters; this happens with probability > 1/2. Let C denote an optimal cover for U. Is C forced to contain a bad cover for U r , for most random strings r? Clearly, C is allowed to pick sets corresponding to portions of many diﬀerent proofs. Using this added capability, can we not construct a cover that is only slightly larger than 2nk ? A set from S helps cover elements from several diﬀerent universes U r , making the rest of the argument more involved. Below we will give a procedure for constructing, from C, a pair of proofs, (y1 , y2 ), in such a way that if |C| is small, then V must accept this pair with high probability. Hence, we will derive the desired lower bound on |C|. Consider the set of answers picked by C for each position of the two proofs. For each position i in P1 , deﬁne A(i) = {a | Si,a ∈ C}, and for each position j in P2 , deﬁne A(j) = {b | Sj,b ∈ C}. Construct proofs y1

29.8

Exercises

329

and y2 by picking for each position i in P1 and j in P2 a random element of A(i) and A(j), respectively. If any of the answer sets is empty, pick an arbitrary answer for that position. Deﬁne B1 = {r | |A(Q1 (r))| > l/2}, B2 = {r | |A(Q2 (r))| > l/2} and G = B1 ∪ B2 . Thus, G is the set of random strings r for which C picks at most l/2 answers each for Q1 (r) and Q2 (r). Hence, C contains a good cover for U r , say Si,a ∪ Sj,b , where a ∈ A(Q1 (r)) and b ∈ A(Q2 (r)). The pair of proofs, (y1 , y2 ), contain a and b in positions Q1 (r) and Q2 (r), respectively, with probability ≥ ( 2 )2 . Hence V , when given proofs (y1 , y2 ), accepts on random l string r with at least this probability. Let fG denote the fraction of random strings contained in G. Then, using Theorem 29.26, fG 2 l

2

≤ Pr[V accepts φ when given proofs (y1 , y2 )] ≤ δ dk .

Hence, fG ≤ δ dk l2 /4. Since l2 is O(log4 n), by picking k = O(log log n) we can ensure that fG < 1/2. As a result, B1 ∪ B2 contains at least half the random strings, and therefore one of these sets contains at least a quarter. Denote this set by Bi . Because of the uniformity property (Remark 29.22), if r is chosen at random, then Q1 (r) is a random position in P1 and Q2 (r) is a random position in P2 (although they will be correlated). Furthermore, r has probability > 1/4 of being in Bi . Therefore, the answer sets of > 1/4 of the positions of Bi are of cardinality > l/2. Hence the size of the cover > lnk /8 = Ω(nk k log n). ✷ As a consequence of Theorem 29.27, inapproximability of set cover modulo NP not being in a one-sided-error complexity class with running time nO(log log n) follows directly. Standard techniques from complexity theory (see Exercise 1.18) lead to the following slightly stronger result. Corollary 29.29 There is a constant b such that if there is a b log n factor approximation algorithm for the cardinality set cover problem, where n is the size of the universal set of the set cover instance, then NP ⊆ ZTIME(nO(log log n) ) (see Section A.4 for deﬁnition).

29.8 Exercises

29.1 Show that PCP(log n, 1) ⊆ NP. Hint: Let L ∈ PCP(log n, 1). The NP machine for accepting L guesses the proof, simulates the veriﬁer for L on all O(log n) length random strings, and accepts iﬀ the veriﬁer accepts on all the random strings.

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Hardness of Approximation

29.2 Show (see Appendix A for deﬁnitions): 1. PCP(0, 0) = PCP(0, log n) = P. 2. PCP(poly(n), 0) = co-RP, where poly(n) = k≥0 nk . 3. PCP(log n, 1) = PCP(log n, poly). Hint: NP ⊆ PCP(log n, 1) ⊆ PCP(log n, poly) ⊆ NP. 29.3 Show the converse of Theorem 29.7, i.e., if there is a gap-introducing reduction from SAT to MAX-3SAT, then NP ⊆ PCP(log n, 1). Hint: Reduce the given SAT formula φ to an instance ψ of MAX-3SAT. The veriﬁer expects, as proof, an optimal truth assignment to ψ. This gives an error probability of 1 − εM . Repeat to decrease the error probability to < 1/2. 29.4 Give a gap-preserving reduction from MAX-3SAT(29) to MAX3SAT(5), with appropriate parameters, to show hardness for the latter problem. Hint: The reduction is similar, though easier, than that in Theorem 29.11. Instead of using an expander graph, use a cycle. Now, an inconsistent assignment can gain as many as 14 clauses corresponding to each old variable x. However, it must leave at least two clauses, corresponding to edges of the cycle of x, unsatisﬁed. 29.5 Complete the proof of Theorem 29.18. 29.6 (Hastad [122]) An important consideration, while obtaining a PCP characterization of NP, is reducing the number of bits of the proof that the veriﬁer needs to query. The following remarkable result reduces it to just 3. Theorem 29.30 For every ε > 0, NP = PCP1−ε, 1 +ε [log n, 1]. 2 Moreover, there is a particularly simple PCP veriﬁer for SAT. It uses the O(log n) random bits to compute three positions in the proof, say i, j and k, and a bit b, and accepts iﬀ y(i) + y(j) + y(k) ≡ b (mod 2).

Here y(i) is the ith bit in the proof y. 1. Consider the restriction of Problem 16.12 (Exercise 16.7), linear equations over GF[2], in which each equation has exactly 3 variables. Use the characterization stated in Theorem 29.30 to give an appropriate gapintroducing reduction from SAT to this problem which shows that if, for any ε > 0, there is a 2 − ε factor approximation algorithm for the latter problem then P = NP.

29.8

Exercises

331

2. Give an appropriate gap-preserving reduction from linear equations over GF[2] to MAX-3SAT which shows that if, for any ε > 0, there is a 8/7 − ε factor approximation algorithm for MAX-3SAT then P = NP. Hint: The equation xi + xj + xk ≡ 0 (mod 2) is transformed into the clauses (xi ∨ xj ∨ xk ) ∧ (xi ∨ xj ∨ xk ) ∧ (xi ∨ xj ∨ xk ) ∧ (xi ∨ xj ∨ xk ). 29.7 Complete the proof of Theorem 29.21, i.e., show that 2P1R1,1−εP (log(n)) ⊆ NP. 29.8 Prove the uniformity conditions stated in Remark 29.22. Hint: Use the standard technique of introducing new Boolean variables. 29.9 Prove Theorem 29.24 using the probabilistic method. Hint: p(m, 2l ) = O(m22l ) suﬃces. Pick each set Ci by including each element of U in it randomly and independently with probability 1/2. 29.10 (Feige [80]) The following stronger hardness result can be established for set cover: Theorem 29.31 For any constant δ > 0, if there is a (1 − δ) ln n factor approximation algorithm for the cardinality set cover problem, where n is the size of the universal set of the set cover instance, then NP ⊆ DTIME(nO(log log n) ), where DTIME(t) is the class of problems for which there is a deterministic algorithm running in time O(t). Consider the maximum coverage problem, Problem 2.18 in Exercise 2.15. Using Theorem 29.31 show that if there is an ε > 0 for which there is a (1 − 1/e + ε) factor approximation algorithm for the maximum coverage problem, then NP ⊆ DTIME(nO(log log n) ). Hint: Use the maximum coverage algorithm to obtain a (1 − δ) ln n factor algorithm for set cover, for some δ > 0, as follows: Guess k, the optimal number of sets needed for the given instance. Run the maximum coverage algorithm, with parameter k, iteratively, until a cover is found. In each iteration, a (1 − 1/e + ε) fraction of the uncovered elements is covered. Therefore, the number of iterations, l, satisﬁes, (1/e − ε)l = 1/n. 29.11 (Jain, Mahdian, and Saberi [138]) Using Theorem 29.31 show that if there is an ε > 0 for which there is a (1 + 2/e − ε) factor approximation algorithm for the metric k-median problem, Problem 25.1, then NP ⊆ DTIME(nO(log log n) ).

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Hardness of Approximation

29.9 Notes

The ﬁrst hardness of approximation result based on probabilistically checkable proofs was due to Feige, Goldwasser, Lov´sz, Safra, and Szegedy [82]. a This work motivated the discovery of the PCP theorem, which additionally builds on work on interactive proof systems (Babai [18] and Goldwasser, Micali, and Rackoﬀ [109]) and program checking (Blum and Kannan [28] and Blum, Luby, and Rubinfeld [29]), and is due to Arora and Safra [14], and Arora, Lund, Motwani, Sudan, and Szegedy [14, 12]. Theorem 29.30, which yields optimal inapproximability results for several problems, is due to Hastad [122]. Before this development, the pioneering work of Papadimitriou and Yannakakis [218] had established evidence of inapproximability of several natural problems using their notion of Max-SNP-completeness. Gap preserving reductions are weaker than their L-reductions. Consequently, the ideas behind their reductions carry over directly to the new development, as in the reductions given in Theorems 29.11 and 29.13. Indeed, one of the motivations for the PCP theorem was that establishing an inapproximability result for MAX SAT would directly yield inapproximability results for all Max-SNP-hard problems. Theorem 29.14 is from Bern and Plassmann [25]. The construction of expander graphs is due to Lubotzky, Phillips, and Sarnak [197]. Theorem 29.17 is due to Impagliazzo and Zuckerman [135]. Theorem 29.19 on hardness of clique follows from [82] and [14, 12]. The current best hardness result for clique, due to Hastad [121], states that it cannot be approximated within a factor of n1−ε for any ε > 0, unless NP = ZPP. This is quite close to the best approximation algorithm, due to Boppana and Holld´rsson [30], achieving a guarantee of O(n/(log2 n)). o Lund and Yannakakis [199] gave the ﬁrst hardness result for set cover, showing that it cannot be approximated within a factor of log n/2 unless NP ⊆ ZTIME(nO(polylog n) ). The improved result, presented in Theorem 29.31, is due to Feige [80]. This enhancement comes about by using a k prover proof system. A deterministic construction of the set system gadget of Theorem 29.24, due to Naor, Schulman, and Srinivasan [211], allows replacing ZTIME by DTIME in the complexity assumption. The two-prover oneround proof system was deﬁned by Ben-or, Goldwasser, Kilian, and Wigderson [24]. Theorem 29.26 is due to Raz [231]. Karloﬀ and Zwick [162] give an algorithm for MAX-3SAT that achieves an approximation guarantee of 8/7 when restricted to satisﬁable formulae. This complements the hardness result stated in Exercise 29.6. For further information on this topic, see the survey by Arora and Lund [11]. For an up-to-date status of the best positive and negative results known for numerous NP-hard optimization problems, see the excellent compendium maintained online at http://www.nada.kth.se/˜viggo/problemlist/compendium.html

29.9

Notes

333

The compendium also appears in Ausiello, Crescenzi, Gambosi, Kann, Marchetti-Spaccamela, and Protasi [17].

30 Open Problems

This chapter is centered around problems and issues currently in vogue in the ﬁeld of approximation algorithms. Important new issues are bound to arise in the future. With each of these problems two questions arise – that of obtaining the best approximation guarantee and a matching hardness result1

30.1 Problems having constant factor algorithms

Since a large number of important open problems in the ﬁeld today involve improving the guarantee for problems for which we already know constant factor algorithms, we found it convenient to present them in a separate section. Of course, we are not looking for small improvements using incremental means. A good model is Goemans and Williamson’s improvement to the MAX-CUT problem, from factor 1/2 to 0.878, which introduced semideﬁnite programming into the repertoire of techniques in this ﬁeld. Most of the problems listed below have the potential of extending known methods in signiﬁcant ways and introducing important new ideas. Vertex cover, Problem 1.1: Improve on factor 2 (see algorithms in Chapters 1, 2, 14, and 15). Semideﬁnite programming may be a possible avenue, see, e.g., the attempt by Goemans and Kleinberg [103]. Set cover, Problem 2.1: This question generalizes the previous one. Consider the restriction of the set cover problem to instances in which the frequency of each element is bounded by a ﬁxed constant f . Improve on factor f (see algorithms in Chapters 2, 14, and 15). The best hardness result known is f 1/19 , assuming P = NP, due to Trevisan [247]. Acyclic subgraph, Problem 1.9: Improve on factor 1/2 (see Exercise 1.1). Semideﬁnite programming may be applicable. Metric TSP, Problem 3.5: As stated in Exercise 23.13, the solution produced by Christoﬁdes’ algorithm (Algorithm 3.10) is within a factor of

1

For an up-to-date status of the best positive and negative results known for numerous NP-hard optimization problems, see the excellent compendium at http://www.nada.kth.se/˜viggo/problemlist/compendium.html

30.1

Problems having constant factor algorithms

335

3/2 of the subtour elimination LP-relaxation for this problem. However, the worst integrality gap example known is (essentially) 4/3. Can a 4/3 factor algorithm be obtained using this relaxation? Christoﬁdes’ algorithm consists of two steps: obtaining an MST and patching up its odd degree vertices. The above stated result follows by bounding the cost of each of these steps individually. It might be a good idea to ﬁrst look for a “one–shot” factor 3/2 algorithm which compares the entire solution to the LP-relaxation. The primal–dual schema may hold the key. Steiner tree, Problem 3.1: The best approximation guarantee known is essentially 5/3 (see Exercise 22.12). A promising avenue for obtaining an improved guarantee is to use the bidirected cut relaxation (22.7). This relaxation is exact for the minimum spanning tree problem. For the Steiner tree problem, the worst integrality gap known is (essentially) 8/7, due to Goemans (see Exercise 22.11). The best upper bound known on the integrality gap is 3/2 for quasi-bipartite graphs (graphs that do not contain edges connecting pairs of Steiner vertices), due to Rajagopalan and Vazirani [226]. Determine the integrality gap of this relaxation and obtain an algorithm achieving this guarantee2 . Recall that in contrast, LP-relaxation (22.2) has an integrality gap of (essentially) 2, not only for this problem, but also for its special case, the minimum spanning tree problem, and its generalization, the Steiner network problem. Steiner network, Problem 23.1: Chapter 23 gives a factor 2 algorithm. However, it uses LP-rounding and has a prohibitive running time. Obtain a factor 2 combinatorial algorithm for this problem. A corollary of Algorithm 23.7 is that the integrality gap of LP-relaxation (23.2) is bounded by 2. Therefore, this relaxation can be used as a lower bound for obtaining a factor 2 combinatorial algorithm. The primal–dual schema appears to be the most promising avenue. A starting point may be determining if the following is true: For each instance of the Steiner forest problem (and more generally, the Steiner network problem) there is an integral primal solution x and dual feasible solution y such that each edge picked by x is tight w.r.t. the dual y and each raised dual S has degree ≤ 2 (≤ 2f (S)). Observe that the dual found by Algorithm 22.3 can have arbitrarily high degree. Multiway cut, Problem 4.1: A 1.5 factor is presented in Chapter 19. As stated, this can be improved to 1.3438. However, the worst integrality gap example known for LP-relaxation (19.1) is (essentially) 8/7. Determine the integrality gap of this relaxation, and obtain an algorithm achieving

2

A more general issue along these lines is to clarify the mysterious connection between the integrality gap of an LP-relaxation and the approximation factor achievable using it.

336

30

Open Problems

this guarantee. A diﬀerent relaxation is presented in Exercise 19.7. How are the two relaxations related? Are they equivalent in that any feasible solution to one be converted to a solution of the other of the same cost? Subset feedback vertex set, Problem 19.15: The best factor known is 8, via a fairly complicated algorithm (see Exercise 19.13). Is a factor 2 algorithm possible, matching several of the other related problems stated in Exercise 19.13?

30.2 Other optimization problems

Shortest vector, Problem 27.1: Obtain a polynomial factor algorithm for this problem. As shown in Chapter 27, the dual lattice helps give a factor n co-NP certiﬁcate for this problem. Is the dual lattice of further algorithmic use? The best hardness result known for this problem, √ of factor 2 − ε, for any ε > 0, assuming RP = NP, is due to Micciancio [204]. Sparsest cut, Problem 21.2: The best approximation factor known is O(log n) (see Chapter 21). However, no hardness of approximation results have been established for this problem – as far as we know a PTAS not yet ruled out. Is there a constant factor algorithm or a PTAS for this problem? Minimum b-balanced cut and minimum bisection cut, Problem 21.27: An O(log2 n) factor algorithm for both these problems was given by Feige and Krauthgamer [83]. As in the case of sparsest cut, a PTAS is not yet ruled out for these problems. Is there a constant factor algorithm or a PTAS for these problems? When restricted to planar graphs, the minimum b-balanced cut problem, for b ≤ 1/3, can be approximated within a factor of 2, see Garg, Saran, and Vazirani [95]. Minimum multicut, 18.1: An O(log n) factor algorithm is given in Chapter 20. A long standing open problem is whether there is a constant factor deterministic algorithm for this problem. Asymmetric TSP, Problem 3.15: The best factor known is O(log n) (see Exercise 3.6). Is there a constant factor algorithm for this problem? Vertex-connectivity network design: This variant of the Steiner network problem (Problem 23.1) asks for a minimum cost subgraph containing ru,v vertex-disjoint paths, instead of edge-disjoint paths, for each pair of vertices u, v ∈ V . No nontrivial approximation algorithms are known for this variant. For the special case when ru,v = k for each pair of vertices u, v ∈ V and the edge costs obey the triangle inequality, a (2 + 2(k−1) ) factor algorithm is given by Khuller and Raghavachari [170]. n A problem of intermediate diﬃculty is the element-connectivity network

30.2

Other optimization problems

337

design problem, in which vertices are partitioned into two sets: terminals and non-terminals. Only edges and non-terminals, referred to as elements, can fail. Only pairs of terminals have connectivity requirements, specifying the number of element-disjoint paths required. An algorithm with an approximation guarantee of factor 2Hk , where k is the largest requirement, is given by Jain, M˘ndoiu, Vazirani, and Williamson [139]. a Maximum integer multicommodity ﬂow, Problem 18.3: Example 18.8 shows that the natural LP-relaxation has an integrality gap of Ω(n). It is easy to get around this diﬃculty while still retaining the essence of the original problem by asking for a maximum half-integral ﬂow. Is there an O(log n) factor algorithm for this latter problem? Metric uncapacitated facility location and k-median, Problems 24.1 and 25.1: Determine the integrality gaps of the LP-relaxations (24.2) and (25.2). Capacitated facility location problem, Exercise 24.8: As stated in Exercise 24.8 the modiﬁcation of LP (24.2) to this problem has unbounded integrality gap. Is there some other lower bounding method that leads to a good approximation algorithm? Directed multicut and sparsest cut: In Chapters 20 and 21 we considered two generalizations of the undirected maximum ﬂow problem and derived approximation algorithms for the corresponding cut problems, multicut and sparsest cut. Not much is known at present about analogous problems in directed graphs. Directed Steiner tree, Problem 3.14: As shown in Exercise 3.3 this problem is unlikely to have a better approximation guarantee than O(log n). Is a guarantee of O(log n) possible? The best guarantee known is nε for any ﬁxed ε > 0, due to Charikar et. al. [37]. Generalizations of this problem to higher connectivity requirements, analogous to the Steiner network problem, also need to be studied. Directed feedback edge (vertex) set: Given a directed graph G = (V, E), a feedback edge (vertex) set is a set of edges (vertices) whose removal leaves an acyclic graph. The problem is to ﬁnd the minimum cardinality such set. More generally, consider the weighted version in which the edges (vertices) have assigned weights, and we want to ﬁnd the minimum weight such set. It is easy to see that the edge and vertex versions are inter-reducible via approximation factor preserving reductions. An O(log n log log n) factor approximation algorithm is known for the weighted version, due to Seymour [238]. Can this be improved to O(log n) or even a constant factor? Cover time: Given an undirected graph G = (V, E), the cover time starting at vertex v ∈ V , C(v) is the expected number of steps taken by a random

338

30

Open Problems

walk on G, which starts at v and visits all vertices. The cover time of G is maxv∈V C(v). Clearly, a randomized algorithm can estimate the cover time to any desired accuracy by empirically simulating the random walk many times and taking the average. Kahn, Kim, Lov´sz, and Vu a [151] have given an O((log log n)2 ) factor deterministic algorithm for this problem. Is a constant factor deterministic algorithm possible?

30.3 Counting problems

For the problems presented below (other than graphs with given degree sequence and triangulations), the decision version is in P, the counting version is #P-complete, and the complexity of approximately counting the number of solutions is unresolved. The complexity of counting the number of graphs with given degree sequence and triangulations is open, though conjectured to be #P-complete. Perfect matchings in general graphs: When restricted to planar graphs, this problem is polynomial time solvable using the classic algorithm of Kastelyn [168]. This result extends to K3,3 -free graphs (graphs that do not contain a subgraph homeomorphic to K3,3 ) as well, see Little [191] and Vazirani [251]. A FPRAS is known for the restriction of this problem to bipartite graphs, which is the same as the problem of evaluating a 0/1 permanent, due to Jerrum, Sinclair, and Vigoda [143] (more generally, this work gives a FPRAS for evaluating the permanent of a square matrix with nonnegative integer entries). Volume of a convex body: Given a convex body in Rn via an oracle, the problem is to estimate its volume. A number of other counting problems can be reduced to this fundamental problem. The ﬁrst FPRAS for this problem was given by Dyer, Frieze, and Kannan [67]. Although polynomial, the running time of this algorithm was exorbitant. It required O∗ (n23 ) oracle calls – the “soft-O” notation of O∗ suppresses factors of log n as well as ε, the error bound. The current best algorithm, due to Kannan, Lov´sz, and Simonovits [155] requires O∗ (n5 ) oracle calls and a O∗ (n7 ) arithmetic operations. Can the running time be further improved? Acyclic orientations: Count the number of acyclic orientations of a given undirected graph G. An orientation of the edges of G is acyclic if the resulting directed graph is acyclic. Several Markov chains on the set of acyclic orientations are known that asymptotically converge to the uniform distribution; however, none of them is known to be rapidly mixing. For instance, say that two orientations are adjacent if one can be obtained from the other by ﬂipping directions of the edges incident at a source or a sink, where a source has all outgoing edges and a sink has all incoming edges. Do a random walk on this graph.

30.3

Counting problems

339

Forests: A forest in an undirected graph is a set of edges that contain no cycles. A maximal forest is a spanning tree (assume the graph is connected). Interestingly enough, the problem of counting the number of spanning trees in a graph is in P – this being one of the very few counting problems known to be polynomial time solvable. This follows as a consequence of the classic matrix tree theorem of Kirchhoﬀ, see [194]. It is worth remarking that elegant polynomial time algorithms are known for generating a random spanning tree in an undirected graph using rapidly mixing Markov chains, due to Aldous [3], Broder [34], and Wilson [259]. On the other hand, the complexity of approximately counting forests in arbitrary graphs is open. The case of dense graphs (each vertex having degree at least αn, for 0 < α < 1) is handled by Annan [8]. Forests and spanning trees are the independent sets and bases, respectively, of the graphic matroid of the given graph. Bases of a matroid: Given an arbitrary matroid via an independence oracle, count the number of bases. Deﬁne the basis exchange graph of a matroid as follows. Its vertices are all bases. Two bases are adjacent iﬀ their symmetric diﬀerence is two elements. The Markov chain deﬁned by a random walk on the basis exchange graph is conjectured to be rapidly mixing by Dagum, Luby, Mihail, and Vazirani [56]. If so, a FPRAS for approximately counting the number of bases will follow. Examples of matroids for which this conjecture has been positively resolved are graphic matroids (see previous problem) and their generalization, balanced matroids. For the latter result, see Feder and Mihail [78]. A positive resolution of this question will also resolve the question of approximately counting forests (since forests of any particular size are bases of a truncation of the graphic matroid). Network reliability: Many versions of the network reliability problem have found practical applications and have been studied in the past. Two basic versions for undirected graphs with edge failure probabilities are s–t reliability, which asks for the probability that special vertices s and t get disconnected, and global reliability, which asks for the probability that any part of the graph gets disconnected. One can deﬁne two analogous problems in directed graphs as well. Of these four problems, only undirected global reliability is settled – a FPRAS for this version is presented in Chapter 28. In addition, for each of these four cases one can also ask for the probability that s–t or the entire graph remain connected. This version is open even for the undirected global case. Euler tours: Count the number of Euler tours of a given undirected graph (a connected graph is Eulerian iﬀ all vertices have even degrees). Interestingly enough, there is a polynomial time algorithm for the analogous problem for directed graphs – again following from Kirchhoﬀ’s Theorem.

340

30

Open Problems

Trees: Given an undirected graph G, count the number of subgraphs of G that are trees. Antichains in a partial order: See Exercise 1.7 for the deﬁnition. For the related problem of counting the number of total orders consistent with a partial order, a FPRAS is known, due to Matthews [202], Karzanov and Khachian [167], and Bubley and Dyer [35]. Graphs with given degree sequence: Given n nonnegative integers d1 , . . . , dn , which represent the degrees of the n vertices, v1 , . . . , vn , of a simple graph, count the number of such graphs. A related problem is to count the number of connected graphs having this degree sequence. In both cases, the question of existence of one such graph can be solved in polynomial time using a matching algorithm. If G is restricted to be a bipartite graph, with the bipartition speciﬁed, then a FPRAS follows from that for 0/1 permanents [143]. Contingency tables: Given the row sums and column sums of an m × n matrix with nonnegative integer entries, count the number of such matrices. A FPRAS is known if the row sums and column sums are all suﬃciently large, being at least (m + n)mn, due to Dyer, Kannan, and Mount [64]. Morris [208] improves this to the case where each row sum is Ω(n3/2 m log m) and each column sum is Ω(m3/2 n log n). If the matrices are constrained to be 0/1, this is same as the degree sequence problem restricted to bipartite graphs, for which a FPRAS follows from that for 0/1 permanents [143]. Triangulations: Compute the number of triangulations of n points on the plane, i.e., the number of ways of putting down non–intersecting line segments connecting pairs of points so that all internal faces are triangles. Consider the graph G on all possible triangulations whose edges are deﬁned as follows: Remove an edge in a triangulation t that is not on the inﬁnite face. If the resulting quadrilateral is convex, let t be the triangulation obtained by adding an edge connecting the other two points of this quadrilateral. Then, G has an edge connecting t and t . A random walk on this graph is conjectured to be rapidly mixing. If the n points form the vertices of a convex n-gon, then the number of triangulations is known to be the Catalan number Cn−2 , and hence polynomial time computable. For this special case, the Markov chain deﬁned above is known to be rapidly mixing, see McShine and Tetali [203]. Stable marriages: An instance of the stable marriage problem consists of n boys and n girls, together with an ordered list of the preferences of each boy and each girl (each boy orders all n girls and each girl orders all n boys). A marriage is a perfect matching of the boys and girls. Boy b and girl g who are not married to each other are said to form a rogue couple if b prefers g to the girl he is married to and g prefers b to the boy she

30.3

Counting problems

341

is married to. The marriage is stable if there are no rogue couples. The complexity of approximately counting the number of stable marriages is unresolved. For numerous structural properties of the set of stable marriages, see Gusﬁeld and Irving [119]. Colorings of a graph: Consider an undirected graph G = (V, E) with maximum degree ∆. Jerrum [144] gave a FPRAS for counting the number of valid k-colorings of G for any k > 2∆, and Vigoda [254] extended this to any k > 11∆/6. Can this be improved to counting the number of valid k-colorings of G for any k ≥ ∆ + 2? (If the number of colors is ≤ ∆ + 1 then the natural Markov chain, that at each step picks a random vertex and recolors it with a random consistent color, may not be connected.) This quantity ﬁnds applications in statistical physics. Hamiltonian cycles: If each vertex of an undirected graph G has degree at least n/2 then G must have a Hamiltonian cycles (see Dirac’s condition in [194]). If the minimum degree is (1/2+ε)n, for ε > 0, Dyer, Frieze, and Jerrum [65] have given a FPRAS for this problem. Can this be extended to ε = 0, i.e., graphs having minimum degree n/2? Independent sets: For graphs having ∆ = 4, a FPRAS was given by Luby and Vigoda [198], where ∆ denotes the maximum degree of the graph. Dyer, Frieze, and Jerrum [66] show that the problem is not approximable for ∆ ≥ 25, assuming RP = NP. They also give an argument to show that the Markov chain Monte Carlo is unlikely to succeed for ∆ ≥ 6. Besides the question of ∆ = 5, this leaves the question of determining whether other methods will work for 6 ≤ ∆ ≤ 24 or whether these cases are also inapproximable. Tutte polynomial: Several of the problems stated above are special cases of evaluating the Tutte polynomial of the given graph G = (V, E) at a particular point of the (x, y)-plane. For A ⊆ E, deﬁne the rank of A, denoted r(A), to be |V | − k(A), where k(A) is the number of connected components in the graph having vertex set V and edge set A. The Tutte polynomial of G at point (x, y) is T (G; x, y) =

A⊆E

(x − 1)r(E)−r(A) (y − 1)|A|−r(A) .

Some of the natural quantities captured by this polynomial are: • At (1, 1), T counts the number of spanning trees in G. • At (2, 1), T counts the number of forests in G. • At (1, 2), T counts the number of connected subgraphs of G. • At (2, 0), T counts the number of acyclic orientations of G. • At (0, 2), T counts the number of orientations of G that form a strongly connected digraph.

342

30

Open Problems

• The chromatic polynomial of G is given by P (G, λ) = (−1)r(E) λk(E) T (G; 1 − λ, 0), where P (G, λ) is the number of colorings of G using λ colors. • If the failure probability of each edge is p, then the probability that G remains connected is given by R(G; p) = q |E|−r(E) pr(E) T (G; 1, 1/(1 − p)). Vertigan and Welsh [253] have shown that other than a few special points and two special hyperbolae (see next problem for deﬁnition), the exact evaluation of the Tutte polynomial is #P-hard. The question of designing FPRAS’s is wide open. Say that a graph is α-dense if each vertex has degree ≥ αn, where 0 < α < 1. Annan [7] and Alon, Frieze, and Welsh [5] have given FPRAS’s for α-dense graphs for the cases y = 1, x ≥ 1 and y > 1, x ≥ 1, respectively. Partition functions of the Ising and Potts models: The hyperbolae Hα deﬁned by Hα = {(x, y) | (x − 1)(y − 1) = α} play a special role in the context of the Tutte polynomial. In particular, along H2 , T gives the partition function of the Ising model for G, and along HQ , for integer Q ≥ 2, T gives the partition function of the Potts model for G. Both these quantities ﬁnd use in statistical physics; see Welsh [256] for precise deﬁnitions and further details (the points on each hyperbola are parametrized by “temperature” and Q represents the number of “color” classes). Jerrum and Sinclair [146] gave a FPRAS for estimating, at any temperature, the partition function of the Ising model of a graph, and Randall and Wilson [228] extended this to a polynomial time sampling procedure. However, because of large exponents in the running times, these algorithms are not practical. The Swendsen-Wang process [245] provides a natural and practically used Markov chain for estimating these quantities. This leads to the question of determining, formally, whether this chain is rapidly mixing. A negative result was provided by Gore and Jerrum [112] who show that this chain is not rapidly mixing for the complete graph, Kn , for Q ≥ 3. Positive results for certain classes of graphs were provided by Cooper and Frieze [53]. Is this chain rapidly mixing for the partition function of the Ising model for an arbitrary graph? Is there some other way of estimating the partition function of the Potts model for an arbitrary graph?

A An Overview of Complexity Theory for the Algorithm Designer

A.1 Certiﬁcates and the class NP

A decision problem is one whose answer is either “yes” or “no”. Two examples are: SAT: Given a Boolean formula in conjunctive normal form, f , is there is a satisfying truth assignment for f ? Cardinality vertex cover: Given an undirected graph G and integer k, does G have a vertex cover of size ≤ k? For any positive integer k, we will denote by kSAT the restriction of SAT to instances in which each clause contains at most k literals. It will be convenient to view a decision problem as a language, i.e., a subset of {0, 1}∗ . The language consists of all strings that encode “yes” instances of the decision problem. A language L ∈ NP if there is a polynomial p and a polynomial time bounded Turing machine M , called the veriﬁer, such that for each string x ∈ {0, 1}∗ : • if x ∈ L, then there is a string y (the certiﬁcate) of polynomially bounded length, i.e., |y| ≤ p(|x|), such that M (x, y) accepts, and • if x ∈ L, then for any string y, such that |y| ≤ p(|x|), M (x, y) rejects. /

M

Certiﬁcate ❄ y

Input and work tape ❄ x

344

An overview of complexity theory

String y that helps ascertain that x is a “yes” instance will be called a Yes certiﬁcate. We will also refer to y as a proof or a solution; in the context of randomized computation, it is also referred to as a witness. Thus, NP is the class of languages that have “short, quickly veriﬁable” Yes certiﬁcates. For example, the veriﬁer for cardinality vertex cover assumes that y speciﬁes a subset of the vertices. It checks whether this subset is indeed a vertex cover and is of the desired size bound. (Observe that no claim has been made about the time needed to actually ﬁnd such a certiﬁcate.) It is also easy to see that the class NP deﬁned above is precisely the class of languages that are decidable by nondeterministic polynomial time Turing machines (see Section A.6 for references), hence the name. A language L belongs to the class co-NP iﬀ L ∈ NP. Thus, co-NP is the class of languages that have “short, quickly veriﬁable” No certiﬁcates. For instance, let L be the language consisting of all prime numbers. This language allows No certiﬁcates: a factorization for number n is proof that n ∈ L. Hence / L ∈ co-NP. Interestingly enough, L ∈ NP as well (see Exercise 1.13), though it is not known to belong to P.

A.2 Reductions and NP-completeness

Next, let us introduce the crucial notion of a polynomial time reduction. Let L1 and L2 be two languages in NP. We will say that L1 reduces to L2 , and write L1 L2 , if there is a polynomial time Turing machine T which given a string x ∈ {0, 1}∗ , outputs string y such that x ∈ L1 iﬀ y ∈ L2 . In general, T does not have to decide whether x is a “yes” or a “no” instance in order to output y. Clearly, if L1 L2 and L2 is polynomial time decidable, then so is L1 . L. A A language L is NP-hard if for every language L ∈ NP, L language L is NP-complete if L ∈ NP, and L is NP-hard. An NP-complete language L is a hardest language in NP, in the sense that a polynomial time algorithm for L implies a polynomial time algorithm for every language in NP, i.e., it implies P = NP. The central theorem of complexity theory gives a proof of NP-hardness for a natural problem, namely SAT. The idea of the proof is as follows. Let L be an arbitrary language in NP. Let M be a nondeterministic polynomial time Turing machine that decides L, and let p be the polynomial bounding the running time of M . The proof involves showing that there is a deterministic polynomial time Turing machine T , that “knows” M and p, and given a string x ∈ {0, 1}∗ , outputs a SAT formula f such that each satisfying truth assignment of f encodes an accepting computation of M on input x. Thus, f is satisﬁable iﬀ there is an accepting computation of M on input x, i.e., iﬀ x ∈ L. Once one problem, namely SAT, has been shown to be NP-hard, the hardness of other natural problems can be established by simply giving poly-

A.3

NP-optimization problems and approximation algorithms

345

nomial time reductions from SAT to these problems (see Exercise 1.11). Perhaps the most impressive feature of the theory of NP-completeness is the ease with which the latter task can be accomplished in most cases, so that with relatively little work, a lot of crucial information is obtained. Other than a handful of (important) problems, most natural problems occurring in NP have been classiﬁed as being either in P or being NP-complete. Indeed, it is remarkable to note that other basic complexity classes, deﬁned using notions of time, space and nondeterminism, also tend to have natural complete problems (under suitably deﬁned reducibilities). Establishing NP-hardness for vertex cover involves giving a polynomial time algorithm that, given a SAT formula f , outputs an instance (G, k) such that G has a vertex cover of size ≤ k iﬀ f is satisﬁable. As a corollary, we get that under the assumption P = NP, there is no polynomial time algorithm that can distinguish “yes” instances of vertex cover from “no” instances. As stated above, this also shows that if P = NP, there is no polynomial time algorithm for solving vertex cover exactly. Considering the large and very diverse collection of NP-complete problems, none of which has yielded to a polynomial time algorithm for so many years, it is widely believed that P = NP, i.e., that there is no polynomial time algorithm for deciding an NP-complete language. The P = NP conjecture has a deep philosophical point to it. The conjecture asserts that the task of ﬁnding a proof for a mathematical statement is qualitatively harder than the task of simply verifying the correctness of a given proof for the statement. To see this, observe that the language L = {(S, 1n ) | statement S has a proof of length ≤ n} is in NP, assuming any reasonable axiomatic system.

A.3 NP-optimization problems and approximation algorithms

Combinatorial optimization problems are problems of picking the “best” solution from a ﬁnite set. An NP-optimization problem, Π, consists of: • A set of valid instances, DΠ , recognizable in polynomial time. We will assume that all numbers speciﬁed in an input are rationals, since our model of computation cannot handle inﬁnite precision arithmetic. The size of an instance I ∈ DΠ , denoted by |I|, is deﬁned as the number of bits needed to write I under the assumption that all numbers occurring in the instance are written in binary. • Each instance I ∈ DΠ has a set of feasible solutions, SΠ (I). We require that SΠ (I) = ∅, and that every solution s ∈ SΠ (I) is of length polynomially bounded in |I|. Furthermore, there is polynomial time algorithm that, given a pair (I, s), decides whether s ∈ SΠ (I).

346

An overview of complexity theory

• There is a polynomial time computable objective function, objΠ , that assigns a nonnegative rational number to each pair (I, s), where I is an instance and s is a feasible solution for I. The objective function is frequently given a physical interpretation, such as cost, length, weight, etc. • Finally, Π is speciﬁed to be either a minimization problem or a maximization problem. The restriction of Π to unit cost instances will be called the cardinality version of Π. An optimal solution for an instance of a minimization (maximization) problem is a feasible solution that achieves the smallest (largest) objective function value. OPTΠ (I) will denote the objective function value of an optimal solution to instance I. We will shorten this to OPT when it is clear that we are referring to a generic instance of the particular problem being studied. With every NP-optimization problem, one can naturally associate a decision problem by giving a bound on the optimal solution. Thus, the decision version of NP-optimization problem Π consist of pairs (I, B), where I is an instance of Π and B is a rational number. If π is a minimization (maximization) problem, then the answer to the decision version is “yes” iﬀ there is a feasible solution to I of cost ≤ B (≥ B). If so, we will say that (I, B) is a “yes” instance; we will call it a “no” instance otherwise. For example, the decision version of cardinality vertex cover is stated in Section A.1. Clearly, a polynomial time algorithm for Π can help solve the decision version – by computing the cost of an optimal solution and comparing it with B. Conversely, hardness established for the decision version carries over to Π. Indeed hardness for an NP-optimization problem is established by showing that its decision version is NP-hard. With a slight abuse of notation, we will also say that the optimization version is NP-hard. An approximation algorithm produces a feasible solution that is “close” to the optimal one, and is time eﬃcient. The formal deﬁnition diﬀers for minimization and maximization problems. Let Π be a minimization (maximization) problem, and let δ be a function, δ : Z+ → Q+ , with δ ≥ 1 (δ ≤ 1). An algorithm A is said to be a factor δ approximation algorithm for Π if, on each instance I, A produces a feasible solution s for I such that fΠ (I, s) ≤ δ(|I|) · OPT(I) (fΠ (I, s) ≥ δ(|I|) · OPT(I)), and the running time of A is bounded by a ﬁxed polynomial in |I|. Clearly, the closer δ is to 1, the better is the approximation algorithm. On occasion we will relax this deﬁnition and will allow A to be randomized, i.e., it will be allowed to use the ﬂips of a fair coin. Assume we have a minimization problem. Then we will say that A is a factor δ randomized approximation algorithm for Π if, on each instance I, A produces a feasible solution s for I such that Pr[fΠ (I, s) ≤ δ(|I|) · OPT(I)] ≥ 1 , 2

A.4

Randomized complexity classes

347

where the probability is over the coin ﬂips. The running time of A is still required to be polynomial in |I|. The deﬁnition for a maximization problem is analogous. Remark A.1 Even though δ has been deﬁned to be a function of the size of the input, we will sometimes pick δ to be a function of a more convenient parameter. For instance, for the set cover problem (Chapter 2), we will pick this parameter to be the number of elements in the ground set. A.3.1 Approximation factor preserving reductions

Typically, polynomial time reductions map optimal solutions to optimal solutions; however, they do not preserve near-optimality of solutions. Indeed, all NP-complete problems are equally hard from the viewpoint of obtaining exact solutions. However, from the viewpoint of obtaining near-optimal solutions, they exhibit the rich set of possibilities alluded to earlier. In this book we will encounter pairs of problems which may look quite diﬀerent superﬁcially, but whose approximability properties are closely linked (e.g., see Exercise 19.13). Let us deﬁne a suitable reducibility in order to formally establish such connections. Several reductions have been deﬁned that preserve constant factor approximability. The reducibility stated below is a stringent version of these, and actually preserves the constant itself. Pair of problems that are linked in this manner are either both minimization problems or both maximization problems. Let Π1 and Π2 be two minimization problems (the deﬁnition for two maximization problems is quite similar). An approximation factor preserving reduction from Π1 to Π2 consists of two polynomial time algorithms, f and g, such that • for any instance I1 of Π1 , I2 = f (I1 ) is an instance of Π2 such that OPTΠ2 (I2 ) ≤ OPTΠ1 (I1 ), and • for any solution t of I2 , s = g(I1 , t) is a solution of I1 such that objΠ1 (I1 , s) ≤ objΠ2 (I2 , t). It is easy to see that this reduction, together with an α factor algorithm for Π2 , gives an α factor algorithm for Π1 (see Exercise 1.16).

A.4 Randomized complexity classes

Certain NP languages1 are characterized by the fact that they possess an abundance of Yes certiﬁcates, which renders them essentially tractable, assuming availability of a source of random bits. Such languages belong to the

1

The deﬁnitions of this section will be useful in Chapter 29.

348

An overview of complexity theory

class RP, short for Randomized Polynomial Time. A language L ∈ RP if there is a polynomial p and a polynomial time bounded Turing machine M such that for each string x ∈ {0, 1}∗ : • if x ∈ L, then M (x, y) accepts for at least half the strings y of length p(|x|), and • if x ∈ L, then for any string y of length p(|x|), M (x, y) rejects. / Clearly, P ⊆ RP ⊆ NP. Suppose language L ∈ RP. On input x, we will pick a random string, y, of length p(|x|) and will run M (x, y). Clearly, the entire computation takes polynomial time. We may erroneously reject x even though x ∈ L. However, the probability of this is at most 1/2. Let us call this the error probability. By the usual trick of making repeated independent runs, we can reduce the error probability to inverse exponential in the number of runs. A language L belongs to the class co-RP iﬀ L ∈ RP. Such a language has an abundance of No certiﬁcates. The corresponding machine may make an error on inputs x ∈ L. Finally, let us deﬁne ZPP, short for Zero-error / Probabilistic Polynomial Time, to be the class of languages for which there is a randomized Turing machine (i.e., a Turing machine equipped with a source of random bits) that always terminates with the correct answer and whose expected running time is polynomial. It is easy to see (Exercise 1.17) that L ∈ ZPP iﬀ L ∈ (RP ∩ co-RP). DTIME(t) denotes the class of problems for which there is a deterministic algorithm running in time O(t). Thus, P = DTIME(poly(n)), where poly(n) = k≥0 nk . ZTIME(t) denotes the class of problems for which there is a randomized algorithm running in expected time O(t). Thus, ZPP = ZTIME(poly(n)).

A.5 Self-reducibility

Most known problems in NP exhibit an interesting property, called selfreducibility, which yields a polynomial time algorithm for ﬁnding a solution (a Yes certiﬁcate), given an oracle for the decision version. A slightly more elaborate version of this property yields an exact polynomial time algorithm for an NP-optimization problem, again given an oracle for the decision version. In a sense this shows that the diﬃcult core of NP and NP-optimization problems is their decision versions (see Section 16.2 and Exercise 28.7 for other fundamental uses of self-reducibility). Perhaps the simplest setting to describe self-reducibility is SAT. Let φ be a SAT formula on n Boolean variables x1 , . . . , xn . We will represent a truth assignment to these n variables as n-bit 0/1 vectors (True = 1 and False = 0). Let S be the set of satisfying truth assignments, i.e., solutions, to φ.

A.5

Self-reducibility

349

The important point is that for the setting of x1 to 0 (1), we can ﬁnd, in polynomial time, a formula φ0 (φ1 ) on the remaining n − 1 variables whose solutions, S0 (S1 ), are precisely solutions of φ having x1 = 0 (x1 = 1). Example A.2 Suppose φ = (x1 ∨x2 ∨x3 )∧(x1 ∨x2 ∨x4 ). Then φ0 = (x2 ∨x3 ) ✷ and φ1 = (x2 ∨ x4 ) Using this property, an oracle for the decision version of SAT can be used to ﬁnd a solution to φ, assuming it is satisﬁable, as follows. First check whether φ0 is satisﬁable. If so, set x0 = 0, and ﬁnd any solution to φ0 . Otherwise, set x1 = 1 (in this case φ1 must be satisﬁable), and ﬁnd a solution to φ1 . In each case the problem has been reduced to a smaller one, and we will be done in n iterations. The following representation will be particularly useful. Let T be a binary tree of depth n whose leaves are all n-bit 0/1 strings, representing truth assignments to the n variables. Leaves that are solutions to φ are marked special. The root of T is labeled with φ and its internal nodes are labeled with formulae whose solutions are in one-to-one correspondence with the marked leaves in the subtree rooted at this node. Thus, the 0th child of the root is labeled with φ0 and the 1st child is labeled with φ1 . Tree T is called the self-reducibility tree for instance φ. φ s ✑ ◗ ◗ ✑ ◗ ✑ ◗ φ ✑ ◗ 1 φ0 ✑ ◗s ✑ s ✁❆ ✁❆ ✁❆ ✁❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ n−1 ❆n n n−1

0 ... 01 10 ... 1

We will formalize the notion of self-reducibility for NP-optimization problems. Formalizing this notion for NP problems is an easier task and is left as Exercise 1.15. First, let us illustrate self-reducibility for cardinality vertex cover. Observe that an oracle for the decision version enables us to compute the size of the optimal cover, OPT(G), by binary search on k. To actually ﬁnd an optimal cover, remove a vertex v together with its incident edges to obtain graph G , and compute OPT(G ). Clearly, v is in an optimal cover iﬀ OPT(G ) = OPT(G) − 1. Furthermore, if v is in an optimal cover, then any optimal cover in G , together with v, is an optimal cover in G. Otherwise, any optimal cover

350

An overview of complexity theory

for G must contain all neighbors, say N (v), of v (in order to cover all edges incident at v). Let G be the graph obtained by removing v and N (v) from G. Any optimal cover in G , together with N (v), is an optimal cover in G. Thus, in both cases, we are left with the problem of ﬁnding an optimal cover in a smaller graph, G or G . Continuing this way, an optimal cover in G can be found in polynomial time. The above-stated reduction from the cardinality vertex cover problem to its decision version works because we could demonstrate polynomial time algorithms for • obtaining the smaller graphs, G and G , • computing the size of the best cover in G, consistent with the atomic decision, and • constructing an optimal cover in G, given an optimal cover in the smaller instance. The exact manner in which self-reducibility manifests itself is quite different for diﬀerent problems. Below we state a fairly general deﬁnition that covers a large number of problems. In the interest of conveying the main idea behind this important concept, we will provide an intuitive, though easily formalizable, deﬁnition. We will assume that solutions to an instance I of NP-optimization problem Π have granularity, i.e., consist of smaller pieces called atoms that are meaningful in the context of the problem. For instance, for cardinality vertex cover, the atoms consist of specifying whether or not a certain vertex is in the cover. Clearly, for vertex cover this can be done using O(log n) bits. Indeed, all problems considered in this book have granularity O(log n). Let us assume this for problem Π. I✲ A α✲ I✲ I✲ α✲ s✲ f s✲

We will say that problem Π is self-reducible if there is a polynomial time algorithm, A, and polynomial time computable functions, f (·, ·, ·) and g(·, ·, ·), satisfying the following conditions. • Given instance I and an atom α of a solution to I, A outputs an instance Iα . We require that |Iα | < |I|. Let S(I | α) represent the set of feasible solutions to I that are consistent with atom α. We require that the feasible solutions of Iα , S(Iα ), are in one-to-one correspondence with S(I | α). This correspondence is given by the polynomial time computable function f (·, ·, ·) as follows.

A.6

Notes

351

f (I, α, ·) : S(Iα ) → S(I | α). • The correspondence f (I, α, ·) preserves order in the objective function values of solutions. Thus, if s1 and s2 are two feasible solutions of Iα with objΠ (Iα , s1 ) ≤ objΠ (Iα , s2 ), and f (I, α, s1 ) = s1 and f (I, α, s2 ) = s2 , then objΠ (I, s1 ) ≤ objΠ (I, s2 ). • Given the cost of an optimal solution to Iα , the cost of the best solution in S(I | α) can be computed eﬃciently, and is given by g(I, α, OPT(Iα )). Theorem A.3 Let Π be an NP-optimization problem that is self-reducible. There is a polynomial time (exact) algorithm for Π, given an oracle, O, for the decision version of Π. Proof: As remarked earlier, via a suitable binary search we can use O to compute the cost of the optimal solution to an instance in polynomial time. We will derive polynomial time algorithm R for solving Π exactly. Assume that A, f , and g are deﬁned as above for the self-reducibility of Π. Let I be an instance of Π. R ﬁrst ﬁnds one atom of an optimal solution to I. An atom, say β, satisﬁes this condition iﬀ g(I, β, OPT(Iβ )) = OPT(I), where Iβ = A(I, β). Since atoms are only O(log n) bits long, ﬁnding such an atom involves simply searching the polynomially many possibilities. Let α be the atom found, and let Iα = A(I, α). R then recursively computes an optimal solution, say s , to Iα . Finally, it outputs f (I, α, s ), which is guaranteed to be an optimal solution to I. Since |Iα | < |I|, the recursion also takes only polynomial time. ✷ Remark A.4 The number of strings of length O(log n) that algorithm R needs to examine for ﬁnding a good atom depends on the speciﬁc problem. For instance, in the case of cardinality vertex cover we picked an arbitrary vertex, say v, and considered only two atoms, that v is or isn’t in the cover.

A.6 Notes

The deﬁnition of an NP-optimization problem is due to Krentel [178]. Approximation factor preserving reductions are a stringent version of Lreducibility from Papadimitriou and Yannakakis [218]. Self-reducibility was ﬁrst deﬁned by Schnorr [234]. See Khuller and Vazirani [171] for a problem that is not self-reducible, assuming P = NP. For further information on NP-completeness and complexity theory see Garey and Johnson [93] and Papadimitriou [216].

B Basic Facts from Probability Theory

Let us recall some useful facts from probability theory. We assume that the reader has already had a detailed exposure to this material (see Section B.4 for references).

B.1 Expectation and moments

Two quantities provide much information about a random variable: the mean, also called expectation, and variance. A key property of the expectation, which often simpliﬁes its evaluation, is called linearity of expectation. It states that if X, X1 , . . . , Xn are random variables such that X = c1 X1 + . . . + cn Xn , where c1 , . . . , cn are constants, then E[X] = c1 E[X1 ] + . . . + cn E[Xn ]. (In particular, the expectation of a sum of random variables is the sum of their expectations.) The usefulness of this property arises from the fact that no assumption is made about independence between the random variables X1 , . . . , Xn . Often a complex random variable can be written as the sum of indicator random variables (i.e., random variables taking on 0/1 values only), thereby simplifying the evaluation of its expectation. The variance of random variable X measures the spread of X from its mean, and is deﬁned as V [X] = E[(X − E[X])2 ] = E[X 2 ] − E[X]2 . Its positive square root is called the standard deviation. The mean and standard deviation of X are denoted by µ(X) and σ(X), respectively. For k ∈ N, the kth moment and kth central moment of X are deﬁned to be E[X k ] and E[(X − E[X])k ], respectively. Thus the variance is the second central moment. In general, the expectation of the product of random variables is not the product of expectations. An important exception is when the random variables are independent. Thus, if X and Y are independent random variables, then E[XY ] = E[X]E[Y ]. This immediately implies that the variance of the sum of independent random variables is the sum of their variances, i.e., for independent random variables X and Y , V [X + Y ] = V [X] + V [Y ].

B.2

Deviations from the mean

353

B.2 Deviations from the mean

If X is a nonnegative random variable with a known expectation, then Markov’s Inequality helps bound the probability of deviations from the mean as follows. For t ∈ R+ , Pr[X ≥ t] ≤ E[X] . t

This obvious inequality ﬁnds surprisingly many uses. For instance, it helps in obtaining a high probability statement from a bound on the expectation (e.g., see Section 14.2). If the variance of a random variable is small, then large deviations from the mean are improbable. This intuitive statement is formalized by Chebyshev’s inequality which states that for any random variable X and a ∈ R+, Pr[|X − E[X]| ≥ a] ≤ σ(X) a

2

.

See Lemma 28.5 for an application. Poisson trials are repeated independent trials, each of which has two possible outcomes, called success and failure. In general, the success probability is allowed to change with the trials. They are called Bernoulli trials if the success probability is the same for each trial. The Chernoﬀ bounds, which provide bounds on the tail probabilities of Poisson trials, are very useful in analyzing algorithms. Let us represent n Poisson trials by indicator random variables X1 , . . . , Xn , with 1 and 0 representing success and failure, respectively. Let Pr[Xi = 1] = pi , where 0 < pi < 1 for n 1 ≤ i ≤ n. Let random variable X = X1 + . . . + Xn and µ = E[X] = i=1 pi . For the bound on the lower tail assume 0 < δ ≤ 1. Then, Pr[X < (1 − δ)µ] < e(−µδ

2

/2)

.

The expression for the upper tail is more involved: for any δ > 0, Pr[X > (1 + δ)µ] < eδ (1 + δ)(1+δ) µ .

It can be simpliﬁed by considering two ranges for δ. For δ > 2e − 1, Pr[X > (1 + δ)µ] < 2−(1+δ)µ , and for δ ≤ 2e − 1, Pr[X > (1 + δ)µ] < e−µδ

2

/4

.

354

B

Basic Facts from Probability Theory

B.3 Basic distributions

Three distributions, of great universality, are deﬁned below. The probability distribution of the number of successes in Bernoulli trials is called the binomial distribution. Consider n Bernoulli trials with probability of success p. The probability of k successes, for 0 ≤ k ≤ n, is given by B(k; n, p) = n k p (1 − p)n−k . k

The Poisson distribution with parameter λ > 0 is as follows. For each nonnegative integer k, the probability of exactly k successes is deﬁned to be p(k; λ) = e−λ λk . k!

The limit of the binomial distribution B(k; n, p) as n → ∞ and np → λ, a constant, is the Poisson distribution p(k; λ). Indeed, in many applications one comes across Bernoulli trials in which n is large, p is small, and the product λ = np is moderate. In these situations, p(k; np) is a good approximation to B(k; n, p). The normal density function with mean µ and standard deviation σ is n(x) =

(x−µ)2 1 √ e− 2σ2 , σ 2π

and the normal distribution function is its integral, N (x) = 1 √ σ 2π x −∞

e−

(y−µ)2 2σ 2

dy.

The normal distribution also approximates the binomial distribution. Let us state this for the case p = 1/2. Let n be even, n = 2ν, say. For −ν ≤ k ≤ ν, deﬁne ak = a−k = B(ν + k; 2ν, 1/2). In the limit as ν → ∞ and k varies in the range 0 < k < approximated by hn(kh), where h =

2 ν

√

ν, ak can be

=

2 √ . n

B.4 Notes

For further information see the books by Feller [85], Motwani and Raghavan [209], Spencer [243], and Alon and Spencer [6].

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Problem Index

2CNF≡ clause deletion Acyclic subgraph 7, 334 Antichain cover 8

176, 179

Cover time 337 Covering integer programs 118 Cycle cover 35, 62 Dominating set 48, 50, 52

112, 116,

Bandwidth minimization 196 Betweenness 267 Bin covering 77 Bin packing 74, 74–78, 80, 124 – with ﬁxed number of object sizes Chain cover 8 Clique 9, 306, 309, 318–322 Closest vector 292 Clustering 243 – 2 253, 254 2 – metric k-cluster 52 Counting problems 294–305 – acyclic orientations 338 – antichains 340 – bases of a matroid 339 – colorings of a graph 341 – contingency tables 340 – DNF solutions 295, 305 – – weighted version 302 – Euler tours 339 – forests 339 – graphs with given degree sequence 340 – Hamiltonian cycles 341 – independent sets 341 – perfect matchings 305, 338 – simple cycles in a directed graph 303 – stable marriages 340 – trees 340 – triangulations 340 – volume of a convex body 338

81

Edge coloring 10 Edge expansion 192 Enumerating cuts 304 Feedback edge set – directed 337 – subset 166, 166, 167 Feedback vertex set 25, 54, 54–60, 129, 166 – directed 337 – subset 166, 166, 167, 336 Graph bipartization by edge deletion 178 Hamiltonian cycle 30, 303

Independent set 48, 51–53 – maximal 239 Knapsack 68, 68–73 138

Linear equations over GF[2]

Matching 3, 104 – b-matching 152, 227 – bipartite 129 – – maximum weight 129 – maximal 3, 8 – – minimum cardinality 8 – maximum 3, 5, 9, 124, 152, 153 – minimum weight 107 – perfect 105, 142, 143

372

Problem Index – uniform parallel machines 140, 145 Minimum spanning tree (MST) 28–31, 105, 206, 207, 212 Multicommodity ﬂow 97, 147, 163 – demands 168, 180, 180–197 – directed 165 – integer 148, 153, 154, 337 – – in trees 146–154 – – in trees of height one 152 – – in unit capacity trees 153 – sum 168, 168–176, 179 – uniform 192, 197 Multicut 146, 153, 168–179, 336 – directed 337 – in trees 146–154, 166 – in trees of height one 152 Multiway cut 38, 38–40, 155–167, 335 – bidirected integer program formulation 164 – directed 165, 166, 167 – fractional 156 – node 160, 160–163, 166 Network design – element connectivity 337 – vertex connectivity 336 Network reliability 297, 304, 305, 339 – s–t reliability 339 – global 339 Point-to-point connection 208

– – minimum weight 32, 35, 62, 105, 230 Matroid intersection 228 Matroid parity 212, 212 MAX k-CUT 23, 138, 267, 269 Maximum antichain 8 Maximum coverage 25 Maximum cut (MAX-CUT) 10, 22, 138, 255, 255, 256, 260–263, 267, 268, 334 – directed 23, 138, 267, 269 Maximum ﬂow 38, 97, 97–100, 168 Maximum satisﬁability (MAX-SAT) 9, 131, 131–139, 263, 306 – MAX k-FUNCTION SAT 312 – MAX-2SAT 131, 263, 268 – MAX-3SAT 131, 309, 311–315, 322, 323, 326, 330, 331 – – with bounded occurrence of variables 313–316, 330 Metric k-center 47, 47–50, 53 – fault-tolerant 52 – weighted 50, 50–52 Metric k-median 243, 243–254, 337 Metric k-MST 252 Metric facility location – capacitated 240, 337 – fault tolerant 240 – metric uncapacitated 242 – prize-collecting 240 – uncapacitated 232, 232–239, 242, 337 Minimum k-connected subgraph – edge 228 – vertex 226 Minimum k-cut 38, 40–44 Minimum bisection 193, 196, 197, 336 Minimum chain cover 8 Minimum cut 38, 298 – b-balanced 193, 193–194, 196, 197, 336 – s–t 38, 98, 97–100, 146 Minimum cut linear arrangement 194, 194–195, 197 Minimum length linear arrangement 178 Minimum makespan scheduling 9, 10, 79, 79–83, 140

Satisﬁability (SAT) 9, 330, 343, 344 – 3SAT 310, 343 Scheduling on unrelated parallel machines 140, 140–145 Semideﬁnite programming 258, 255–269 Set cover VIII, 11, 15, 15–26, 34, 108–122, 124–130, 239, 251, 306, 309, 322–329, 334 – constrained set multicover 112, 116, 118 – multiset multicover 112, 116, 117, 123 – set multicover 24, 112, 116, 123 – with concave costs 117 Shortest superstring 9, 20, 19–22, 26, 61–67 – variants 25, 67

Problem Index Shortest vector 273, 273–293, 336 Sparsest cut 180, 180–197, 336, 337 Steiner arborescence – rectilinear 35 Steiner forest 198, 198–213 Steiner network 213, 213–231, 335 Steiner tree 27, 27–30, 33, 37, 198, 213, 306, 309, 335 – directed 34, 337 – Euclidean 89 – prize-collecting 208, 252 Subset sum 291 Subset-sum ratio problem 72 Survivable network design see Steiner network and network design Traveling salesman problem (TSP) 30, 229, 231 – asymmetric 34, 336 – Euclidean 84, 84–89 – metric 30–33, 37, 229, 231, 334 – – lengths one and two 34 – – variants 34 Tutte polynomial 341

373

Vertex coloring 23 – k-coloring 267, 269 Vertex cover 1, 15, 17–19, 23, 24, 104, 122–124, 129, 146, 152, 166, 306, 307, 309, 334 – cardinality 1, 2–5, 8, 152 Zero-skew tree – rectilinear 36, 37

Subject Index

α-min cut 304 #P 294, 305 1-tree 230 Active set 200, 209 Approximation algorithm 2, 345–347 – approximation factor 346 – randomized 346 Approximation scheme 68 – fully polynomial randomized (FPRAS) 295, 295, 297, 300, 302, 303, 305, 338–340 – fully polynomial time (FPTAS) 68, 69–70, 72, 77, 83 – polynomial time (PTAS) 68, 80–89, 140, 145, 311, 336 – – asymptotic 75, 74–78 Arborescence 228 Arithmetic-geometric mean inequality 135 Basis of a lattice 274 – Gauss reduced 281, 290 – KZ reduced 290 – Lovasz reduced 283 – weakly reduced 283, 290 Bernoulli trials 190, 353 Catalan numbers 86, 340 Certiﬁcate – co-NP 336 – Yes/No 5–7, 93, 96, 294, 343–344, 348 – – approximate 274, 288 Chebyshev’s inequality 297, 353 Chernoﬀ bounds 9, 190, 353 Christoﬁdes’ algorithm 37, 229, 334 Chromatic polynomial 342

co-NP 344 co-RP 10, 330, 348 Complementary slackness conditions 97, 100, 105, 125, 149, 161, 178, 199, 233 – relaxed 126, 129, 130, 146, 149, 199, 234 Compression 64 Concave function 135 Convex combination 258, 259 Convex set 259 Cost-eﬀectiveness of a set 16, 113 Counting problems VII, 294–305, 338–342 – #P-complete VII, 294, 294, 305, 338 Covering LP 109 Crossing sets 215, 219 Cut packing 183–191 – approximate 184 Cut requirement function 213 Cycle space 54 – cyclomatic number 54 Cyclomatic weighted graphs 54–57 Decision problem 343 – NP-complete 344 – well-characterized 6, 5–7, 10, 93 – Yes/No certiﬁcate – – approximate 7 Deﬁciency of a set 226 Demand graph 182 Derandomization 132–134, 138, 248–250, 268 Determinant of a lattice 274 Dilworth’s theorem 8 Divide-and-conquer algorithm 179, 193

376

Subject Index Integrality gap 102, 101–103, 111, 129, 137, 151, 164, 167, 207, 210, 211, 218, 229, 254, 262, 335, 337 Integrality ratio see Integrality gap Interactive proof systems 332 Ising model 342 Isolating cut 38 Kirchhoﬀ’s theorem 339 Konig-Egervary theorem 5, 104 Kruskal’s algorithm 105, 206 Lagrangian relaxation 250–252 Laminar family of sets 219 Layering 17–19, 25, 57, 60, 129 Linearity of expectation 136, 352 Local search 23, 253 Lower bounding OPT 2, 17, 31, 32, 39, 47, 62, 79, 89, 108, 206, 278–280 Lowest common ancestor 149 LP-duality – theorem 6, 95, 93–97, 100, 106, 107, 148, 183 – – weak 96, 148, 169 – theory 6, 17, 29, 97, 101, 108, 147 Mader’s theorem 227, 231 Markov chain 192, 338, 339 – conductance 192–193, 197 – Markov chain Monte Carlo method VIII, 294 – rapidly mixing 305, 339, 340 – stationary probability distribution 192 – Swendsen-Wang process 342 – transition matrix 192 Markov’s inequality 88, 353 Matroid 339 – balanced 339 – basis exchange graph 339 – graphic 339 – independent sets 212 Max-ﬂow min-cut theorem 97, 103, 168, 207 – approximate version for demands multicommodity ﬂow 191 – approximate version for uniform multicommodity ﬂow 197 MAX-SNP-completeness 332

DTIME 331, 332, 348 Dual ﬁtting 101, 108–118, 241 Dual growing – synchronized 198 Dual lattice 284, 284–288 Dynamic programming 69, 81, 153 Edge expansion 192 Edge-disjoint s–t paths 103, 336 Eigenvalue 257 Eigenvector 257 Ellipsoid algorithm 170, 214, 255, 259 Euclid’s algorithm 273, 276–278 Euler tour 28, 32 Eulerian graph 28, 31 Expander graph 175, 179, 192, 320, 332 Expander graphs 314 Extreme point solution 100, 102–104, 119, 122, 141–145, 214, 219–221 First-ﬁt algorithm 74, 77 Flow-equivalent tree 44 Forward delete 153 Frequency of an element 15, 119 Function – degree-weighted 17 – proper 208 Fundamental cycle 54 Game – two-person zero-sum 106 Gauss’ algorithm 273, 276–278, 288 Gomory–Hu tree 40, 44, 46 Gram–Schmidt lower bound 287, 288 Gram–Schmidt orthogonalization 278, 278–280, 282, 285 Greedy algorithm 8, 16–17, 24, 44, 60, 64, 72, 108, 138, 241 Half-integrality 119, 122–124, 153, 160–163, 165, 213–221 Hall’s theorem 144 Hamiltonian cycle 29, 214 Hardness of approximation VIII, 306–333 Hungarian method 129

Subject Index Maximum weight spanning tree 44 Menger’s theorem 103 Method of conditional expectation 131–134, 138, 139, 248 Metric 183–191 – 1 -embedding 183–191 – – β-distortion 185 – – isometric 185, 186 – 2 -embedding 196 – 2 -embedding 2 – – isometric 195 – – optimal distortion 197, 266 – p 185 Min–max relation 5–7, 11, 97–100, 168 – approximate 7, 151 Minkovski’s theorem 287 Moments of a random variable 352 – central 352 Monte Carlo sampling 297, 301 Near-minimum cuts 298–299 Next-ﬁt algorithm 77 Norm 185 – p 185 NP 343 Odd set cover 6 Optimization problem 2, 345, 351 – NP-complete 10 – NP-hard 68, 344 – strongly NP-hard 71 Orthogonality defect 275, 279 Overlap graph 66 P=NP conjecture VII, 10, 68, 71, 345 Packing LP 110 Parametric pruning 47–52, 140–141, 252 Parsimonious property 229, 230 Partial ordering 8 PCP theorem VIII, 306, 308–311, 323, 332 Petersen graph 6, 214 Poisson trials 353 Positive semideﬁnite matrix 257, 257–258 Potts model 342 Preﬁx graph 62

377

Primal–dual schema VII, 101, 125–130, 149–152, 235–236, 335 – with synchronization 199–204 Primitive root 9 Primitive vector 275, 285, 286, 290 Principal submatrix 265 Probabilistic argument 179 Probabilistic method 324 Probabilistically checkable proof system (PCP) 309, 332 – completeness 319 – parallel repetition 325–326 – soundness 319 – two-prover one round 322–324, 332 Probability distribution – binomial 354 – normal 261, 266, 354 – Poisson 354 – spherically symmetric 261 Probability theory 352–354 Pseudo-approximation algorithm 193–195, 197 Pseudo-forest 143 Pseudo-polynomial time algorithm 69, 69, 71–73 Pseudo-tree 143 Quadratic forms 292 Quadratic program 255 – strict 255, 255–257, 267, 268 Random contraction algorithm 298, 304 Random walk 320, 338–340 Reduction – L- 332, 351 – approximation factor preserving 24, 27, 34, 60, 152, 160, 166, 196, 242, 347, 351 – gap-introducing 307 – gap-preserving 307 – polynomial time 344 – randomized 293 Region growing 171–175 Relaxation – convex 269 – exact 102 – – for maximum weight bipartite matching 129

378

Subject Index Supermodular function – weakly supermodular 216

– – for MST 212, 230 – LP- VII, 39, 99, 100–106, 109, 111, 113, 119, 120, 122, 124, 125, 134, 147, 153, 155–157, 160, 164, 165, 179, 199, 206, 209, 211, 213–221, 224, 229–231, 233, 240, 244, 251, 335, 337 – – bidirected cut relaxation for Steiner tree 210, 335 – – subtour elimination relaxation for TSP 229, 229–231 Reverse delete 149, 210 – dynamic 209 Rounding VII, 101, 119–124, 134–136, 170–175, 191 – iterated 213, 217–218 – randomized 120–122, 124, 157–160, 164, 247–248, 260–263 RP 348 Scaling and rounding 73, 117 Self-reducibility IX, 9, 10, 303, 348–351 – tree 10, 303, 349 Semideﬁnite program 197, 266, 267 – duality theory 268 Separating hyperplane 259 Separation oracle 102, 107, 170, 179, 217 Short-cutting 29, 31, 32, 85, 241 Simplex 155 Sparsity of a cut 181 Spread of an edge 196 Square of a graph 48 Standard deviation 352 Steiner tree 316–318 Sublattice 285, 290 Submodular function 215, 224

Throughput 180, 182 Tight example IX, 4, 8, 17, 19, 23–25, 29, 31, 33, 39, 43, 49, 51, 59, 80, 83, 120, 123, 128, 137, 144, 153, 165, 175, 206, 218, 238, 239, 249, 268 Totally unimodular matrix 104 Tournament 25 Traveling salesman tour – maximum weight 66 – minimum weight 62 Triangle inequality 27, 51, 52, 178 – directed 34 Unbiased estimator 295 Uncrossable function 224 Uniform generator 302, 303 – almost uniform 303 Unimodular matrix 274, 274–276, 288 Unit sphere 260 Upper bounding OPT 256 Vector program 256, 255–257, 266, 267 Veriﬁer 309 Vertex cover 316–318 Vertex-disjoint s–t paths 103, 336 VLSI design 178 – clock routing 36 von Neumann’s minimax theorem 106 Witness family 225

ZPP 10, 348 ZTIME 329, 332, 348…...

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