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Paradigms of Combinatorial Optimization

Combinatorial Optimization volume 2


Problems and New Approaches

Edited by Vangelis Th Paschos

First published in Great Britain and the United States in 2010 by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from Optimisation combinatoire volumes 1 to 5 published 2005-2007 in France by Hermes ScienceLavoisier copy LAVOISIER 2005 2006 2007 Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom copy ISTE Ltd 2010 The rights of Vangelis Th Paschos to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data Combinatorial optimization edited by Vangelis Th Paschos v cm Includes bibliographical references and index Contents v 1 Concepts of combinatorial optimization ISBN 978-1-84821-146-9 (set of 3 vols) -- ISBN 978-1-84821-148-3 (v 2) 1 Combinatorial optimization 2 Programming (Mathematics) I Paschos Vangelis Th QA4025C545123 2010 51964--dc22

2010018423 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-146-9 (Set of 3 volumes) ISBN 978-1-84821-148-3 (Volume 2)

Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

Table of Contents

Preface xvii Vangelis Th PASCHOS

PART I PARADIGMATIC PROBLEMS 1

Chapter 1 Optimal Satisfiability 3 Cristina BAZGAN

11 Introduction 3 12 Preliminaries 5

121 Constraint satisfaction problems decision and optimization versions 6 122 Constraint types 8

13 Complexity of decision problems 10 14 Complexity and approximation of optimization problems 13

141 Maximization problems 13 142 Minimization problems 20

15 Particular instances of constraint satisfaction problems 20 151 Planar instances 21 152 Dense instances 22 153 Instances with a bounded number of occurrences 24

16 Satisfiability problems under global constraints 25 17 Conclusion 27 18 Bibliography 27

Chapter 2 Scheduling Problems 33 Philippe CHREacuteTIENNE and Christophe PICOULEAU

21 Introduction 33 22 New techniques for approximation 34

vi Combinatorial Optimization 2

221 Linear programming and scheduling 35 222 An approximation scheme for P||Cmax 40

23 Constraints and scheduling 41 231 The monomachine constraint 41 232 The cumulative constraint 44 233 Energetic reasoning 45

24 Non-regular criteria 46 241 PERT with convex costs 47

242 Minimizing the earlyndashtardy cost on one machine 52 25 Bibliography 57

Chapter 3 Location Problems 61 Aristotelis GIANNAKOS

31 Introduction 61 311 Weberrsquos problem 62 312 A classification 64

32 Continuous problems 65 321 Complete covering 65 322 Maximal covering 66 323 Empty covering 67 324 Bicriteria models 69 325 Covering with multiple resources 69

33 Discrete problems 70 331 p-Center 70 332 p-Dispersion 70 333 p-Median 71 334 Hub 73 335 p-Maxisum 73

34 On-line problems 74 35 The quadratic assignment problem 77

351 Definitions and formulations of the problem 77 352 Complexity 79 353 Relaxations and lower bounds 79

36 Conclusion 82 37 Bibliography 83

Chapter 4 MiniMax Algorithms and Games 89 Michel KOSKAS

41 Introduction 89 42 Games of no chance the simple cases 91 43 The case of complex no chance games 94

431 Approximative evaluation 95

Table of Contents vii

432 Horizon effect 97 433 α-β pruning 97

44 Quiescence search 99 441 Other refinements of the MiniMax algorithm 102

45 Case of games using chance 103 46 Conclusion 103 47 Bibliography 106

Chapter 5 Two-dimensional Bin Packing Problems 107 Andrea LODI Silvano MARTELLO Michele MONACI and Daniele VIGO

51 Introduction 107 52 Models 108

521 ILP models for level packing 109 53 Upper bounds 112

531 Strip packing 112 532 Bin packing two-phase heuristics 113 533 Bin packing one-phase level heuristics 115 534 Bin packing one-phase non-level heuristics 116 535 Metaheuristics 116 536 Approximation algorithms 118

54 Lower bounds 119 541 Lower bounds for level packing 123

55 Exact algorithms 123 56 Acknowledgements 125 57 Bibliography 125

Chapter 6 The Maximum Cut Problem 131 Walid BEN-AMEUR Ali Ridha MAHJOUB and Joseacute NETO

61 Introduction 131 62 Complexity and polynomial cases 133 63 Applications 134

631 Spin glass models 134 632 Unconstrained 0ndash1 quadratic programming 135 633 The via minimization problem 136

64 The cut polytope 137 641 Valid inequalities and separation 137 642 Branch-and-cut algorithms 142 643 The cut polyhedron 144

65 Semi-definite programming (SDP) and the maximum cut problem 145 651 Semi-definite formulation of the MAX-CUT problem 146 652 Quality of the semi-definite formulation 147 653 Existing works in the literature 150

viii Combinatorial Optimization 2

66 The cut cone and applications 152 661 The cut cone 152 662 Relationship to the cut polytope 152 663 The semi-metric cone 153 664 Applications to the multiflow problem 155

67 Approximation methods 157 671 Methods with performance guarantees 157 672 Methods with no guarantees 158

68 Related problems 159 681 Unconstrained 0ndash1 quadratic programming 159 682 The maximum even (odd) cut problem 160 683 The equipartition problem 161 684 Other problems 162


Chapter 7 The Traveling Salesman Problem and its Variations 173 Jeacuterocircme MONNOT and Sophie TOULOUSE

71 Introduction 173 72 Elementary properties and various subproblems 174

721 Elementary properties 174 722 Various subproblems 175

73 Exact solving algorithms 177 731 A dynamic programming algorithm 177 732 A branch-and-bound algorithm 179

74 Approximation algorithm for max TSP 184 741 An algorithm based on 2-matching 186 742 Algorithm mixing 2-matching and matching 189

75 Approximation algorithm for min TSP 192 751 Algorithm based on the spanning tree and matching 196 752 Local search algorithm 197

76 Constructive algorithms 201 761 Nearest neighbor algorithm 201 762 Nearest insertion algorithm 207


Chapter 8 0ndash1 Knapsack Problems 215 Geacuterard PLATEAU and Anass NAGIH

81 General solution principle 215 82 Relaxation 217 83 Heuristic 222

Table of Contents ix

84 Variable fixing 222 85 Dynamic programming 226

851 General principle 227 852 Managing feasible combinations of objects 230

86 Solution search by hybridization of branch-and-bound and dynamic programming 234

861 Principle of hybridization 235 862 Illustration of hybridization 237


Chapter 9 Integer Quadratic Knapsack Problems 243 Dominique QUADRI Eric SOUTIF and Pierre TOLLA

91 Introduction 243 911 Problem formulation 243 912 Significance of the problem 244

92 Solution methods 246 921 The convex separable problem 246 922 The non-convex separable problem 252 923 The convex non-separable problem 254 924 The non-convex non-separable problem 256

93 Numerical experiments 259 931 The convex and separable integer quadratic knapsack problem 260 932 The convex and separable integer quadratic multi-knapsack problem 260


Chapter 10 Graph Coloring Problems 265 Dominique DE WERRA and Daniel KOBLER

101 Basic notions of colorings 265 102 Complexity of coloring 269 103 Sequential methods of coloring 270 104 An exact coloring algorithm 272 105 Tabu search 276 106 Perfect graphs 280 107 Chromatic scheduling 285 108 Interval coloring 287 109 T-colorings 289 1010 List colorings 292 1011 Coloring with cardinality constraints 295 1012 Other extensions 298

x Combinatorial Optimization 2

1013 Edge coloring 299 10131 f-Coloring of edge 300 10132 [g f]-Colorings of edges 301 10133 A model of hypergraph coloring 303


PART II NEW APPROACHES 311

Chapter 11 Polynomial Approximation 313 Marc DEMANGE and Vangelis Th PASCHOS

111 What is polynomial approximation 313 1111 Efficiently solving a difficult problem 314 1112 Approximation measures 314

112 Some first examples of analysis constant approximation ratios 316 1121 An example of classical approximation the metric traveling salesman 316 1122 Examples of the differential ratio case 317

113 Approximation schemes 323 1131 Non-complete schemes 323 1132 Complete approximation schemes ndash example of the Boolean knapsack 333

114 Analyses depending on the instance 336 1141 Set covering and classical ratios 336 1142 Set covering and differential ratios 337 1143 The maximum stable set problem 338

115 Conclusion methods and issues of approximation 339 1151 The types of algorithms a few great classics 340 1152 Approximation classes structuring the class NPO 341 1153 Reductions in approximation 344 1154 Issues 345

116 Bibliography 346

Chapter 12 Approximation Preserving Reductions 351 Giorgio AUSIELLO and Vangelis Th PASCHOS

121 Introduction 351 122 Strict and continuous reductions 353

1221 Strict reductions 353 1222 Continuous reduction 357

123 AP-reduction and completeness in the classes NPO and APX 359 1231 Completeness in NPO 360

Table of Contents xi

1232 Completeness in APX 362 1233 Using completeness to derive negative results 365

124 L-reduction and completeness in the classes Max-SNP and APX 366 1241 The L-reduction and the class Max-SNP 366 1242 Examples of L-reductions 367 1243 Completeness in Max-SNP and APX 370

125 Affine reduction 371 126 A few words on GAP-reduction 373 127 History and comment 374 128 Bibliography 378

Chapter 13 Inapproximability of Combinatorial Optimization Problems 381 Luca TREVISAN

131 Introduction 381 1311 A brief historical overview 382 1312 Organization of this chapter 385 1313 Further reading 386

132 Some technical preliminaries 387 133 Probabilistically checkable proofs 389

1331 PCP and the approximability of Max SAT 390 134 Basic reductions 392

1341 Max 3SAT with bounded occurrences 392 1342 Vertex Cover and Independent Set 394 1343 Steiner tree problem 396 1344 More about Independent Set 398

135 Optimized reductions and PCP constructions 400 1351 PCPs optimized for Max SAT and Max CUT 400 1352 PCPs optimized for Independent Set 402 1353 The Unique Games Conjecture 403

136 An overview of known inapproximability results 404 1361 Lattice problems 404 1362 Decoding linear error-correcting codes 406 1363 The traveling salesman problem 407 1364 Coloring problems 409 1365 Covering problems 409 1366 Graph partitioning problems 411

137 Integrality gap results 412 1371 Convex relaxations of the Sparsest Cut problem 413 1372 Families of relaxations 413 1373 Integrality gaps versus Unique Games 415

138 Other topics 416

xii Combinatorial Optimization 2

1381 Complexity classes of optimization problems 416 1382 Average-case complexity and approximability 418 1383 Witness length in PCP constructions 419 1384 Typical and unusual approximation factors 419

139 Conclusions 421 1310 Prove optimal results for 2-query PCPs 422 1311 Settle the Unique Games Conjecture 422 1312 Prove a strong inapproximability result for Metric TSP 422 1313 Acknowledgements 423 1314 Bibliography 423

Chapter 14 Local Search Complexity and Approximation 435 Eric ANGEL Petros CHRISTOPOULOS and Vassilis ZISSIMOPOULOS

141 Introduction 435 142 Formal framework 437 143 A few familiar optimization problems and their neighborhoods 439

1431 Graph partitioning problem 439 1432 The maximum cut problem 439 1433 The traveling salesman problem 440 1434 Clause satisfaction problems 441 1435 Stable configurations in a Hopfield-type neural network 441

144 The PLS class 442 145 Complexity of the standard local search algorithm 447 146 The quality of local optima 449 147 Approximation results 450

1471 The MAX k-SAT problem 451 1472 The MAX CUT problem 452 1473 Other problems on graphs 454 1474 The traveling salesman problem 456 1475 The quadratic assignment problem 457 1476 Classification problems 460 1477 Facility location problems 462

148 Conclusion and open problems 465 149 Bibliography 467

Chapter 15 On-line Algorithms 473 Giorgio AUSIELLO and Luca BECCHETTI

151 Introduction 473 152 Some classical on-line problem 475

1521 List updating 476 1522 Paging 477 1523 The traveling salesman problem 480

Table of Contents xiii

1524 Load balancing 482 153 Competitive analysis of deterministic algorithms 483

1531 Competitive analysis of list updating 484 1532 Competitive analysis of paging algorithms 486 1533 Competitive analysis of on-line TSP 488 1534 Competitive analysis of on-line load balancing 494

154 Randomization 496 1541 Randomized paging 497 1542 Lower bounds Yaorsquos lemma and its application to paging algorithm 499

155 Extensions of competitive analysis 501 1551 Ad hoc techniques the case of paging 502 1552 General techniques 503


Chapter 16 Polynomial Approximation for Multicriteria Combinatorial Optimization Problems 511 Eric ANGEL Evripidis BAMPIS and Laurent GOURVEgraveS

161 Introduction 511 162 Presentation of multicriteria combinatorial problems 513

1621 Multicriteria combinatorial problems 513 1622 Optimality 514 1623 Complexity of multicriteria combinatorial problems 517

163 Polynomial approximation and performance guarantee 521 1631 Criteria weighting approach 521 1632 Simultaneous approach 524 1633 Budget approach 527 1634 Pareto curve approach 531

164 Bibliographical notes 541 1641 Presentation of multicriteria combinatorial problems 541 1642 Polynomial approximation with performance guarantees 541


Chapter 17 An Introduction to Inverse Combinatorial Problems 547 Marc DEMANGE and Jeacuterocircme MONNOT

171 Introduction 547 172 Definitions and notation 549 173 Polynomial inverse problems and solution techniques 552

1731 The linear programming case 553 1732 Inverse maximum flow problem 562 1733 A class of polynomial inverse problems 564

xiv Combinatorial Optimization 2

1734 Avenues to explore the example of the inverse bivalent variable maximum weight matching problem 567

174 Hard inverse problems 569 1741 Inverse NP-hard problems 569 1742 Facility location problem 572 1743 A partial inverse problem the minimum capacity cut 575 1744 Maximum weight matching problem 578


Chapter 18 Probabilistic Combinatorial Optimization 587 Ceacutecile MURAT and Vangelis Th PASCHOS

181 Motivations and applications 587 182 The issues formalism and methodology 589 183 Problem complexity 593

1831 Membership of NP is not given 593 1832 Links between the deterministic and probabilistic frameworks from the complexity point of view 599

184 Solving problems 601 1841 Characterization of optimal solutions 602 1842 Polynomial solution of certain instances 605 1843 Effective solution 607

185 Approximation 608 186 Bibliography 611

Chapter 19 Robust Shortest Path Problems 615 Virginie GABREL and Ceacutecile MURAT

191 Introduction 615 192 Taking uncertainty into account the various models 616

1921 The interval model 617 1922 The discrete scenario mode 617

193 Measures of robustness 619 1931 Classical criterion derived from decision-making theory 619 1932 Methodology inspired by mathematical programming 622 1933 Methodology inspired by multicriteria analysis 623

194 Complexity and solution of robust shortest path problems in the interval mode 625

1941 With the worst-case criterion 625 1942 With the maximum regret criterion 626 1943 With the mathematical programming inspired approach 630 1944 With the multicriteria analysis inspired approach 632

Table of Contents xv

195 Complexity and solution of robust shortest path problems in a discrete set of scenarios model 635

1951 With the worst-case criterion 635 1952 With the maximum regret criterion 636 1953 With the multicriteria analysis inspired approach 637


Chapter 20 Algorithmic Games 641 Aristotelis GIANNAKOS and Vangelis PASCHOS

201 Preliminaries 642 2011 Basic notions of games 642 2012 The classes of complexity covered in this chapter 645

202 Nash equilibria 647 203 Mixed extension of a game and Nash equilibria 649 204 Algorithmic problems 650

2041 Succinct description game 651 2042 Results on the complexity of computing a mixed equilibrium 651 2043 Counting the number of equilibria in a mixed strategy game 657

205 Potential games 657 2051 Definitions 657 2052 Properties 658

206 Congestion games 662 2061 Rosenthalrsquos model 662 2062 Complexity of congestion games (Rosenthalrsquos model) 665 2063 Other models 666

207 Final notes 670 208 Bibliography 670

List of Authors 675

Index 681

Summary of Other Volumes in the Series 689

Th

Preface

Paradigms of Combinatorial Optimization is the second volume of the Combinatorial Optimization series It deals with advanced concepts as well as a series of problems studies and research which have made and continue to make their mark on the evolution of this discipline This work is divided into two parts

ndash Part I Paradigmatic Problems

ndash Part II New Approaches

Part I contains the following chapters

ndash Optimal Satisfiability by Cristina Bazgan

ndash Scheduling Problems by Philippe Chreacutetienne and Christophe Picouleau

ndash Location Problems by Aristotelis Giannakos

ndash Min Max Algorithms and Games by Michel Koskas

ndash Two-dimensional Bin Packing Problems by Andrea Lodi Silvano Martello Michele Monaci and Daniele Vigo

ndash The Maximum Cut Problem by Walid Ben-Ameur Ali Ridha Mahjoub and Joseacute Neto

ndash The Traveling Salesman Problem and its Variations by Jeacuterocircme Monnot and Sophie Toulouse

ndash 0ndash1 Knapsack Problems by Geacuterard Plateau and Anass Nagih

ndash Integer Quadratic Knapsack Problems by Dominique Quadri Eric Soutif and Pierre Tolla

ndash Graph Coloring Problems by Dominique De Werra and Daniel Kobler

i

xviii Combinatorial Optimization 2

All these chapters not only deal with the problems in question but also highlight various tools and methods from combinatorial optimization and operations research Obviously this list is very limited and does not pretend to cover all the flagship problems in combinatorial optimization

It is best to view the problems in this book as a sample that testifies to the richness of the themes and problems that can be tackled by combinatorial optimization and of the tools developed by this discipline

Part II includes the following chapters

ndash Polynomial Approximation by Marc Demange and Vangelis Th Paschos

ndash Approximation Preserving Reductions by Giorgio Ausiello and Vangelis Th Paschos

ndash Inapproximability of Combinatorial Optimization Problems by Luca Trevisan

ndash Local Search Complexity and Approximation by Eric Angel Petros Christopoulos and Vassilis Zissimopoulos

ndash On-line Algorithms by Giorgio Ausiello and Luca Becchetti

ndash Polynomial Approximation for Multicriteria Combinatorial Optimization Problems by Eric Angel Evripidis Bampis and Laurent Gourvegraves

ndash An Introduction to Inverse Combinatorial Problems by Marc Demange and Jeacuterocircme Monnot

ndash Probabilistic Combinatorial Optimization by Ceacutecile Murat and Vangelis Th Paschos

ndash Robust Shortest Path Problems by Virginie Gabrel and Ceacutecile Murat

ndash Algorithmic Games by Aristotelis Giannakos and Vangelis Th Paschos

The themes of this part are at the border between research operations and combinatorial optimization theoretical computer science and discrete mathematics Nevertheless all these subjects have their rightful place in the vast scientific field that we call combinatorial optimization They are developed at least in part at the heart of this discipline fertilize it widen its renown and enrich its models

For this volume my thanks go firstly to the authors who have agreed to participate in the book This work could never have come into being without the

Preface xix

original proposal of Jean-Charles Pomerol Vice President of the scientific committee at Hermes and Sami Meacutenasceacute and Raphaeumll Meacutenasceacute the heads of publications at ISTE I give my warmest thanks to them for their insistence and encouragement It is a pleasure to work with them as well as with Rupert Heywood who has ingeniously translated this bookrsquos material from the original French

Vangelis Th PASCHOS June 2010

PART I

Paradigmatic Problems

Chapter 1

Optimal Satisfiability

11 Introduction

Given a set of constraints defined on Boolean variables a satisfiability problemalso called a Boolean constraint satisfaction problem consists of deciding whetherthere exists an assignment of values to the variables that satisfies all the constraints(and possibly establishing such an assignment) Often such an assignment does notexist and in this case it is natural to seek an assignment that satisfies a maximumnumber of constraints or minimizes the number of non-satisfied constraints

An example of a Boolean constraint satisfaction problem is the problem knownas SAT which consists of deciding whether a propositional formula (expressed as aconjunction of disjunctions) is satisfiable or not SAT was the first problem shownto be NP-complete by Cook [COO 71] and Levin [LEV 73] and it has remained acentral problem in the study of NP-hardness of optimization problems [GAR 79] TheNP-completeness of SAT asserts that no algorithm for this problem can be efficient inthe worst case under the hypothesis P =NP Nevertheless in practice many efficientalgorithms exist for solving the SAT problem

Satisfiability problems have direct applications in various domains such as opera-tions research artificial intelligence and system architecture For example in opera-tions research the graph-coloring problem can be modeled as an instance of SAT Todecide whether a graph with n vertices can be colored with k colors we consider ktimesnBoolean variables xij i = 1 n j = 1 k where xij takes the value true if

Chapter written by Cristina BAZGAN

4 Combinatorial Optimization 2

and only if the vertex i is assigned the color j Hoos [HOO 98] studied the effective-ness of various modelings of the graph-coloring problem as a satisfiability problemwhere we apply a specific local search algorithm to the instance of the obtained sat-isfiability problem The Steiner tree problem widely studied in operations researchcontributes to network design and routing applications In [JIA 95] the authors re-duced this problem to a problem that consists of finding an assignment that maximizesthe number of satisfied constraints Certain scheduling problems have been solved byusing modeling in terms of a satisfiability problem [CRA 94] Testing various proper-ties of graphs or hypergraphs is also a problem that reduces to a satisfiability problemIn artificial intelligence an interesting application is the planning problem that can berepresented as a set of constraints such that every satisfying assignment correspondsto a valid plan (see [KAU 92] for such a modeling) Other applications in artificialintelligence are learning from examples establishing the coherence of a system ofrules of a knowledge base and constructing inferences in a knowledge base In thedesign of electrical circuits we generally wish to construct a circuit with certain func-tionalities (described by a Boolean function) that satisfy various constraints justifiedby technological considerations of reliability or availability such as minimizing thenumber of gates used minimizing the depth of the circuit or only using certain typesof gates

Satisfiability problems also have other applications in automatic reasoning com-puter vision databases robotics and computer-assisted design Gu Purdom Francoand Wah wrote an overview article [GU 97] that cites many applications of satisfiabil-ity problems (about 250 references)

Faced with a satisfiability problem we can either study it from the theoreticalpoint of view (establish its exact or approximate complexity construct algorithms thatguarantee an exact or approximate solution) or solve it from the practical point ofview Among the most effective methods for the practical solution of satisfiabilityproblems are local search Tabu search and simulated annealing For further detailsrefer to [GU 97] and [GEN 99] which offer a summary of the majority of practicalalgorithms for satisfiability problems

In this chapter we present the principal results of exact and approximation com-plexity for satisfiability problems according to the type of Boolean functions that par-ticipate in the constraints Our goal is not to present exhaustively all the results thatexist in the literature but rather to identify the most studied problems and to introducethe basic concepts and algorithms The majority of satisfiability problems are hardIt is therefore advantageous both from the theoretical and practical points of view toidentify some specific cases that are easier We have chosen to present the most stud-ied specific cases planar instances instances with a bounded number of occurrencesof each variable and dense instances Several optimization problems can be modeledas a satisfiability problem with an additional global constraint on the set of feasible so-lutions In particular the MIN BISECTION problem whose approximation complexity

Optimal Satisfiability 5

has not yet been established can be modeled as a satisfiability problem where the setof feasible solutions is the set of the balanced assignments (with as many variables setto 0 as to 1) We also present a few results obtained on satisfiability problems underthis global constraint

Readers who wish to acquire a deeper knowledge of the complexity of satisfiabilityproblems should consult the monograph by Creignou Khanna and Sudan [CRE 01]where the proofs of the majority of important results in this domain can be found andthat cover besides the results presented here other aspects such as counting complex-ity and function representation complexity as well as other satisfiability problemsNote also that there is an electronic compendium by Crescenzi and Kann [CRE 95b]which regroups known results of approximation complexity for optimization prob-lems in particular for satisfiability problems

This chapter is structured as follows In section 12 we introduce the types ofBoolean functions that we will use and we define the decision and optimization prob-lems considered In section 13 we study decision problems and in section 14 max-imization and minimization problems We then discuss a few specific instances of sat-isfiability problems planar instances (section 151) dense instances (section 152)and instances with a bounded number of occurrences of each variable (section 153)We also present the complexity of satisfiability problems when the set of feasible so-lutions is restricted to balanced assignments (section 16) We close our chapter witha brief conclusion (section 17)

12 Preliminaries

An instance of a satisfiability problem is a set ofm constraintsC1 Cm definedon a set of n variables x1 xn A constraint Cj is the application of a Booleanfunction f 0 1k rarr 0 1 to a subset of variables xi1 xik

where i1 ik isin1 n This constraint is also expressed as f(xi1 xik

) An assignment xi =vi for i = 1 n where vi isin 0 1 satisfies the constraint f(xi1 xik

) if andonly if f(vi1 vik

) = 1

A literal is a variable xi (positive literal) or its negation xi (negative literal)

EXAMPLE 11ndash A few examples of Boolean functions used to define constraints

ndash T (x) = x F (x) = x

ndash ORki (x1 xk) = x1 or or xi or xi+1 or or xk where i k represents the

number of negative literals in the disjunction

ndash ANDki (x1 xk) = x1 and and xi and xi+1 and and xk where i k represents

the number of negative literals in the conjunction

ndash XORk(x1 xk) = x1 oplus oplus xk where oplus represents the ldquoexclusive orrdquooperation (0 oplus 0 = 0 1oplus 0 = 1 0oplus 1 = 1 1oplus 1 = 0)


ndash XNORk(x1 xk) = x1 oplus oplus xk

A constraint can also be represented as a Boolean expression that can be in variousforms An expression is in conjunctive normal form (CNF) if it is in the form c1 and andcm where each ct is a disjunctive clause that is in the form t1 or or tp whereti i = 1 p are literals An expression is in disjunctive normal form (DNF) if itis in the form c1 or or cm where each ct is a conjunctive clause that is in the formt1 and and tp where ti i = 1 p are literals A kCNF (or kDNF) expression isa CNF (or DNF) expression in which each clause contains at most k literals

Note that if each constraint of a satisfiability problem is represented by a CNFexpression the set of constraints of the problem can itself be represented by a CNFexpression that corresponds to the conjunction of the previous expressions

We consider various satisfiability problems according to the type of Boolean func-tions used to define the constraints Let F be a finite set of Boolean functions AF -set of constraints is a set of constraints that only use functions that belong to F Anassignment satisfies an F -set of constraints if and only if it satisfies each constraint inthe constraint set

121 Constraint satisfaction problems decision and optimization versions

In this section we define the classes of problems that we are going to study Thisconcerns decision and optimization versions of satisfiability problems

The decision version of a problem consists of establishing whether this problemallows at least one solution its search version consists of finding a solution if any existThe optimization version of a problem consists of finding a solution that maximizesor minimizes a suitable function

DEFINITION 11ndash The satisfiability problem SAT(F ) consists of deciding whetherthere exists an assignment that satisfies an F -set of constraints The search problemassociated with the decision problem SAT(F ) consists of finding an assignment thatsatisfies an F -set of constraints if such an assignment exists or then returning ldquonordquootherwise

In this chapter we will see that whenever we can solve the decision problemSAT(F ) in polynomial time we can also find a solution for the satisfiable instancesand therefore solve the associated search problem in polynomial time

It is normal practice to distinguish certain variants of SAT(F ) where each functionof F depends on at most (or exactly) k variables These variants are expressed askSAT(F ) (or EkSAT(F ))


We now present a few classical decision problems as well as the correspondingsatisfiability problem SAT(F )

ndash SAT is the problem that consists of deciding whether a set of disjunctive clausesdefined on n Boolean variables is satisfiable It corresponds to the SAT(F ) problemwhere F is the set of ORk

i functions for k n

ndash CONJ is the problem that consists of deciding whether a set of conjunctiveclauses defined on n Boolean variables is satisfiable It corresponds to the SAT(F )problem where F is the set of ANDk

i functions for k n

ndash LIN2 is the problem that consists of deciding whether a set of linear equationsdefined on n Boolean variables is satisfiable It corresponds to the SAT(F ) problemwhere F is the set of XORk XNORk functions for k n

ndash 2SAT is the version of SAT where each disjunctive clause has at most two literalsand it corresponds to 2SAT(F ) where F is the set of ORk

i functions for k 2

ndash E3SAT is the version of SAT where each disjunctive clause has exactly threeliterals and it corresponds to SAT(OR3

0 OR31 OR3

2 OR33)

DEFINITION 12ndash The maximization problem MAX SAT(F ) consists of establish-ing an assignment that satisfies a maximum number of constraints from an F -set ofconstraints

For example the MAX CUT problem which consists of partitioning the set ofvertices of a graph into two parts such that the number of edges whose extremitiesbelong to different parts is maximum can be formulated as a problem of the type MAX

SAT(XOR2) as follows Considering a graph G an instance of MAX CUT weassociate with each vertex i a variable xi and with each edge (i j) of G the constraintXOR2(xi xj)

DEFINITION 13ndash The minimization problem MIN SAT DELETION(F ) consists ofestablishing an assignment that minimizes the number of non-satisfied constraintsfrom an F -set of constraints which corresponds with the minimum number of con-straints to remove so that the remaining constraints are satisfied

MIN SAT DELETION(F ) allows us to model certain minimization problems natu-rally For example the s-t MIN CUT problem in a non-directed graph which consistsof partitioning the set of vertices of a graph into two parts such that s and t belong todifferent parts and such that the number of edges whose extremities belong to differ-ent parts is minimum can be formulated as a problem of the type MIN SAT DELE-TION(XNOR2cupT F) as follows Considering a graphG an instance of s-t MIN

CUT we associate with each vertex i a variable xi and with each edge (i j) of G theconstraint XNOR2(xi xj) Furthermore we add the constraints T (xs) and F (xt)


COMMENT 11ndash

1) The problems MAX SAT(F ) and MIN SAT DELETION(F ) are clearly relatedIndeed considering an instance I of MAX SAT(F ) with m constraints an optimalsolution for the instance I of MAX SAT(F ) of value optMAX SAT(F)(I) is alsoan optimal solution of the instance I of the MIN SAT DELETION(F ) problem ofvalue optMIN SAT DELETION(F)(I)= m-optMAX SAT(F)(I) Therefore the exactcomplexities of MAX SAT(F ) and MIN SAT DELETION(F ) coincide However theapproximation complexities of the two problems can be very different as we will seein what follows

2) In the literature we also define the MIN SAT(F ) problem that consists of es-tablishing an assignment that minimizes the number of satisfied constraints For ex-ample in the compendium of Crescenzi and Kann [CRE 95b] the MIN SAT prob-lem consists of establishing an assignment that minimizes the number of satisfiedclauses from a set of disjunctive clauses Note that MIN SAT(F ) is equivalent fromthe exact and approximation point of view to MIN SAT DELETION(F prime) where F prime

is the set of functions that are complementary to the functions of F For examplefinding an assignment that minimizes the number of constraints satisfied among theconstraints x1 or x2 x1 or x3 x2 or x3 x1 or x2 is equivalent to finding an assign-ment that minimizes the number of non-satisfied constraints among the constraintsx1 and x2 x1 and x3 x2 and x3 x1 and x2 Thus the MIN SAT problem is equivalent toMIN SAT DELETION(F ) where the constraints are conjunctive clauses (the problemcalled MIN CONJ DELETION) In what follows we will consider only the MIN SATDELETION(F ) problem

122 Constraint types

The complexity of SAT(F ) as well as the exact and approximation complexitiesof MAX SAT(F ) and MIN SAT DELETION(F ) depend on the types of Boolean func-tions of the set F In this section we describe the types of Boolean functions that havebeen most studied and that we will discuss in the rest of the chapter

A Boolean function f is

ndash 0-valid if f(0 0) = 1


ndash Horn if it can be expressed as a CNF expression that has at most one positiveliteral in each clause

ndash anti-Horn if it can be expressed as a CNF expression that has at most one nega-tive literal in each clause

ndash affine if it can be expressed as a conjunction of linear equations on the Galoisbody GF (2) that is as a conjunction of equations of type xi1 oplus oplus xip = 0 orxj1 oplus oplus xjq = 1











Table of Contents























































































































































List of Authors 675

Index 681


Th

Preface















i

















Preface xix



PART I


Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash























































































































































List of Authors 675

Index 681


Th

Preface















i

















Preface xix



PART I


Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash












Preface















i

















Preface xix



PART I


Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash




























Preface xix



PART I


Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash












PART I


Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash












Chapter 1


11 Introduction














12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash





















12 Preliminaries





) = 1























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash


























i functions for k n


i functions for k n



i functions for k 2


0 OR31 OR3

2 OR33)








COMMENT 11ndash













COMMENT 11ndash












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