The Design of Approximation Algorithms

517 Pages·2011·8.08 MB·English

by  David P. Williamson

TheDesignofApproximationAlgorithms
Discreteoptimizationproblemsareeverywhere,fromtraditionaloperationsresearchplan-
ning problems, such as scheduling, facility location, and network design; to computer
scienceproblemsindatabases;toadvertisingissuesinviralmarketing.Yetmostsuchprob-
lems are NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal
solutions to such problems. This book shows how to design approximation algorithms:
efficientalgorithmsthatfindprovablynear-optimalsolutions.
The book is organized around central algorithmic techniques for designing approxi-
mationalgorithms,includinggreedyandlocalsearchalgorithms,dynamicprogramming,
linearandsemidefiniteprogramming,andrandomization.Eachchapterinthefirstpartof
the book is devoted to a single algorithmic technique, which is then applied to several
different problems. The second part revisits the techniques but offers more sophisticated
treatmentsofthem.Thebookalsocoversmethodsforprovingthatoptimizationproblems
arehardtoapproximate.
Designedasatextbookforgraduate-levelalgorithmscourses,thebookwillalsoserve
as a reference for researchers interested in the heuristic solution of discrete optimization
problems.
David P. Williamson is a Professor at Cornell University with a joint appointment in the
School of Operations Research and Information Engineering and in the Department of
Information Science. Prior to joining Cornell, he was a Research Staff Member at the
IBM T.J. Watson Research Center and a Senior Manager at the IBM Almaden Research
Center. He has won several awards for his work on approximation algorithms, including
the2000FulkersonPrize,sponsoredbytheAmericanMathematicalSocietyandtheMath-
ematicalProgrammingSociety.Hehasservedonseveraleditorialboards,includingACM
Transactions onAlgorithms,Mathematics ofOperations Research,theSIAMJournalon
Computing,andtheSIAMJournalonDiscreteMathematics.
DavidB.ShmoysisaProfessoratCornellUniversitywithappointmentsinboththeSchool
ofOperationsResearchandInformationEngineeringandtheDepartmentofComputerSci-
ence. Heis currently Associate Director of the Institutefor Computational Sustainability
atCornell.HeisaFellowoftheACM,wasanNSFPresidentialYoungInvestigator,and
has served onnumerous editorialboards, including Mathematics ofOperations Research
(for which he is currently an associate editor), Operations Research, ORSA Journal on
Computing, Mathematical Programming, and both the SIAM Journal on Computing and
JournalonDiscreteMathematics;healsoservedaseditor-in-chiefforthelatter.

The Design of
Approximation Algorithms
David P. Williamson
CornellUniversity
David B. Shmoys
CornellUniversity
cambridgeuniversitypress
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Williamson,DavidP.
Thedesignofapproximationalgorithms/DavidP.Williamson,DavidB.Shmoys.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-19527-0(hardback)
1.Approximationtheory. 2.Mathematicaloptimization.
I.Shmoys,DavidBernard. II.Title.
QA221.W55 2011
518(cid:3).5–dc22 2011001945
ISBN978-0-521-19527-0Hardback
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Contents
Preface pageix
I AnIntroductiontotheTechniques
1 AnIntroductiontoApproximationAlgorithms 3
1.1 TheWhatsandWhysofApproximationAlgorithms 3
1.2 AnIntroductiontotheTechniquesandtoLinearProgramming:
TheSetCoverProblem 6
1.3 ADeterministicRoundingAlgorithm 9
1.4 RoundingaDualSolution 10
1.5 ConstructingaDualSolution:ThePrimal-DualMethod 14
1.6 AGreedyAlgorithm 15
1.7 ARandomizedRoundingAlgorithm 19
Exercises 22
ChapterNotes 24
2 GreedyAlgorithmsandLocalSearch 27
2.1 SchedulingJobswithDeadlinesonaSingleMachine 28
2.2 Thek-CenterProblem 30
2.3 SchedulingJobsonIdenticalParallelMachines 32
2.4 TheTravelingSalesmanProblem 35
2.5 MaximizingFloatinBankAccounts 40
2.6 FindingMinimum-DegreeSpanningTrees 43
2.7 EdgeColoring 47
Exercises 51
ChapterNotes 54
3 RoundingDataandDynamicProgramming 57
3.1 TheKnapsackProblem 57
3.2 SchedulingJobsonIdenticalParallelMachines 61
v
vi contents
3.3 TheBin-PackingProblem 66
Exercises 70
ChapterNotes 71
4 DeterministicRoundingofLinearPrograms 73
4.1 MinimizingtheSumofCompletionTimesonaSingleMachine 74
4.2 MinimizingtheWeightedSumofCompletionTimesona
SingleMachine 76
4.3 SolvingLargeLinearProgramsinPolynomialTimevia
theEllipsoidMethod 79
4.4 ThePrize-CollectingSteinerTreeProblem 80
4.5 TheUncapacitatedFacilityLocationProblem 83
4.6 TheBin-PackingProblem 88
Exercises 93
ChapterNotes 96
5 RandomSamplingandRandomizedRoundingofLinearPrograms 99
5.1 SimpleAlgorithmsforMAXSATandMAXCUT 100
5.2 Derandomization 103
5.3 FlippingBiasedCoins 104
5.4 RandomizedRounding 106
5.5 ChoosingtheBetterofTwoSolutions 109
5.6 NonlinearRandomizedRounding 111
5.7 ThePrize-CollectingSteinerTreeProblem 113
5.8 TheUncapacitatedFacilityLocationProblem 116
5.9 SchedulingaSingleMachinewithReleaseDates 119
5.10 ChernoffBounds 124
5.11 IntegerMulticommodityFlows 128
5.12 RandomSamplingandColoringDense3-ColorableGraphs 130
Exercises 132
ChapterNotes 135
6 RandomizedRoundingofSemidefinitePrograms 137
6.1 ABriefIntroductiontoSemidefiniteProgramming 137
6.2 FindingLargeCuts 139
6.3 ApproximatingQuadraticPrograms 143
6.4 FindingaCorrelationClustering 146
6.5 Coloring3-ColorableGraphs 149
Exercises 152
ChapterNotes 155
7 ThePrimal-DualMethod 157
7.1 TheSetCoverProblem:AReview 157
7.2 ChoosingVariablestoIncrease:TheFeedbackVertexSetProblem
inUndirectedGraphs 160
7.3 CleaningUpthePrimalSolution:TheShortests-t PathProblem 164
7.4 IncreasingMultipleVariablesatOnce:TheGeneralizedSteiner
TreeProblem 167
contents vii
7.5 StrengtheningInequalities:TheMinimumKnapsackProblem 174
7.6 TheUncapacitatedFacilityLocationProblem 177
7.7 LagrangeanRelaxationandthek-MedianProblem 182
Exercises 188
ChapterNotes 191
8 CutsandMetrics 193
8.1 TheMultiwayCutProblemandaMinimum-Cut–BasedAlgorithm 194
8.2 TheMultiwayCutProblemandanLPRoundingAlgorithm 195
8.3 TheMulticutProblem 201
8.4 BalancedCuts 206
8.5 ProbabilisticApproximationofMetricsbyTreeMetrics 210
8.6 AnApplicationofTreeMetrics:Buy-at-BulkNetworkDesign 215
8.7 SpreadingMetrics,TreeMetrics,andLinearArrangement 218
Exercises 224
ChapterNotes 227
II FurtherUsesoftheTechniques
9 FurtherUsesofGreedyandLocalSearchAlgorithms 231
9.1 ALocalSearchAlgorithmfortheUncapacitatedFacility
LocationProblem 232
9.2 ALocalSearchAlgorithmforthek-MedianProblem 238
9.3 Minimum-DegreeSpanningTrees 242
9.4 AGreedyAlgorithmfortheUncapacitatedFacility
LocationProblem 246
Exercises 251
ChapterNotes 253
10 FurtherUsesofRoundingDataandDynamicProgramming 255
10.1 TheEuclideanTravelingSalesmanProblem 256
10.2 TheMaximumIndependentSetProbleminPlanarGraphs 267
Exercises 275
ChapterNotes 277
11 FurtherUsesofDeterministicRoundingofLinearPrograms 279
11.1 TheGeneralizedAssignmentProblem 280
11.2 Minimum-CostBounded-DegreeSpanningTrees 284
11.3 SurvivableNetworkDesignandIteratedRounding 296
Exercises 303
ChapterNotes 307
12 FurtherUsesofRandomSamplingandRandomizedRounding
ofLinearPrograms 309
12.1 TheUncapacitatedFacilityLocationProblem 310
12.2 TheSingle-SourceRent-or-BuyProblem 314
12.3 TheSteinerTreeProblem 316
viii contents
12.4 EverythingatOnce:FindingaLargeCutinaDenseGraph 324
Exercises 329
ChapterNotes 331
13 FurtherUsesofRandomizedRoundingofSemidefinitePrograms 333
13.1 ApproximatingQuadraticPrograms 334
13.2 Coloring3-ColorableGraphs 341
13.3 UniqueGames 345
Exercises 352
ChapterNotes 354
14 FurtherUsesofthePrimal-DualMethod 355
14.1 ThePrize-CollectingSteinerTreeProblem 355
14.2 TheFeedbackVertexSetProbleminUndirectedGraphs 360
Exercises 366
ChapterNotes 368
15 FurtherUsesofCutsandMetrics 369
15.1 Low-DistortionEmbeddingsandtheSparsestCutProblem 370
15.2 ObliviousRoutingandCut-TreePackings 377
15.3 Cut-TreePackingsandtheMinimumBisectionProblem 383
15.4 TheUniformSparsestCutProblem 387
Exercises 406
ChapterNotes 408
16 TechniquesinProvingtheHardnessofApproximation 409
16.1 ReductionsfromNP-CompleteProblems 410
16.2 ReductionsthatPreserveApproximation 414
16.3 ReductionsfromProbabilisticallyCheckableProofs 424
16.4 ReductionsfromLabelCover 429
16.5 ReductionsfromUniqueGames 442
ChapterNotes 450
17 OpenProblems 453
AppendixA:LinearProgramming 459
AppendixB:NP-Completeness 465
Bibliography 469
AuthorIndex 485
SubjectIndex 491
Preface
This book is designed to be a textbook for graduate-level courses in approximation
algorithms.Aftersomeexperienceteachingminicoursesintheareainthemid-1990s,
we sat down and wrote out an outline of the book. Then one of us (DPW), who was
atthetimeanIBMResearchStaffMember,taughtseveraliterationsofthecoursefol-
lowingtheoutlinewehaddevised,inColumbiaUniversity’sDepartmentofIndustrial
EngineeringandOperationsResearchinSpring1998,inCornellUniversity’sSchoolof
OperationsResearchandIndustrialEngineeringinFall1998,andattheMassachusetts
InstituteofTechnology’sLaboratoryforComputerScienceinSpring2000.Thelecture
notesfromthesecoursesweremadeavailable,andwegotenoughpositivefeedbackon
themfromstudentsandfromprofessorsteachingsuchcourseselsewherethatwefelt
wewereontherighttrack.Sincethen,therehavebeenmanyexcitingdevelopmentsin
thearea,andwehaveaddedmanyofthemtothebook;wetaughtadditionaliterations
of the course at Cornell in Fall 2006 and Fall 2009 in order to field test some of the
writingofthenewerresults.
Thecoursesweredevelopedforstudentswhohavealreadyhadaclass,undergraduate
or graduate, in algorithms, and who were comfortable with the idea of mathematical
proofsaboutthecorrectnessofalgorithms.Thebookassumesthislevelofpreparation.
Thebookalsoassumessomebasicknowledgeofprobabilitytheory(forinstance,how
tocomputetheexpectedvalueofadiscreterandomvariable).Finally,weassumethat
the reader knows something about NP-completeness, at least enough to know that
theremightbegoodreasonforwantingfast,approximatesolutionstoNP-harddiscrete
optimizationproblems.Atoneortwopointsinthebook,wedoanNP-completeness
reductiontoshowthatitcanbehardtofindapproximatesolutionstosuchproblems;we
includeashortappendixontheproblemclassNPandthenotionofNP-completeness
for those unfamiliar with the concepts. However, the reader unfamiliar with such
reductionscanalsosafelyskipoversuchproofs.
Inadditiontoservingasagraduatetextbook,thisbookisawayforstudentstoget
the background to read current research in the area of approximation algorithms. In
particular,wewantedabookthatwecouldhandourownPh.D.studentsjuststarting
inthefieldandsay,“Here,readthis.”
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