Design and Analysis of Approximation Algorithms

453 Pages·2012·2.87 MB·English

Springer Optimization and Its Applications
VOLUME62
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor–CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(PrincetonUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(McMasterUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimization has been expanding in all directions at an astonishing rate
during the last few decades. New algorithmic and theoretical techniques
have been developed, the diffusion into other disciplines has proceeded at
a rapid pace, and our knowledge of all aspects of the field has grown even
more profound. At the same time, one of the most striking trends in opti-
mization is the constantly increasing emphasis on the interdisciplinary na-
ture of the field. Optimization has been a basic tool in all areas of applied
mathematics,engineering,medicine,economics,andothersciences.
The series Springer Optimization and Its Applications publishes under-
graduate and graduate textbooks, monographs and state-of-the-art exposi-
tory work that focus on algorithms for solving optimization problems and
alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered
includenonlinearoptimization(convexandnonconvex),networkflowprob-
lems,stochasticoptimization,optimalcontrol,discreteoptimization,multi-
objective programming, description of software packages, approximation
techniquesandheuristicapproaches.
Forfurthervolumes:
http://www.springer.com/series/7393

Ding-Zhu Du • Ker-I Ko • Xiaodong Hu
Design and Analysis
of Approximation Algorithms
Ding-Zhu Du Ker-I Ko
Department of Computer Science Department of Computer Science
University of Texas at Dallas State University of New York at Stony Brook
Richardson, TX 75080 Stony Brook, NY 11794
USA USA
[email protected] [email protected]

Xiaodong Hu
Institute of Applied Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100190
China
[email protected]
ISSN 1931-6828
ISBN 978-1-4614-1700-2 e-ISBN 978-1-4614-1701-9
DOI 10.1007/978-1-4614-1701-9
Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011942512

(cid:164) Springer Science+Business Media, LLC 2012
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Preface
Anapproximationalgorithmisanefficientalgorithmthatproducessolutionstoan
optimizationproblemthat are guaranteed tobe withina fixed ratio of the optimal
solution. Instead of spending an exponential amount of time finding the optimal
solution,anapproximationalgorithmsettlesfornear-optimalsolutionswithinpoly-
nomialtimeintheinputsize.Approximationalgorithmshavebeenstudiedsincethe
mid-1960s.Their importance was, however, notfullyunderstooduntilthe discov-
eryoftheNP-completenesstheory.Manywell-knownoptimizationproblemshave
been proved, under reasonable assumptionsin thistheory,to be intractable,in the
sense that optimalsolutionsto these problems are not computable withinpolyno-
mial time. As a consequence, near-optimal approximationalgorithms are the best
onecanexpectwhentryingtosolvetheseproblems.
Inthepastdecade, thearea ofapproximationalgorithmshasexperiencedanex-
plosiverateofgrowth.Thisgrowthrateispartlyduetothedevelopmentofrelated
research areas, suchas datamining,communicationnetworks,bioinformatics,and
computationalgametheory.Thesenewlyestablishedresearchareasgeneratealarge
numberofnew,intractableoptimizationproblems,mostofwhichhavedirectappli-
cationstoreal-worldproblems,and so efficientapproximatesolutionstothem are
activelysoughtafter.
Inadditiontotheexternal,practicalneedforefficientapproximationalgorithms,
there is also an intrinsic,theoretical motive behindthe research of approximation
algorithms.Inthedesignofan exact-solutionalgorithm,themain, andoftenonly,
measureofthealgorithm’sperformanceisitsrunningtime.Thisfixedmeasure of-
tenlimitsourchoiceoftechniquesinthealgorithm’sdesign.Foranapproximation
algorithm,however, thereisanequallyimportantsecond measure, thatis,theper-
formance ratioof the algorithm,whichmeasures howclose theapproximational-
v
vi Preface
gorithm’soutputistotheoptimalsolution.Thismeasure addsanewdimensionto
thedesignandanalysisofapproximationalgorithms.Namely,wecannowstudythe
tradeoffbetweentherunningtimeandtheperformanceratioofapproximationalgo-
rithms,andapplydifferentdesigntechniquestoachievedifferenttradeoffsbetween
thesetwomeasures. Inaddition,newtheoreticalissuesabouttheapproximationto
anoptimizationproblemneedtobeaddressed:Whatistheperformanceratioofan
approximationalgorithmforthisproblembasedoncertaintypesofdesignstrategy?
Whatisthebestperformanceratioofanypolynomial-timeapproximationalgorithm
forthisproblem?Doestheproblemhaveapolynomial-timeapproximationscheme
orafullypolynomial-timeapproximationscheme? Thesequestionsarenotonlyof
significanceinpracticeforthedesignofapproximationalgorithms;theyarealsoof
greattheoreticalinterest, withintriguingconnectionstotheNP-completeness the-
ory.
Motivatedbythesetheoreticalquestionsandthegreatnumberofnewlydiscov-
ered optimization problems, people have developed many new design techniques
forapproximationalgorithms,includingthegreedystrategy,therestrictionmethod,
therelaxationmethod,partition,localsearch,powergraphs,andlinearandsemidef-
initeprogramming. A comprehensive survey of all these methods and results in a
singlebookisnotpossible.Weinsteadprovideinthisbookanintensivestudyofthe
mainmethods,withabundantapplicationsfollowingourdiscussionofeachmethod.
Indeed, thisbookis organizedaccording to designmethods insteadof application
problems. Thus, one can study approximation algorithms of the same nature to-
gether,andlearnaboutthedesigntechniquesinamoreunifiedway.Tothisend,the
bookisarrangedinthefollowingway:First,inChapter1,wegiveabriefintroduc-
tiontotheconceptofNP-completeness andapproximationalgorithms.InChapter
2,wegiveanin-depthanalysisofthegreedystrategy,includinggreedyalgorithms
withsubmodularpotentialfunctionsandthosewithnonsubmodularpotentialfunc-
tions.InChapters3,4,and5,wecover variousrestrictionmethods,includingpar-
titionand Guillotinecut methods, withapplicationsto many geometric problems.
Inthenextfourchapters,westudytherelaxationmethods.Inadditiontoageneral
discussionof the relaxationmethod in Chapter 6, we devote three chapters toap-
proximationalgorithms based on linear and semidefinite programming, including
theprimal-dualschema anditsequivalencewiththelocalratiomethod.Finally,in
Chapter10, we present various inapproximabilityresults based on recent workin
theNP-completenesstheory.Anumberofexamplesandexercises areprovidedfor
each design technique. They are drawn from diverse areas of research, including
communicationnetworkdesign,opticalnetworks,wirelessadhocnetworks,sensor
networks,bioinformatics,socialnetworks,industrialengineering,andinformation
managementsystems.
This bookhas grownout oflecture notes used bythe authorsat the University
ofMinnesota,UniversityofTexas atDallas,TsinghuaUniversity,GraduateSchool
ofChinese Academy ofSciences, Xi’an JiaotongUniversity,Zhejiang University,
EastChinaNormalUniversity,DalianUniversityofTechnology,XinjiangUniver-
sity,NankaiUniversity,LanzhouJiaotongUniversity,XidianUniversity,andHarbin
InstituteofTechnology. Ina typicalone-semester class forfirst-year graduatestu-
Preface vii
dents,one may cover the first twochapters, one ortwochapters onthe restriction
method,twoorthreechaptersontherelaxationmethod,andChapter10.Withmore
advanced students, one may also teach a seminar course focusing on one of the
greedy, restriction,or relaxationmethods, based on the correspondingchapters of
thisbookandsupplementarymaterialfrom recent research papers. Forinstance, a
seminaroncombinatorialoptimizationemphasizingapproximationsbasedonlinear
andsemidefiniteprogrammingcanbeorganizedusingChapters7,8,and9.
Thisbookhasbenefitedmuchfromthehelpofourfriends,colleagues,andstu-
dents.WeareindebtedtoPeng-JunWan,WeiliWu,XiuzhenCheng,JieWang,Yin-
fengXu,ZhaoZhang,DeyingLi,HejiaoHuang,HongZhu,GuochuanZhang,Wei
Wang,ShugangGao,XiaofengGao,FengZou,LingDing,XianyueLi,MyT.Thai,
DonghyunKim,J.K.Willson,andRoozbehEbrahimiSoorchaei,whomademuch-
valued suggestions and corrections to the earlier drafts of the book. We are also
gratefultoProfessorsFrancesYao,RichardKarp,RonaldGraham,andFanChung
fortheirencouragement. Special thanksare dueto ProfessorAndrew Yao and the
InstituteforTheoretical Computer Science, Tsinghua University,for the generous
supportandstimulatingenvironmenttheyprovidedforthefirsttwoauthorsduring
theirnumerousvisitstoTsinghuaUniversity.
Dallas,Texas Ding-ZhuDu
StonyBrook,NewYork Ker-IKo
Beijing,China XiaodongHu
August2011

Contents
Preface v
1 Introduction 1
1.1 OpenSesame 1
1.2 DesignTechniquesforApproximationAlgorithms 8
1.3 HeuristicsVersusApproximation 13
1.4 NotionsinComputationalComplexity 14
1.5 NP-CompleteProblems 17
1.6 PerformanceRatios 23
Exercises 28
HistoricalNotes 33
2 GreedyStrategy 35
2.1 IndependentSystems 35
2.2 Matroids 40
2.3 QuadrilateralConditiononCostFunctions 43
2.4 SubmodularPotentialFunctions 49
2.5 Applications 59
2.6 NonsubmodularPotentialFunctions 66
Exercises 75
HistoricalNotes 80
3 Restriction 81
3.1 SteinerTreesandSpanningTrees 82
3.2 k-RestrictedSteinerTrees 86
3.3 Greedyk-RestrictedSteinerTrees 89
ix