Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications PDF
Preview Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications
GRAPH THEORY, COMBINATORICS AND ALGORITHMS INTERDISCIPLINARY APPLICATIONS GRAPH THEORY, COMBINATORICS AND ALGORITHMS INTERDISCIPLINARY APPLICATIONS Edited by Martin Charles Golumbic Irith Ben-Arroyo Hartman MartinCharlesGolumbic IrithBen-ArroyoHartman UniversityofHaifa,Israel UniversityofHaifa,Israel LibraryofCongressCataloging-in-PublicationData Graphtheory,combinatorics,andalgorithms/[edited]byMartinCharlesGolumbic, IrithBen-ArroyoHartman. p.cm. Includesbibliographicalreferences. ISBN-10:0-387-24347-X ISBN-13:978-0387-24347-4 e-ISBN0-387-25036-0 1.Graphtheory. 2.Combinatorialanalysis. 3.Graphtheory—Dataprocessing. I.Golumbic,MartinCharles. II.Hartman,IrithBen-Arroyo. QA166.G71672005 511(cid:1).5—dc22 2005042555 Copyright(cid:2)C 2005bySpringerScience+BusinessMedia,Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 SPIN11374107 springeronline.com Contents Foreword............................................................. vii Chapter1 OptimizationProblemsRelatedtoInternetCongestion Control RichardKarp ............................................. 1 Chapter2 ProblemsinDataStructuresandAlgorithms RobertTarjan............................................. 17 Chapter3 AlgorithmicGraphTheoryanditsApplications MartinCharlesGolumbic................................... 41 Chapter4 DecompositionsandForcingRelationsinGraphsandOther CombinatorialStructures RossMcConnell........................................... 63 Chapter5 TheLocalRatioTechniqueanditsApplicationtoScheduling andResourceAllocationProblems ReuvenBar-Yehuda,KerenBendel,AriFreundandDrorRawitz 107 Chapter6 DominationAnalysisofCombinatorialOptimization AlgorithmsandProblems GregoryGutinandAndersYeo.............................. 145 Chapter7 OnMulti-ObjectAuctionsandMatchingTheory: AlgorithmicAspects MichalPennandMosheTennenholtz........................ 173 Chapter8 StrategiesforSearchingGraphs ShmuelGal............................................... 189 Chapter9 RecentTrendsinArcRouting AlainHertz............................................... 215 Chapter10 SoftwareandHardwareTestingUsingCombinatorial CoveringSuites AlanHartman............................................. 237 Chapter11 Incidences JanosPachandMichaSharir ............................... 267 Foreword The Haifa Workshops on Interdisciplinary Applications of Graph Theory, Combina- torics and Algorithms have been held at the Caesarea Rothschild Institute (C.R.I.), University of Haifa, every year since 2001. This volume consists of survey chapters basedonpresentationsgivenatthe2001and2002Workshops,aswellasothercollo- quiagivenatC.R.I.TheRothschildLecturesofRichardKarp(Berkeley)andRobert Tarjan(Princeton),bothTuringawardwinners,werethehighlightsoftheWorkshops. Two chapters based on these talks are included. Other chapters were submitted by selectedauthorsandwerepeerreviewedandedited.Thisvolume,writtenbyvarious expertsinthefield,focusesondiscretemathematicsandcombinatorialalgorithmsand theirapplicationstorealworldproblemsincomputerscienceandengineering.Abrief summaryofeachchapterisgivenbelow. RichardKarp’soverview,OptimizationProblemsRelatedtoInternetCongestion Control,presentssomeofthemajorchallengesandnewresultsrelatedtocontrolling congestion in the Internet. Large data sets are broken down into smaller packets, all competingforcommunicationresourcesonanimperfectchannel.Thetheoreticalissues addressedbyProf.Karpleadtoadeeperunderstandingofthestrategiesformanaging thetransmissionofpacketsandtheretransmissionoflostpackets. Robert Tarjan’s lecture, Problems in Data Structures and Algorithms, provides an overview of some data structures and algorithms discovered by Tarjan during the courseofhiscareer.Tarjangivesaclearexpositionofthealgorithmicapplicationsof basicstructureslikesearchtreesandself-adjustingsearchtrees,alsoknownassplay trees. Some open problems related to these structures and to the minimum spanning treeproblemarealsodiscussed. ThethirdchapterbyMartinCharlesGolumbic,AlgorithmicGraphTheoryandits Applications,isbasedonasurveylecturegivenatClemsonUniversity.Thischapteris aimedatthereaderwithlittlebasicknowledgeofgraphtheory,anditintroducesthe readertotheconceptsofintervalgraphsandotherfamiliesofintersectiongraphs.The lectureincludesdemonstrationsoftheseconceptstakenfromreallifeexamples. ThechapterDecompositionsandForcingRelationsinGraphsandotherCombi- natorialStructuresbyRossMcConnelldealswithproblemsrelatedtoclassesofinter- section graphs, including interval graphs, circular-arc graphs, probe interval graphs, permutationgraphs,andothers.McConnellpointstoageneralstructurecalledmodu- lardecompositionwhichhelpstoobtainlinearboundsforrecognizingsomeofthese graphs,andsolvingotherproblemsrelatedtothesespecialgraphclasses. viii Foreword IntheirchapterTheLocalRatioTechniqueanditsApplicationtoSchedulingand ResourceAllocationProblems,Bar-Yehuda,Bendel,FreundandRawitzgiveasurvey ofthelocalratiotechniqueforapproximationalgorithms.Anapproximationalgorithm efficientlyfindsafeasiblesolutiontoanintractableproblemwhosevalueapproximates theoptimum.Therearenumerousreallifeintractableproblems,suchasthescheduling problem,whichcanbeapproachedonlythroughheuristicsorapproximationalgorithms. This chapter contains a comprehensive survey of approximation algorithms for such problems. DominationAnalysisofCombinatorialOptimizationAlgorithmsandProblemsby GutinandYeoprovidesanalternativeandacomplementtoapproximationanalysis.One ofthegoalsofdominationanalysisistoanalyzethedominationratioofvariousheuristic algorithms. Given a problem P and a heuristic H, the ratio between the number of feasiblesolutionsthatarenotbetterthanasolutionproducedbyH,andthetotalnumber of feasible solutions to P, is the domination ratio. The chapter discusses domination analysesofvariousheuristicsforthewell-knowntravelingsalesmanproblem,aswellas otherintractablecombinatorialoptimizationproblems,suchastheminimumpartition problem,multiprocessorscheduling,maximumcut,k-satisfiability,andothers. Anotherreal-lifeproblemisthedesignofauctions.IntheirchapterOnMulti-Object Auctions and Matching Theory: Algorithmic Aspects, Penn and Tennenholtz use b- matchingtechniquestoconstructefficientalgorithmsforcombinatorialandconstrained auction problems. The typical auction problem can be described as the problem of designingamechanismforsellingasetofobjectstoasetofpotentialbuyers.Inthe combinatorial auction problem bids for bundles of goods are allowed, and the buyer may evaluate a bundle of goods for a different value than the sum of the values of eachgood.Inconstrainedauctionssomerestrictionsareimposeduponthesetfeasible solutions,suchastheguaranteethataparticularbuyerwillgetatleastonegoodfrom a given set. Both combinatorial and constrained auction problems are NP-complete problems, however, the authors explore special tractable instances where b-matching techniquescanbeusedsuccessfully. ShmuelGal’schapterStrategiesforSearchingGraphsisrelatedtotheproblemof detectinganobjectsuchasaperson,avehicle,orabombhidinginagraph(onanedge oratavertex).Itisgenerallyassumedthatthereisnoknowledgeabouttheprobability distributionofthetarget’slocationand,insomecases,eventhestructureofthegraphis notknown.Galusesprobabilisticmethodstofindoptimalsearchstrategiesthatassure findingthetargetinminimumexpectedtime. The chapter Recent Trends in Arc Routing by Alain Hertz studies the problem of finding a least cost tour of a graph, with demands on the edges, using a fleet of identical vehicles. This problem and other related problems are intractable, and the chapterreportsonrecentexactandheuristicalgorithms.Theproblemhasapplications in garbage collection, mail delivery, snow clearing, network maintenance, and many others. Foreword ix Software and Hardware Testing Using Combinatorial Covering Suites by Alan Hartmanisanexampleoftheinterplaybetweenpuremathematics,computerscience, andtheappliedproblemsgeneratedbysoftwareandhardwareengineers.Theconstruc- tionofefficientcombinatorialcoveringsuiteshasimportantapplicationsinthetesting ofsoftwareandhardwaresystems.Thischapterdiscussesthelowerboundsonthesize ofcoveringsuites,andgivesaseriesofconstructionsthatachievetheseboundsasymp- totically.Theseconstructionsinvolvetheuseoffinitefieldtheory,extremalsettheory, group theory, coding theory, combinatorial recursive techniques, and other areas of computerscienceandmathematics. JanosPachandMichaSharir’schapter,Incidences,relatestothefollowinggeneral problemincombinatorialgeometry:Whatisthemaximumnumberofincidencesbe- tweenmpointsandnmembersofafamilyofcurvesorsurfacesind-space?Resultsof thiskindhavenumerousapplicationstogeometricproblemsrelatedtothedistribution ofdistancesamongpoints,toquestionsinadditivenumbertheory,inanalysis,andin computationalgeometry. Wewouldliketothanktheauthorsfortheirenthusiasticresponsetothechallenge of writing a chapter in this book. We also thank the referees for their comments and suggestions.Finally,thisbook,andmanyworkshops,internationalvisits,coursesand projectsatCRI,aretheresultsofagenerousgrantfromtheCaesareaEdmondBenjamin deRothschildFoundation.Wearegreatlyindebtedfortheirsupportthroughoutthelast fouryears. MartinCharlesGolumbic IrithBen-ArroyoHartman CaesareaEdmondBenjamin deRothschildFoundationInstitutefor InterdisciplinaryApplicationsofComputerScience UniversityofHaifa,Israel 1 Optimization Problems Related to Internet Congestion Control Richard Karp DepartmentofElectricalEngineeringandComputerSciences UniversityofCalifornia,Berkeley Introduction I’m going to be talking about a paper by Elias Koutsoupias, Christos Papadim- itriou,ScottShenkerandmyself,thatwaspresentedatthe2000FOCSConference[1] relatedtoInternet-congestioncontrol.Somepeopleduringthecoffeebreakexpressed surprisethatI’mworkinginthisarea,becauseoverthelastseveralyears,Ihavebeen concentratingmoreoncomputationalbiology,theareaonwhichRonShamirreported soeloquentlyinthelastlecture.Iwashavingtroubleexplaining,eventomyself,howit isthatI’vebeenworkinginthesetwoveryseparatefields,untilRonPinterjustexplained ittome,afewminutesago.Hepointedouttomethatimprovingtheperformanceof thewebiscruciallyimportantforbioinformatics,becauseafterall,peoplespendmost oftheirtimeconsultingdistributeddatabases.Sothisismyexplanation,afterthefact, forworkinginthesetwofields. TheModel InordertosetthestagefortheproblemsI’mgoingtodiscuss,let’stalkinslightly oversimplified terms about how information is transmitted over the Internet. We’ll consider the simplest case of what’s called unicast—the transmission of message or file D from one Internet host, or node, A to another node B. The data D, that host A wishes to send to host B is broken up into packets of equal size which are assigned consecutiveserialnumbers.Thesepacketsformaflowpassingthroughaseriesoflinks androutersontheInternet.Asthepacketsflowthroughsomepathoflinksandrouters, theypassthroughqueues.Eachlinkhasoneormorequeuesoffinitecapacityinwhich packets are buffered as they pass through the routers. Because these buffers have a finite capacity, the queues may sometimes overflow. In that case, a choice has to be 2 RichardKarp madeastowhichpacketsshallbedropped.Therearevariousqueuedisciplines.The onemostcommonlyused,becauseitisthesimplest,isasimplefirst-in-first-out(FIFO) discipline.Inthatcase,whenpacketshavetobedropped,thelastpackettoarrivewillbe thefirsttobedropped.Theotherswillpassthroughthequeueinfirst-in-first-outorder. The Internet Made Simple • A wishes to send data to B • D is broken into equal packets with consecutive serial numbers • The packets form a flow passing through a sequence of links and routers. • Each link has one or more queues of finite capacity. When a packet arrives at a full queue, it is dropped. First-in-first-outdisciplines,aswewillsee,havecertaindisadvantages.Therefore, peopletalkaboutfairqueuingwhereseveral,morecomplicateddatastructuresareused inordertotreatallofthedataflowsmorefairly,andinordertotransmitapproximately thesamenumberofpacketsfromeachflow.Butinpractice,theoverheadoffairqueuing istoolarge,althoughsomeapproximationstoithavebeencontemplated.Andso,this first-in-first-outqueuingisthemostcommonqueuingdisciplineinpracticaluse. Now, since not all packets reach their destination, there has to be a mechanism forthereceivertoletthesenderknowwhetherpacketshavebeenreceived,andwhich packetshavebeenreceived,sothatthesendercanretransmitdroppedpackets.Thus, whenthereceiverBreceivesthepackets,itsendsbackanacknowledgementtoA.There arevariousconventionsaboutsendingacknowledgements.ThesimplestoneiswhenB simplyletsAknowtheserialnumberofthefirstpacketnotyetreceived.InthatcaseA willknowthatconsecutivepacketsuptosomepointhavebeenreceived,butwon’tknow aboutthepacketsafterthatpointwhichmayhavebeenreceivedsporadically.Depending onthisflowofacknowledgementsbacktoA,Awilldetectthatsomepacketshavebeen droppedbecauseanacknowledgementhasn’tbeenreceivedwithinareasonabletime, andwillretransmitcertainofthesepackets. The most undesirable situation is when the various flows are transmitting too rapidly.Inthatcase,thedisasterofcongestioncollapsemayoccur,inwhichsomany packets are being sent that most of them never get through—they get dropped. The acknowledgementtellsthesenderthatthepackethasbeendropped.Thesendersends
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