Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications

296 Pages·2005·14.74 MB·English

by  Martin Charles Golumbic

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