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Maximum independent set and related problems, with applications

University of Florida Institutional Repository

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Firstofall,IwouldliketothankPanosPardalos,whohasbeenagreatsupervisorandmentorthroughoutmyfouryearsattheUniversityofFlorida.Hisinspiringenthusiasmandenergyhavebeencontagious,andhisadviceandconstantsupporthavebeenextremelyhelpful. SpecialthanksgotoStanUryasev,whorecruitedmetotheUniversityofFloridaandwasalwaysverysupportive.IwouldalsoliketoacknowledgemycommitteemembersWilliamHager,EdwinRomeijn,andMaxShenfortheirtimeandguidance. IamgratefultomycollaboratorsJamesAbello,VladimirBoginski,StasBusygin,XiuzhenCheng,Ding-ZhuDu,AlexanderGolodnikov,CarlosOliveira,MauricioG.C.Resende,IvanV.Sergienko,VladimirShylo,OlegProkopyev,PetroStetsyuk,andVitaliyYatsenko,whohavebeenapleasuretoworkwith. IwouldliketothankDonaldHearnforfacilitatingmygraduatestudiesbyprovidingnancialaswellasmoralsupport.Iamalsogratefultothefaculty,sta,andstudentsoftheIndustrialandSystemsEngineeringDepartmentattheUniversityofFloridaforhelpingmakemyexperiencehereunforgettable. Finally,myutmostappreciationgoestomyfamilymembers,andespeciallymywifeJoanna,whoselove,understandingandfaithinmemadeitakeyingredientofthiswork. iii

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vii LISTOFFIGURES ................................ ix ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 1.1DenitionsandNotations ....................... 4 1.2ComplexityResults .......................... 7 1.3LowerandUpperBounds ....................... 7 1.4ExactAlgorithms ........................... 9 1.5Heuristics ............................... 11 1.5.1SimulatedAnnealing ..................... 11 1.5.2NeuralNetworks ........................ 13 1.5.3GeneticAlgorithms ...................... 14 1.5.4GreedyRandomizedAdaptiveSearchProcedures ...... 14 1.5.5TabuSearch .......................... 15 1.5.6HeuristicsBasedonContinuousOptimization ........ 15 1.6Applications .............................. 16 1.6.1MatchingMolecularStructuresbyCliqueDetection .... 16 1.6.2MacromolecularDocking ................... 17 1.6.3IntegrationofGenomeMappingData ............ 17 1.6.4ComparativeModelingofProteinStructure ......... 17 1.6.5CoveringLocationUsingCliquePartition .......... 18 1.7OrganizationoftheDissertation ................... 19 2MATHEMATICALPROGRAMMINGFORMULATIONS ........ 21 2.1IntegerProgrammingFormulations ................. 21 2.2ContinuousFormulations ....................... 22 2.3PolynomialFormulationsOvertheUnitHypercube ........ 26 2.3.1Degree(+1)PolynomialFormulation ........... 26 2.3.2QuadraticPolynomialFormulation .............. 29 2.3.3RelationBetween(P2)andMotzkin-StrausQP ....... 31 2.4AGeneralizationforDominatingSets ................ 33 iv

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37 3.1HeuristicBasedonFormulation(P1) ................ 37 3.2HeuristicBasedonFormulation(P2) ................ 39 3.3Examples ............................... 40 3.3.1Example1 ........................... 40 3.3.2Example2 ........................... 41 3.3.3Example3 ........................... 42 3.3.4ComputationalExperiments ................. 43 3.4HeuristicBasedonOptimizationofaQuadraticOveraSphere .. 46 3.4.1OptimizationofaQuadraticFunctionOveraSphere .... 49 3.4.2TheHeuristic ......................... 52 3.4.3ComputationalExperiments ................. 54 3.5ConcludingRemarks ......................... 57 4APPLICATIONSINMASSIVEDATASETS ............... 59 4.1ModelingandOptimizationinMassiveGraphs ........... 59 4.1.1ExamplesofMassiveGraphs ................. 60 4.1.2ExternalMemoryAlgorithms ................. 68 4.1.3ModelingMassiveGraphs ................... 69 4.1.4OptimizationinRandomMassiveGraphs .......... 79 4.1.5Remarks ............................ 82 4.2TheMarketGraph .......................... 83 4.2.1ConstructingtheMarketGraph ............... 83 4.2.2ConnectivityoftheMarketGraph .............. 85 4.2.3DegreeDistributionsintheMarketGraph ......... 87 4.2.4CliquesandIndependentSetsintheMarketGraph .... 90 4.2.5InstrumentsCorrespondingtoHigh-DegreeVertices .... 93 4.3EvolutionoftheMarketGraph ................... 94 4.4Conclusion ............................... 100 5APPLICATIONSINCODINGTHEORY ................. 103 5.1Introduction .............................. 103 5.2FindingLowerBoundsandExactSizesoftheLargestCodes ... 105 5.2.1FindingtheLargestCorrectingCodes ............ 107 5.3LowerBoundsforCodesCorrectingOneErrorontheZ-Channel 113 5.3.1ThePartitioningMethod ................... 114 5.3.2ThePartitioningAlgorithm .................. 116 5.3.3ImprovedLowerBoundsforCodeSizes ........... 117 5.4Conclusion ............................... 119 6APPLICATIONSINWIRELESSNETWORKS .............. 121 6.1Introduction .............................. 121 6.2An8-ApproximateAlgorithmtoComputeCDS .......... 125 v

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..................... 125 6.2.2PerformanceAnalysis ..................... 127 6.3NumericalExperiments ........................ 130 6.4Conclusion ............................... 130 7CONCLUSIONSANDFUTURERESEARCH ............... 134 7.1ExtensionstotheMAX-CUTProblem ............... 134 7.2CriticalSetsandtheMaximumIndependentSetProblem ..... 136 7.2.1Results ............................. 137 7.3Applications .............................. 138 REFERENCES ................................... 140 BIOGRAPHICALSKETCH ............................ 155 vi

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LISTOFTABLES Table page 3{1Resultsonbenchmarkinstances:Algorithm1,random x 0 ....... 44 3{2Resultsonbenchmarkinstances:Algorithm2,random x 0 ....... 45 3{3Resultsonbenchmarkinstances:Algorithm3,random x 0 ....... 46 3{4Resultsonbenchmarkinstances:Algorithms1{3, x 0 i =0,for i =1 ;:::;n 47 3{5Resultsonbenchmarkinstances:comparisonwithothercontinuous basedapproaches. x 0 i =0 ;i =1 ;:::;n ................ 47 3{6Resultsonbenchmarkinstances,partI. ................. 55 3{7Resultsonbenchmarkinstances,partII. ................. 56 3{8Comparisonoftheresultsonbenchmarkinstances. ........... 57 4{1Clusteringcoecientsofthemarketgraph( -complementarygraph) 90 4{2Sizesofcliquesfoundusingthegreedyalgorithmandsizesofgraphs remainingafterapplyingthepreprocessingtechnique ........ 91 4{3Sizesofthemaximumcliquesinthemarketgraphwithdierentvalues ofthecorrelationthreshold ....................... 92 4{4Sizesofindependentsetsfoundusingthegreedyalgorithm ...... 93 4{5Top25instrumentswithhighestdegrees( =0 : 6). .......... 95 4{6Datescorrespondingtoeach500-dayshift. ............... 97 4{7Numberofverticesandnumberofedgesinthemarketgraph( =0 : 5) fordierentperiods. ........................... 99 4{8Verticeswiththehighestdegreesinthemarketgraphfordierent periods( =0 : 5). ............................ 101 5{1Lowerboundsobtained. .......................... 107 5{2Exactalgorithm:computationalresults. ................. 112 5{3Exactsolutionsfound. ........................... 113 5{4Lowerbounds. ............................... 115 vii

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.................. 118 5{6Partitionsofconstantweightcodes. ................... 119 5{7Newlowerbounds. ............................. 119 6{1Performancecomparisonofthealgorithms ................ 129 viii

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LISTOFFIGURES Figure page 3{1IllustrationtoExample1 ......................... 41 3{2IllustrationtoExample2 ......................... 41 3{3IllustrationtoExample3 ......................... 42 4{1FrequenciesofcliquesizesinthecallgraphfoundbyAbelloetal. .. 61 4{2Numberofverticeswithvariousout-degrees(a)andin-degrees(b); thenumberofconnectedcomponentsofvarioussizes(c)inthecall graph,duetoAielloetal. ....................... 62 4{3NumberofInternethostsfortheperiod01/1991-01/2002.Databy InternetSoftwareConsortium. ..................... 63 4{4AsampleofpathsofthephysicalnetworkofInternetcablescreated byW.CheswickandH.Burch ..................... 64 4{5Numberofverticeswithvariousout-degrees(left)anddistributionof sizesofstronglyconnectedcomponents(right)inWebgraph .... 65 4{6ConnectivityoftheWebduetoBroderetal. .............. 67 4{7Distributionofcorrelationcoecientsinthestockmarket ....... 84 4{8Edgedensityofthemarketgraphfordierentvaluesofthecorrelation threshold. ................................ 85 4{9Plotofthesizeofthelargestconnectedcomponentinthemarketgraph asafunctionofcorrelationthreshold ................ 86 4{10Degreedistributionofthemarketgraphfor(a) =0 : 2;(b) =0 : 3; (c) =0 : 4;(d) =0 : 5 ......................... 88 4{11Degreedistributionofthecomplementarymarketgraphfor(a) = 0 : 15;(b) = 0 : 2;(c) = 0 : 25 .................. 89 4{12Timeshiftsusedforstudyingtheevolutionofthemarketgraphstructure. 96 4{13Distributionofthecorrelationcoecientsbetweenallconsideredpairs ofstocksinthemarket,forodd-numberedtimeshifts. ....... 97 ix

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....... 98 4{15Growthdynamicsoftheedgedensityofthemarketgraphovertime. 99 5{1AschemeoftheZ-channel. ........................ 113 5{2Algorithmforndingindependentsetpartitions. ............ 117 6{1Approximatingthevirtualbackbonewithaconnecteddominatingsetinaunit-diskgraph .......................... 123 6{2AveragedresultsforR=15inrandomgraphs. ............. 131 6{3AveragedresultsforR=25inrandomgraphs. ............. 131 6{4AveragedresultsforR=50inrandomgraphs. ............. 132 6{5AveragedresultsforR=15inuniformgraphs. ............. 132 6{6AveragedresultsforR=25inuniformgraphs. ............. 133 6{7AveragedresultsforR=50inuniformgraphs. ............. 133 x

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Thisdissertationdevelopsnovelapproachestosolvingcomputationallydicultcombinatorialoptimizationproblemsongraphs,namelymaximumindependentset,maximumclique,graphcoloring,minimumdominatingsetsandrelatedproblems.Theapplicationareasoftheconsideredproblemsincludeinformationretrieval,classicationtheory,economics,scheduling,experimentaldesign,andcomputervisionamongmanyothers. Themaximumindependentsetandrelatedproblemsareformulatedasnonlinearprograms,andnewmethodsforndinggoodqualityapproximatesolutionsinreasonablecomputationaltimesareintroduced.Allalgorithmsareimplementedandsuccessfullytestedonanumberofexamplesfromdiverseapplicationareas.Theproposedmethodsfavorablycomparewithothercompetingapproaches. Alargepartofthisdissertationisdevotedtodetailedstudiesofselectedapplicationsoftheproblemsofinterest.NoveltechniquesforanalyzingthestructureofnancialmarketsbasedontheirnetworkrepresentationareproposedandveriedusingmassivedatasetsgeneratedbytheU.S.stockmarkets.Thenetwork xi

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Inanotherapplication,newexactvaluesandestimatesofsizeofthelargesterrorcorrectingcodesarecomputedusingoptimizationinspeciallyconstructedgraphs.Errorcorrectingcodeslieintheheartofdigitaltechnology,makingcellphones,compactdiskplayersandmodemspossible.TheyarealsoofaspecialsignicanceduetoincreasingimportanceofreliabilityissuesinInternettransmissions. Finally,ecientapproximatealgorithmsforconstructionofvirtualbackboneinwirelessnetworksbymeansofsolvingtheminimumconnecteddominatingsetprobleminunit-diskgraphsaredevelopedandtested. xii

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Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthelastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped,thediusionintootherdisciplineshasproceededatarapidpace,andourknowledgeofallaspectsoftheeldhasgrownevenmoreprofound[ 88 169 ].Atthesametime,oneofthemoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisontheinterdisciplinarynatureoftheeld.Optimizationtodayisabasicresearchtoolinallareasofengineering,medicineandthesciences.Thedecisionmakingtoolsbasedonoptimizationproceduresaresuccessfullyappliedinawiderangeofpracticalproblemsarisinginvirtuallyanysphereofhumanactivities,includingbiomedicine,energymanagement,aerospaceresearch,telecommunicationsandnance. Havingbeenappliedinmanydiverseareas,theproblemsstudiedinthisthesisprovideaperfectillustrationoftheinterdisciplinarydevelopmentsinoptimization.Forinstance,themaximumindependentsetproblemputforwardinthetitleofthisthesishasapplicationsinnumerouselds,includinginformationretrieval,signaltransmissionanalysis,classicationtheory,economics,scheduling,andbiomedicalengineering.Newndingsconcerningsomeoftheseapplicationswillbediscussedinalatterpartofthisthesis.Butbeforewewillpresentaformaldescriptionofresultsobtainedinthiswork,letusgentlytouchthehistoricalgroundsofoptimizationtheoryandmathematicalprogramming. Theproblemsofndingthe\best"andthe\worst"havealwaysbeenofgreatinterest.Forexample,givennsites,whatisthefastestwaytovisitallofthemconsecutively?Inmanufacturing,howshouldonecutplatesofamaterialsothatthewasteisminimized?Someoftherstoptimizationproblemsweresolvedinancient 1

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Greeceandareregardedamongthemostsignicantdiscoveriesofthattime.IntherstcenturyA.D.,theAlexandrianmathematicianHeronsolvedtheproblemofndingtheshortestpathbetweentwopointsbywayofthemirror.Thisresult,alsoknownastheHeron'stheoremofthelightray,canbeviewedastheoriginofthetheoryofgeometricaloptics.Theproblemofndingextremevaluesgainedaspecialimportanceintheseventeenthcenturywhenitservedasoneofmotivationsintheinventionofdierentialcalculus.Thesoon-after-developedcalculusofvariationsandthetheoryofstationaryvalueslieinthefoundationofthemodernmathematicaltheoryofoptimization. Theinventionofthedigitalcomputerservedasapowerfulspurtotheeldofnumericaloptimization.DuringWorldWarIIoptimizationalgorithmswereusedtosolvemilitarylogisticsandoperationsproblems.Themilitaryapplicationsmotivatedthedevelopmentoflinearprogramming(LP),whichstudiesoptimizationproblemswithlinearobjectivefunctionandconstraints.In1947GeorgeDantziginventedthesimplexmethodforsolvinglinearprogramsarisinginU.S.AirForceoperations.Linearprogramminghasbecomeoneofthemostpopularandwellstudiedoptimizationtopicseversince. Despiteaneperformanceofthesimplexmethodonawidevarietyofpracticalinstances,ithasanexponentialworst-casetimecomplexityandthereforeisunacceptablyslowinsomelarge-scalecases.ThequestionofexistenceofatheoreticallyecientalgorithmforLPremainedopenuntil1979,whenLeonidKhachianpublishedhispolynomialtimeellipsoidalgorithmforlinearprogramming[ 138 ].ThistheoreticalbreakthroughwasfollowedbytheinteriorpointalgorithmofNarendraKarmarkar[ 137 ]publishedin1984.Notonlydoesthisalgorithmhaveapolynomialtimetheoreticalcomplexity,itisalsoextremelyecientpractically,allowingforsolvinglargerinstancesoflinearprograms.Nowadays,

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variousversionsofinteriorpointmethodsareanintegralpartofthestate-of-the-artoptimizationsolvers. Linearprogrammingcanbeconsideredasaspecialcaseofabroadoptimizationareacalledcombinatorialoptimization,inwhichathefeasibleregionisanite,butusuallyverylarge,set.Alloftheproblemsstudiedinthisworkareessentiallycombinatorialoptimizationproblems.Innonlinearoptimization,onedealswithoptimizinganonlinearfunctionoverafeasibledomaindescribedbyasetof,ingeneral,nonlinearfunctions.Itisfascinatingtoobservehownaturallynonlinearandcombinatorialoptimizationarebridgedwitheachothertoyieldnew,betteroptimizationtechniques.Combiningthetechniquesforsolvingcombinatorialproblemswithnonlinearoptimizationapproachesisespeciallypromisingsinceitprovidesanalternativepointofviewandleadstonewcharacterizationsoftheconsideredproblems.Theseideasalsogiveafreshinsightintothecomplexityissuesandfrequentlyprovideaguidetothediscoveryofnontrivialconnectionsbetweenproblemsofseeminglydierentnature.Forexample,theellipsoidandinteriorpointmethodsforlinearprogrammingmentionedabovearebasedonnonlinearoptimizationtechniques.Letusalsomentionthatanintegralityconstraintoftheformx2f0;1gisequivalenttothenonconvexquadraticconstraintx2x=0.Thisstraightforwardfactsuggeststhatitisthepresenceofnonconvexity,notintegralitythatmakesanoptimizationproblemdicult[ 123 ].Theremarkableassociationbetweencombinatorialandnonlinearoptimizationcanalsobeobservedthroughoutthisthesis.InChapter 2 weproveseveralcontinuousformulationsofthemaximumindependentsetproblem.Consequently,inChapter 3 wederiveecientalgorithmsforndinglargeindependentsetsbasedontheseformulations. Asaresultofongoingenhancementoftheoptimizationmethodologyandofimprovementofavailablecomputationalfacilities,thescaleoftheproblemssolvabletooptimalityiscontinuouslyrising.However,manylarge-scaleoptimizationproblems

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encounteredinpracticecannotbesolvedusingtraditionaloptimizationtechniques.Avarietyofnewcomputationalapproaches,calledheuristics,havebeenproposedforndinggoodsub-optimalsolutionstodicultoptimizationproblems.Etymologically,theword\heuristic"comesfromtheGreekheuriskein(tond).Recallthefamous\Eureka!Eureka!"(Ihavefoundit!Ihavefoundit!)byArchimedes(287-212B.C.).Aheuristicinoptimizationisanymethodthatndsan\acceptable"feasiblesolution.Manyclassicalheuristicsarebasedonlocalsearchprocedures,whichiterativelymovetoabettersolution(ifsuchsolutionexists)inaneighborhoodofthecurrentsolution.Aprocedureofthistypeusuallyterminateswhentherstlocaloptimumisobtained.Randomizationandrestartingapproachesusedtoovercomepoorqualitylocalsolutionsareoftenineective.Moregeneralstrategiesknownasmetaheuristicsusuallycombinesomeheuristicapproachesanddirectthemtowardssolutionsofbetterqualitythanthosefoundbylocalsearchheuristics.Heuristicsandmetaheuristicsplayakeyroleinthesolutionoflargedicultappliedoptimizationproblems.Newecientheuristicsforthemaximumindependentsetandrelatedproblemshavebeendevelopedandsuccessfullytestedinthiswork. Intheremainderofthischapterwewillrstformallyintroducedenitionsandnotationsusedinthisthesis.Thenwewillstatesomefundamentalfactsconcerningtheconsideredproblems.Finally,inSection 1.7 wewilloutlinetheorganizationoftheremainingchaptersofthisthesis. 1.1 DenitionsandNotations LetG=(V;E)beasimpleundirectedgraphwithvertexsetV=f1;:::;ngandsetofedgesE.ThecomplementgraphofGisthegraphG=(V;E),whereEisthecomplementofE.ForasubsetWVletG(W)denotethesubgraphinducedbyWonG.N(i)willdenotethesetofneighborsofvertexianddi=jN(i)jisthedegreeofvertexi.Wedenoteby(G)themaximumdegreeofG.

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AsubsetIViscalledanindependentset(stableset,vertexpacking)iftheedgesetofthesubgraphinducedbyIisempty.Anindependentsetismaximalifitisnotasubsetofanylargerindependentset,andmaximumiftherearenolargerindependentsetsinthegraph.Theindependencenumber(G)(alsocalledthestabilitynumber)isthecardinalityofamaximumindependentsetinG. Alongwiththemaximumindependentsetproblem,weconsidersomeotherrelatedproblemsforgraphs,includingthemaximumclique,theminimumvertexcover,andthemaximummatchingproblems.AcliqueCisasubsetofVsuchthatthesubgraphG(C)inducedbyConGiscomplete.Themaximumcliqueproblemistondacliqueofmaximumcardinality.Thecliquenumber!(G)isthecardinalityofamaximumcliqueinG.AvertexcoverV0isasubsetofV,suchthateveryedge(i;j)2EhasatleastoneendpointinV0.Theminimumvertexcoverproblemistondavertexcoverofminimumcardinality.Twoedgesinagraphareincidentiftheyhaveanendpointincommon.Asetofedgesisindependentifnotwoofthemareincident.AsetMofindependentedgesinagraphG=(V;E)iscalledamatching.Themaximummatchingproblemistondamatchingofmaximumcardinality. ItiseasytoseethatIisamaximumindependentsetofGifandonlyifIisamaximumcliqueofGandifandonlyifVnIisaminimumvertexcoverofG.ThelastfactyieldsGallai'sidentity[ 92 ] whereSisaminimumvertexcoverofthegraphG.Duetothecloserelationbetweenthemaximumindependentsetandmaximumcliqueproblems,wewilloperatewithbothproblemswhiledescribingthepropertiesandalgorithmsforthemaximumindependentsetproblem.Inthiscase,itisclearthataresultholdingforthemaximumcliqueprobleminGwillalsobetrueforthemaximumindependentsetprobleminG.

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Sometimes,eachvertexi2Visassociatedwithapositiveweightwi.ThenforasubsetSVitsweightW(S)isdenedasthesumofweightsofallverticesinS:W(S)=Xi2Swi: Thefollowingdenitionsgeneralizetheconceptofclique.Namely,insomeapplications,insteadofcliquesoneisinterestedindensesubgraphs,orquasi-cliques.A-cliqueC,alsocalledaquasi-clique,isasubsetofVsuchthatG(C)hasatleastbq(q1)=2cedges,whereqisthecardinalityofC. Alegal(proper)coloringofGisanassignmentofcolorstoitsverticessothatnopairofadjacentverticeshasthesamecolor.Acoloringinducesnaturallyapartitionofthevertexsetsuchthattheelementsofeachsetinthepartitionarepairwisenonadjacent.Thesesetsarepreciselythesubsetsofverticesbeingassignedthesamecolor,whichareindependentsetsbydenition.IfthereexistsacoloringofGthatusesnomorethankcolors,wesaythatGadmitsak-coloring(Gisk-colorable).TheminimalkforwhichGadmitsak-coloringiscalledthechromaticnumberandisdenotedby(G).Thegraphcoloringproblemistond(G)aswellasthepartitionofverticesinducedbya(G)-coloring. Minimumcliquepartitionproblem,whichistopartitionverticesintominimumnumberofcliques,isanalogoustographcoloringproblem.Infact,anypropercoloringinGisacliquepartitioninG. Forotherstandarddenitionswhichareusedinthispaperthereaderisreferredtoastandardtextbookingraphtheory[ 31 ]. Konig'stheorem(seepage30inDiestel[ 74 ]andRizzi[ 178 ]forashortproof)statesthatthemaximumcardinalityofamatchinginabipartitegraphGisequaltotheminimumcardinalityofavertexcover.

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1.2 ComplexityResults ThemaximumcliqueandthegraphcoloringproblemsareNP-hard[ 96 ];moreover,theyareassociatedwithaseriesofrecentresultsabouthardnessofapproximations.Thediscoveryofaremarkableconnectionbetweenprobabilisticallycheckableproofsandapproximabilityofcombinatorialoptimizationproblems[ 20 21 83 ]yieldednewhardnessofapproximationresultsformanyproblems.AroraandSafra[ 21 ]provedthatforsomepositivetheapproximationofthemaximumcliquewithinafactorofnisNP-hard.Recently,Hastad[ 116 ]hasshownthatinfactforany>0themaximumcliqueishardtoapproximateinpolynomialtimewithinafactorn1.Similarapproximationcomplexityresultsholdforthegraphcoloringproblemaswell.GareyandJohnson[ 95 ]haveshownthatobtainingcoloringsusings(G)colors,wheres<2,isNP-hard.IthasbeenshownbyLundandYannakakis[ 152 ]that(G)ishardtoapproximatewithinnforsome>0,andfollowingHastad[ 116 ],FeigeandKilian[ 84 ]haveshownthatforany>0thechromaticnumberishardtoapproximatewithinafactorofn1,unlessNPZPP.Alloftheabovefactstogetherwithpracticalevidence[ 136 ]suggestthatthemaximumcliqueandcoloringproblemsarehardtosolveeveningraphsofmoderatesizes.Thetheoreticalcomplexityandthehugesizesofdatamaketheseproblemsespeciallyhardinmassivegraphs. Alternatively,themaximummatchingproblemcanbesolvedinpolynomialtimeevenfortheweightedcase(see,forinstance,PapadimitriouandSteiglitz[ 168 ]). 1.3 LowerandUpperBounds Inthissectionwebrieyreviewsomewell-knownlowerandupperboundsontheindependenceandcliquenumbers. PerhapsthebestknownlowerboundbasedondegreesofverticesisgivenbyCaroandTuza[ 57 ],andWei[ 200 ]:(G)Xi2V1

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In1967,Wilf[ 201 ]showedthat!(G)(AG)+1; DenotebyN1thenumberofeigenvaluesofAGthatdonotexceed1,andbyN0thenumberofzeroeigenvalues.AminandHakimi[ 16 ]provedthat!(G)N1+1
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Sincetheabovesemideniteprogramcanbesolvedinpolynomialtime,thesandwichtheoremalsoshowsthatinaperfectgraphG(suchthat!(G)=(G)[ 154 ]),thecliquenumbercanbecomputedinpolynomialtime. 1.4 ExactAlgorithms In1957,HararyandRoss[ 114 ]published,allegedly,therstalgorithmforenumeratingallcliquesinagraph.Interestingly,theirworkwasmotivatedbyanapplicationinsociometry.Theideaoftheirmethodistoreducetheproblemongeneralgraphstoaspecialcasewithgraphshavingatmostthreecliques,andthensolvetheproblemforthisspecialcase.Thisworkwasfollowedbymanyotheralgorithms.PaullandUnger[ 174 ],andMarcus[ 158 ]proposedalgorithmsforminimizingthenumberofstatesinsequentialswitchingfunctions.TheworkofBonner[ 41 ]wasmotivatedbyclusteringproblem.BednarekandTaulbee[ 29 ]werelookingforallmaximalchainsinasetwithagivenbinaryrelation.Althoughalltheseapproachesweredesignedtosolveproblemsarisingindierentapplications,theirideaisessentiallytoenumerateallcliquesinsomegraph. Thedevelopmentofcomputertechnologyin1960'smadeitpossibletotestthealgorithmsongraphsoflargersizes.Asaresult,intheearly1970's,manynewenumerativealgorithmswereproposedandtested.PerhapsthemostnotableofthesealgorithmswasthebacktrackingmethodbyBronandKerbosch[ 45 ].Theadvantagesoftheirapproachincludeitspolynomialstoragerequirementandexclusionofthepossibilityofgeneratingthesamecliquetwice.Thealgorithmwassuccessfullytestedongraphswith10to50verticesandwithedgedensityintherangebetween10%and95%.AmodicationofthisapproachconsideredbyTomitaetal.[ 192 ]wasclaimedtohavethetimecomplexityofO(3n=3),whichisthebestpossibleforenumerativealgorithmsduetoexistenceofgraphswith3n=3maximalcliques[ 163 ].Recently,theBron-Kerboschalgorithmwasshowntobequiteecientwithgraphsarisinginmatching3-dimensionalmolecularstructures[ 93 ].

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Allofthealgorithmsmentionedaboveinthissectionweredesignedforndingallmaximalcliquesinagraph.However,tosolvethemaximumcliqueproblem,oneneedstondonlythecliquenumberandthecorrespondingmaximumclique.Therearemanyexactalgorithmsforthemaximumcliqueandrelatedproblemsavailableintheliterature.Mostofthemarevariationsofthebranchandboundmethod,whichcanbedenedbydierenttechniquesfordetermininglowerandupperboundsandbyproperbranchingstrategies.TarjanandTrojanowski[ 191 ]proposedarecursivealgorithmforthemaximumindependentsetproblemwiththetimecomplexityofO(2n=3).Later,thisresultwasimprovedbyRobson[ 179 ],whomodiedthealgorithmofTarjanandTrojanowskitoobtainthetimecomplexityofO(20:276n). AnotherimportantalgorithmwasdevelopedbyBalasandYuin1986[ 25 ].Usinganinterestingnewimplementationoftheimplicitenumeration,theywereabletocomputemaximumcliquesingraphswithupto400verticesand30,000edges.CarraghanandPardalos[ 58 ]proposedyetanotherimplicitenumerativealgorithmforthemaximumcliqueproblembasedonexaminingverticesintheordercorrespondingtothenondecreasingorderoftheirdegrees.Despiteitssimplicity,theapproachprovedtobeveryecient,especiallyforsparsegraphs.Apubliclyavailableimplementationofthisalgorithmcurrentlyservesasabenchmarkforcomparingdierentalgorithms[ 136 ].Recently,Ostergard[ 167 ]proposedabranch-and-boundalgorithmwhichanalyzesverticesinorderdenedbytheircoloringandemploysanewpruningstrategy.HecomparedtheperformanceofthisalgorithmwithseveralotherapproachesonrandomgraphsandDIMACSbenchmarkinstancesandclaimedthathisalgorithmissuperiorinmanycases. Asmostcombinatorialoptimizationproblems,themaximumindependentsetproblemcanbeformulatedasanintegerprogram.SeveralsuchformulationswillbementionedinChapter 2 .ThemostpowerfulintegerprogrammingsolversusedbymodernoptimizationpackagessuchasCPLEXofILOG[ 125 ]andXpressofDash

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Optimization[ 68 ]usuallycombinebranch-and-boundalgorithmswithcuttingplanemethods,ecientpreprocessingschemes,includingfastheuristics,andsophisticateddecompositiontechniquesinordertondanexactsolution. 1.5 Heuristics Althoughexactapproachesprovideanoptimalsolution,theybecomeimpractical(tooslow)evenongraphswithseveralhundredsofvertices.Therefore,whenonedealswithmaximumindependentsetproblemonverylargegraphs,theexactapproachescannotbeapplied,andheuristicsprovidetheonlyavailableoption. Perhapsthesimplestheuristicsforthemaximumindependentsetproblemaresequentialgreedyheuristicswhichrepeatedlyaddavertextoanintermediateindependentset(\Bestin")orremoveavertexfromasetofverticeswhichisnotanindependentset(\Worstout"),basedonsomecriterion,saythedegreesofvertices[ 135 144 ]. Sometimesinasearchforecientheuristicspeopleturntonature,whichseemstoalwaysndthebestsolutions.Intherecentdecades,newtypesofoptimizationalgorithmshavebeendevelopedandsuccessfullytested,whichessentiallyattempttoimitatecertainnaturalprocesses.Thenaturalphenomenaobservedinannealingprocesses,nervoussystemsandnaturalevolutionwereadoptedbyoptimizersandledtodesignofthesimulatedannealing[ 139 ],neuralnetworks[ 121 ]andevolutionarycomputation[ 119 ]methodsintheareaofoptimization.OtherpopularmetaheuristicsincludegreedyrandomizedadaptivesearchproceduresorGRASP[ 86 ]andtabusearch[ 101 ].Variousversionsoftheseheuristicshavebeensuccessfullyappliedtomanyimportantcombinatorialoptimizationproblems,includingthemaximumindependentsetproblem.Belowwewillbrieyreviewsomeofsuchheuristics. 1.5.1 SimulatedAnnealing Theannealingprocessisusedtoobtainapurelatticestructureinphysics.Itconsistsofrstmeltingasolidbyheatingitup,andthensolidifyingitbyslowly

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coolingitdowntoalow-energystate.Asaresultofthisprocess,thesystem'sfreeenergyisminimized.Thispropertyoftheannealingprocessisusedforoptimizationpurposesinthesimulatedannealingmethod,whereeachfeasiblesolutionembodiesastateofthehypotheticalphysicalsystem,andtheobjectivefunctionrepresentstheenergycorrespondingtothestate.Thebasicideaofsimulatedannealingisthefollowing.Generateaninitialfeasiblesolutionx(0).Atiterationk+1,givenafeasiblesolutionx(k),acceptitsneighborx(k+1)asthenextfeasiblesolutionwithprobabilitypk+1=8><>:1;iff(xk+1)
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1.5.2 NeuralNetworks Articialneuralnetworks(orsimply,neuralnetworks)representanattempttoimitatesomeoftheusefulpropertiesofbiologicalnervoussystems,suchasadaptivebiologicallearning.Aneuralnetworkconsistsofalargenumberofhighlyinterconnectedprocessingelementsemulatingneurons,whicharetiedtogetherwithweightedconnectionsanalogoustosynapses.Justlikebiologicalnervoussystems,neuralnetworksarecharacterizedbymassiveparallelismandahighinterconnectedness. Althoughneuralnetworkshadbeenintroducedasearlyasinthelate1950's,theywerenotwidelyapplieduntilthemid-1980's,whensucientlyadvancedrelatedmethodologyhasbeendeveloped.HopeldandTank[ 122 ]provedthatcertainneuralnetworkmodelscanbeusedtoapproximatelysolvesomedicultcombinatorialoptimizationproblems.Nowadays,neuralnetworksareappliedtomanycomplexreal-worldproblems.Moredetailonneuralnetworksinoptimizationcanbefound,forexample,inZhang[ 205 ]. Variousversionsofneuralnetworkshavebeenappliedtothemaximumindependentsetproblembymanyresearcherssincethelate1980's.Theeciencyofearlyattemptsisdiculttoevaluateduetothelackofexperimentalresults.Therefore,wewillmentionsomeofthemorerecentcontributions.Grossman[ 107 ]consideredadiscreteversionoftheHopeldmodelforthemaximumclique.TheresultsofcomputationalexperimentswiththisapproachonDIMACSbenchmarksweresatisfactorybutinferiortoothercompetingmethods,suchassimulatedannealing.Jagota[ 127 ]proposedsevaraldierentdiscreteandcontinuousversionsoftheHopeldmodelforthemaximumcliqueproblem.Later,Jagotaetal.[ 128 ]improvedsomeofthesealgorithmstoobtainconsiderablylargercliquesthanthosefoundbysimplerheuristics,whichranonlyslightlyfaster.

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1.5.3 GeneticAlgorithms Geneticalgorithmsinoptimizationweremotivatedbyevolutionprocessesinnaturalsystems.Optimizationinageneticalgorithmiscarriedoutonapopulationofpoints,whicharealsocalledindividualsorchromosomes.Inthesimplestversion,thechromosomesarerepresentedbybinaryvectors.Eachchromosomeisassociatedwithatnessvalue,whichistheprobabilitythattheindividualdenedbythischromosomeinthenextgenerationwillsurvivetotheadulthood.Individualtnessvaluesareusedtocomputevaluesoftheobjectivefunctionrepresentingthetotaltnessofthepopulation.Thefundamentallawofnaturalselectionstatingthatthetotaltnessofthepopulationisnon-decreasingfromgenerationtogeneration,canbeusedasabasisforanoptimizationprocedure.Initssimplestversion,thegeneticalgorithmstartswithapopulationchosenrandomly,andcomputesanewpopulationusingoneofthethreebasicmechanisms:reproduction,crossover,ormutation[ 104 ].Reproductionoperatorchoosesthechromosomesusedinthenextgenerationaccordingtotheprobabilitygivenbytheirtness.Thecrossoveroperatorisappliedtoproducenewchildrenfrompairsofindividuals.Finally,themutationoperatorreversesthevalueofeachbitinachromosomewithagivenprobability. Earlyattemptstoapplygeneticalgorithmstothemaximumindependentsetandmaximumcliqueproblemsweremadeinthebeginningof1990's.Manysuccessfulimplementationshaveappearedintheliteratureeversince[ 50 118 157 ].Mostofthegeneticalgorithmscanbeeasilyparallelized. 1.5.4 GreedyRandomizedAdaptiveSearchProcedures Agreedyrandomizedadaptivesearchprocedure(GRASP)isaniterativerandomizedsamplingtechniqueinwhicheachiterationprovidesanheuristicsolutiontotheproblemathand[ 86 ].ThebestsolutionoverallGRASPiterationsiskeptasthenalresult.TherearetwophaseswithineachGRASPiteration:therstconstructsalistofsolutionscalledrestrictedcandidatelist(RCL)viaanadaptiverandomized

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greedyfunction;thesecondappliesalocalsearchtechniquetotheconstructedsolutioninhopeofndinganimprovement.GRASPhasbeenappliedsuccessfullytoavarietyofcombinatorialoptimizationproblemsincludingthemaximumindependentsetproblem[ 85 ]. ArecentadditiontoGRASP,theso-calledpathrelinking[ 102 ]hasbeenproposedinordertoenhancetheperformanceoftheheuristicbylinkinggoodqualitysolutionswithapathofintermediatefeasiblepoints.Itappearsthatinmanycasessomeofthesefeasiblepointsprovideabetterqualitythanthesolutionsusedastheendpointsinthepath.GRASPprocedurewithpathrelinkingcanbeappliedforthemaximumindependentsetandthegraphcoloringproblems. 1.5.5 TabuSearch Tabusearch[ 99 100 ]isavariationoflocalsearchalgorithms,whichusesatabustrategyinordertoavoidrepetitionswhilesearchingthespaceoffeasiblesolutions.Suchastrategyisimplementedbymaintainingasetoftabulists,containingthepathsoffeasiblesolutionspreviouslyexaminedbythealgorithm.Thenextvisitedsolutionisdenedasthebestlegal(notprohibitedbythetabustrategy)solutionintheneighborhoodofthecurrentsolution,evenifitisworsethanthecurrentsolution. Variousversionsoftabusearchhavebeensuccessfullyappliedtothemaximumindependentsetandmaximumcliqueproblems[ 28 90 155 189 ]. 1.5.6 HeuristicsBasedonContinuousOptimization Asitwasdiscussedabove,continuousapproachestocombinatorialoptimizationproblemsturnouttobeespeciallyattractive.Muchoftherecenteortsindevelopingcontinuous-basedheuristicsforthemaximumindependentsetandmaximumcliqueproblemsfocusedaroundtheMotzkin-Straus[ 165 ]formulationwhichrelatesthecliquenumberofagraphtosomequadraticprogram.ThisformulationwillbeprovedanddiscussedinmoredetailinChapter 2 ofthiswork.Wewillalsoprovesomeother

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continuousformulationsoftheconsideredproblemsanddevelopandtestecientheuristicsbasedontheseformulations. Recently,wewitnessedfoundationandrapiddevelopmentofsemideniteprogrammingtechniques,whichareessentiallycontinuous.TheremarkableresultbyGoemansandWilliamson[ 103 ]servedasamajorstepforwardindevelopmentofapproximationalgorithmsandprovedaspecialimportanceofsemideniteprogrammingforcombinatorialoptimization.Bureretal.[ 52 ]consideredrank-oneandrank-tworelaxationsoftheLovaszsemideniteprogram[ 108 149 ]andderivedtwocontinuousoptimizationformulationsforthemaximumindependentsetproblem.Basedontheseformulations,theydevelopedandtestednewheuristicsforndinglargeindependentsets. 1.6 Applications Practicalapplicationsoftheconsideredoptimizationproblemsareabundant.Theyappearininformationretrieval,signaltransmissionanalysis,classicationtheory,economics,scheduling,experimentaldesign,andcomputervision[ 4 23 25 30 38 72 171 193 ].Inthissection,wewillbrieydescribeselectedapplications.Ourworkconcerningapplicationsinmassivegraphs,codingtheoryandwirelessnetworkswillbepresentedinChapters 4 6 1.6.1 MatchingMolecularStructuresbyCliqueDetection TwographsG1andG2arecalledisomorphicifthereexistsaone-to-onecorrespondencebetweentheirvertices,suchthatadjacentpairsofverticesinG1aremappedtoadjacentpairsofverticesinG2.AcommonsubgraphoftwographsG1andG2consistsofsubgraphsG01andG02ofG1andG2,respectively,suchthatG01isisomorphictoG02.Thelargestsuchcommonsubgraphisthemaximumcommonsubgraph(MCS).Forapairofgraphs,G1=(V1;E1)andG2=(V2;E2),theircorrespondencegraphChasallpossiblepairs(v1;v2),wherevi2Vi;i=1;2,as

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itssetofvertices;twovertices(v1;v2)and(v01;v02)areconnectedinCifthevaluesoftheedgesfromv1tov01inG1andfromv2tov02inG2arethesame. Forapairofthree-dimensionalchemicalmoleculestheMCSisdenedasthelargestsetofatomsthathavematchingdistancesbetweenatoms(givenuser-denedtolerancevalues).ItcanbeshownthattheproblemofndingtheMCScanbesolvedecientlyusingclique-detectionalgorithmsappliedtothecorrespondencegraph[ 93 ]. 1.6.2 MacromolecularDocking Giventwoproteins,theproteindockingproblemistondwhethertheyinteracttoformastablecomplex,and,iftheydo,thenhow.Thisproblemisfundamentaltoallaspectsofbiologicalfunction,anddevelopmentofreliabletheoreticalproteindockingtechniquesisanimportantgoal.Oneoftheapproachestothemacromoleculardockingproblemconsistsinrepresentingeachoftwoproteinsasasetofpotentialhydrogenbonddonorsandacceptorsandusingaclique-detectionalgorithmtondmaximallycomplementarysetsofdonor/acceptorpairs[ 94 ]. 1.6.3 IntegrationofGenomeMappingData Duetodierencesinthemethodsusedforanalyzingoverlapandprobedata,theintegrationofthesedatabecomesanimportantproblem.Itappearsthatoverlapdatacanbeeectivelyconvertedtoprobe-likedataelementsbyndingmaximalsetsofmutuallyoverlappingclones[ 115 ].Eachsetdeterminesasiteinthegenomecorrespondingtotheregionwhichiscommonamongtheclonesoftheset;thereforethesesetsarecalledvirtualprobes.Findingthevirtualprobesisequivalenttondingthemaximalcliquesofagraph. 1.6.4 ComparativeModelingofProteinStructure Therapidlygrowingnumberofknownproteinstructuresrequirestheconstructionofaccuratecomparativemodels.Thiscanbedoneusingaclique-detectionapproachasfollows.Weconstructagraphinwhichverticescorrespondtoeachpossibleconformationofaresidueinanaminoacidsequence,andedges

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connectpairsofresidueconformations(vertices)thatareconsistentwitheachother(i.e.,clash-freeandsatisfyinggeometricalconstraints).Basedontheinteractionbetweentheatomscorrespondingtothetwovertices,weightsareassignedtotheedges.Thenthecliqueswiththelargestweightsintheconstructedgraphrepresenttheoptimalcombinationofthevariousmain-chainandside-chainpossibilities,takingtherespectiveenvironmentsintoaccount[ 180 ]. 1.6.5 CoveringLocationUsingCliquePartition Coveringproblemsariseinlocationtheory.Givenasetofdemandpointsandasetofpotentialsitesforlocatingfacilities,ademandpointlocatedwithinaprespecieddistancefromafacilityissaidtobecoveredbythatfacility.Coveringproblemsareusuallyclassiedintotwomainclasses[ 69 ]:mandatorycoverageproblemsaimtocoveralldemandpointswiththeminimumnumberoffacilities;andmaximalcoverageproblemsintendtocoveramaximumnumberofdemandpointswithagivennumberoffacilities. Locationproblemsarealsoclassiedbythenatureofsetsofdemandpointsandpotentialsites,eachofwhichcanbediscreteorcontinuous.Inmostapplicationsbothsetsarediscrete,butinsomecasesatleastoneofthemiscontinuous.Forexample,Brotcorneetal.[ 46 ]consideranapplicationofmandatorycoverageproblemarisingincytologicalscreeningtestsforcervicalcancer,inwhichthesetofdemandpointsisdiscreteandthesetofpotentialsitesiscontinuous.Inthisapplication,aservicalspecimenonaglassslidehastobeviewedbyascreener,whichisrelocatedontheglassslideinordertoexploretheentirespecimen.Thegoalistomaximizeeciencybyminimizingthenumberofviewinglocations.Theareacoveredbythescreenercanbesquareorcircular.Tobespecic,considerthecaseofthesquarescreener,whichcanmoveinanyoffourdirectionsparalleltothesidesoftherectangularglassslide.Inthiscase,weneedtocovertheviewingareaoftheslidebysquarescalledtiles.Tolocateatileitissucestospecifythepositionofitscenteronthetile.

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Letusformulatetheprobleminmathematicalterms.Givenndemandpointsinsidearectangle,wewanttocoverthesepointswithaminimumnumberofsquareswithsides(tiles). 1. Thereexistsacoveringofndemandpointsintherectangleusingktiles. 2. Givenntilescenteredinthedemandpoints,thereexistkpointsintherectanglesuchthateachofthetilescontainsatleastoneofthem. 1.7 OrganizationoftheDissertation Organizationally,thisthesisisdividedintosevenchapters.Chapter 2 introducesmathematicalprogrammingformulationsofthemaximumindependentsetandrelatedproblems.Inparticular,weproveseveralcontinuousoptimizationformulations.TheseformulationsandsomeclassicalresultsconcerningtheproblemofoptimizingaquadraticoverasphereareutilizedinChapter 3 ,whereweproposeseveralheuristicsforthemaximumindependentsetproblem.Todemonstratetheireciency,weprovidetheresultsofnumericalexperimentswithourapproaches,aswellascomparisonwithotherheuristics. Chapter 4 discussesapplicationsoftheconsideredgraphproblemsinstudyingmassivedatasets.First,wereviewrecentadvancesandchallengesinmodelingandoptimizationofmassivegraphs.Thenweintroducethenotionoftheso-calledmarket

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graph,inwhichnancialinstrumentsarerepresentedbyvertices,andedgesareconstructedbasedonthevaluesofpairwisecross-correlationsofthepriceuctuationscomputedoveracertaintimeperiod.Weanalyzestructuralpropertiesofthisgraphandsuggestsomepracticalimplicationsoftheobtainedresults. NewexactsolutionsandimprovedlowerboundsfortheproblemofndingthelargestcorrectingcodesdealingwithcertaintypesoferrorsarereportedinChapter 5 .Theseresultswereobtainedusingecienttechniquesforsolvingthemaximumindependentsetandgraphcoloringproblemsinspeciallyconstructedgraphs.YetanotherapplicationoftheconsideredgraphoptimizationproblemsispresentedinChapter 6 ,inwhichwesolvetheproblemofconstructingavirtualbackboneinadhocwirelessnetworksbyapproximatingitwiththeminimumconnecteddominatingsetprobleminunit-diskgraphs.Theproposeddistributedalgorithmhasaxedperformanceratioandconsistsoftwostages.Namely,itrstcomputesamaximalindependentsetandthenconnectsitusingsomeadditionalvertices.Theresultsofnumericalsimulationarealsoincluded.Finally,weconcludewithsomeremarksandideasforthefutureresearchinChapter 7

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Themaximumindependentsetproblemhasmanyequivalentformulationsasanintegerprogrammingproblemandasacontinuousnonconvexoptimizationproblem[ 38 171 ].Inthischapterwewillreviewsomeexistingapproachesandwillpresentproofsforsomeoftheformulations.Wewillstartwithmentioningsomeintegerprogrammingformulationsinthenextsection.InSections 2.2 and 2.3 ,wewillconsiderseveralcontinuousformulationsofthemaximumindependentsetproblemandwillgiveproofsforsomeofthem.ThiswillfollowbyageneralizationofoneoftheformulationstodominatingsetsinSection 2.4 .MostresultspresentedinthischapterpreviouslyappearedinAbelloetal.[ 3 ]. 2.1 IntegerProgrammingFormulations Givenavectorw2Rnofpositiveweightswi(associatedwitheachvertexi;i=1;:::;n),themaximumweightindependentsetproblemasksforindependentsetsofmaximumweight.Obviously,itisageneralizationofthemaximumindependentsetproblem.Oneofthesimplestformulationsofthemaximumweightindependentsetproblemisthefollowingedgeformulation:maxf(x)=nXi=1wixi; subjecttoxi+xj1;8(i;j)2E; 2.1 a)xi2f0;1g;i=1;:::;n: 2.1 b) 21

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Analternativeformulationofthisproblemisthefollowingcliqueformulation[ 108 ]:maxf(x)=nXi=1wixi; subjecttoXi2Sxi1;8S2CfmaximalcliquesofGg; 2.2 a)xi2f0;1g;i=1;:::;n: 2.2 b) Theadvantageofformulation( 2.2 )over( 2.1 )isasmallergapbetweentheoptimalvaluesof( 2.2 )anditslinearrelaxation.However,sincethereisanexponentialnumberofconstraintsin( 2.2 a),ndinganecientsolutionof( 2.2 )isdicult. Herewementiononemoreintegerprogrammingformulation.LetAGbetheadjacencymatrixofagraphG,andletJdenotethennidentitymatrix.Themaximumindependentsetproblemisequivalenttotheglobalquadraticzero-oneproblemmaxf(x)=xTAx; subjecttoxi2f0;1g;i=1;:::;n; 2.3 a) whereA=JAG:Ifxisasolutionto( 2.3 ),thenthesetIdenedbyI=fi2V:xi=1gisamaximumindependentsetofGwithjIj=f(x).SeePardalosandRodgers[ 170 ]fordetails. 2.2 ContinuousFormulations Shor[ 183 ]consideredaninterestingformulationofthemaximumweightindependentsetproblembynoticingthatformulation( 2.1 )isequivalenttothe

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quadraticallyconstrainedglobaloptimizationproblemmaxf(x)=nXi=1wixi; subjecttoxixj=0;8(i;j)2E; 2.4 a)xi2xi=0;i=1;2;:::;n: 2.4 b) Applyingdualquadraticestimates,ShorreportedgoodcomputationalresultsandpresentedanewwaytocomputetheLovasznumberofagraph[ 149 ]. MotzkinandStraus[ 165 ]establishedaremarkableconnectionbetweenthemaximumcliqueproblemandacertainstandardquadraticprogrammingproblem.TheoriginalproofoftheMotzkin{Straustheoremwasbyinduction.Belowwepresentanewproof.LetAGbetheadjacencymatrixofGandletebethen-dimensionalvectorwithallcomponentsequalto1. 2xTAGx; 2.5 a)x0: 2.5 b) 211

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wherea1;a2;:::;anarepositivenumbers.Equalitytakesplaceifandonlyifa1=a2=:::=an: 2.5 ).LetJdenotethennidentitymatrixandletObethennmatrixofallones.ThenAG=OJAG LetR(x)=xTJx+xTAGx.Program( 2.7 )isequivalentto( 2.5 ).minR(x)=xTJx+xTAGx; s.t.eTx=1; x0. Tocheckthattherealwaysexistsanoptimalsolutionxof( 2.7 )suchthatxTAGx=0,consideranyoptimalsolution^xof( 2.7 ).Assumethat^xTAG^x>0:Thenthereexistsapair(i;j)2Esuchthat^xi^xj>0.ConsiderthefollowingrepresentationforR(x):R(x)=Rij(x)+Rij(x);

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Withoutlossofgenerality,assumethatP(i;k)2E;k6=i^xkP(j;k)2E;k6=j^xk.Then,ifweset~xk=8>>>><>>>>:^xi+^xj;ifk=i;0;ifk=j;^xk;otherwise, wehave:R(~x)=Rij(~x)+Rij(~x)=(^xi+^xj)2+2(^xi+^xj)X(i;k)2E;k6=i^xk^x2i+^x2j+2^xi^xj+2^xiX(i;k)2E;k6=j^xk+2^xjX(j;k)2E;k6=i^xk=R(^x): 2.7 )withjZ(~x)j0gthenCisaclique. Withoutlossofgenerality,assumethatxi>0fori=1;2;:::;mandxi=0form+1in:Considertheobjectivefunctionof( 2.7 ),R(x)=xTJx=mXi=1xi2: 2.6 )andthefeasibilityofxfor( 2.7 ),nXi=1xi2nPi=1xi2

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Ontheotherhand,ifweconsiderxk=8><>:1 whereCisamaximumcliqueofG,thenxisfeasibleandR(x)=1 2.7 ).Returningbacktotheoriginalquadraticprogram,theresultofthetheoremfollows. ThisresultisextendedbyGibbonsetal.[ 98 ],whoprovidedacharacterizationofmaximalcliquesintermsoflocalsolutions.Moreover,optimalityconditionsoftheMotzkin-StrausprogramhavebeenstudiedandpropertiesofanewlyintroducedparametrizationofthecorrespondingQPhavebeeninvestigated.SosandStraus[ 190 ]furthergeneralizedthesametheoremtohypergraphs. 2.3 PolynomialFormulationsOvertheUnitHypercube InthissectionweconsidersomeofthecontinuousformulationsoriginallyprovedbyHarantetal.[ 112 113 ]usingprobabilisticmethods.WeprovedeterministicallythattheindependencenumberofagraphGcanbecharacterizedasanoptimizationproblembasedontheseformulations.Weconsidertwopolynomialformulations,adegree(+1)formulationandaquadraticformulation. 2.3.1 Degree(+1)PolynomialFormulation Considerthedegree(+1)polynomialofnvariablesF(x)=nXi=1(1xi)Y(i;j)2Exj;x2[0;1]n:

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Wewanttoshowthat(G)=f(G). Firstweshowthat( 2.8 )alwayshasanoptimal0-1solution.ThisissobecauseF(x)isacontinuousfunctionand[0;1]n=f(x1;x2;:::;xn):0xi1;i=1;:::;ngisacompactset.Hence,therealwaysexistsx2[0;1]nsuchthatF(x)=max0xi1;i=1;:::;nF(x). Now,xanyi2V.WecanrewriteF(x)intheform whereAi(x)=Y(i;j)2Exj;

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Expressions( 2.10 2.12 )canbeinterpretedintermsofneighborhoods.Ai(x)andBi(x)characterizetherst-andthesecond-orderneighborhoodsofvertexi,respectively,andCi(x)iscomplementarytoBi(x)withrespecttoiinthesensethatitdescribesneighborhoodsofallvertices,otherthani,whicharenotcharacterizedbyBi(x). Noticethatxiisabsentin( 2.10 { 2.12 ),andthereforeF(x)islinearwithrespecttoeachvariable.Itisalsoclearfromtheaboverepresentationthatifxisanyoptimalsolutionof( 2.8 ),thenxi=0ifAi(x)>Bi(x),andxi=1,ifAi(x)
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Eachtermin( 2.14 )iseither0or1.ThereforekmandthereexistsasubsetIf1;:::;kgsuchthatjIj=mand8i2I:Y(i;j)2Exj=1: 2.2 andthefactthatanycliqueinGisanindependentsetforG. 2.2 andGallai'sidentity( 1.1 ). 2.2 andKonig'stheorem. 2.3.2 QuadraticPolynomialFormulation ConsidernowthequadraticpolynomialH(x)=nXi=1xiX(i;j)2Exixj;

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denedforx2[0;1]n. ThefollowingtheoremcharacterizestheindependencenumberofthegraphGasthemaximizationofH(x)overthen-dimensionalhypercube. andletIbeamaximumindependentset.Toprovethath(G)(G),let SinceIisanindependentsetandxi=0fori=2I,thenP(i;j)2Exixj=0.Furthermore,nPi=1xi=jIj=(G).Thisyieldsh(G)H(x)=(G). Tocompletetheproof,weneedtoshowthath(G)(G).Assumeh(G)=m.SinceH(x)islinearwithrespecttoeachvariable,problem( 2.15 )alwayshasanoptimal0-1solution.Takeanyoptimal0-1solutionxof( 2.15 ).Suppose,thatthereexists(i0;j0)2Esuchthatxi0=xj0=1.Changingxi0to0decreasesnPi=1xiby1anddecreasesP(i;j)2Exixjbyatleast1.Thus,theobjectivefunctionwillnotdecrease.Doingthisforallsuchpairs(i0;j0)willnallyleadtoanoptimalsolutionxsuch

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that8(i;j)2E:xixj=0,andanindependentsetI=fi:xi=1gofcardinalityh(G).Thisyieldsh(G)(G)andthetheoremisproved. 2.3.3 RelationBetween(P2)andMotzkin-StrausQP ThenextstatementisareformulationtheMotzkin-Straustheoremforthemaximumindependentsetproblem.Weshowhowitcanbeobtainedfromtheformulation(P2). 2xTAGx; (2.18) x0. isgivenby1 211 Proof. 2xTAGx: 2(G)2yTAGy

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s.t.eTy=1;y0: isanoptimalsolutionforthelastprogram. Wehave:AG=OJAG 2.19 )isequivalenttothefollowing:(G)=maxy2F(G)+1 2(G)2(1+yTOy+yTAGy

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whichyieldsthefollowing:~yTJ~y<1 2.4 AGeneralizationforDominatingSets ForagraphG=(V;E)withV=f1;:::;ng,letl=(k1;:::;kn)beavectorofintegerssuchthat1kidifori2V,wherediisthedegreeofvertexi2V.Anl-dominatingset[ 113 ]isasetDlVsuchthateveryvertexi2VnDlhasatleastkineighborsinDl.Thel-dominationnumberl(G)ofGisthecardinalityofasmallestl-dominatingsetofG. Fork1==kn=1,l-dominationcorrespondstotheusualdenitionofdomination.Thedominationnumber(G)ofGisthecardinalityofasmallestdominatingsetofG.Ifki=difori=1;:::;n,thenI=VnDlisanindependentsetandd(G)=n(G)withd=(d1;:::;dn). Thefollowingtheoremcharacterizesthedominationnumber.

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Wewanttoshowthatl(G)=g(G). Werstshowthat( 2.21 )alwayshasanoptimal0-1solution.Sincefl(x)isacontinuousfunctionand[0;1]n=f(x1;x2;:::;xn):0xi1;i=1;:::;ngisacompactset,therealwaysexistsx2[0;1]nsuchthatfl(x)=min0xi1;i=1;:::;nfl(x).Thestatementfollowsfromlinearityoffl(x)withrespecttoeachvariable. Nowweshowthatg(G)l(G).Assumel(G)=m.Setxli=8>><>>:1;ifi2Dl;0;otherwise. Then,g(G)=min0xi1;i=1;:::;nfl(x)fl(xl)=m=l(G): 2.21 )alwayshasanoptimal0-1solution,theng(G)mustbeinteger.Assumeg(G)=m.Takeanoptimal0-1solutionxof( 2.20 ),suchthatthenumberof1'sismaximumamongall0-1optimalsolutions.Withoutlossofgeneralitywecanassumethatthissolutionisx1=x2==xr=1;xr+1=xr+2==xn=0,forsomer.LetQi(x)=ki1Xp=0Xfi1;:::ipgN(i)Ym2fi1;:::;ipgxmYm2N(i)nfi1;:::;ipg(1xm);Q(x)=nXi=r+1(1xi)0@ki1Xp=0Xfi1;:::ipgN(i)Ym2fi1;:::;ipgxmYm2N(i)nfi1;:::;ipg(1xm)1A:

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LetDl=fi:xi=1g: WewanttoshowthatDlisanl-dominatingset.Assumethatitisnot.LetS=fi2VnDl:jN(i)\Dlj><>>:1;ifi2Dl;0;otherwise. Thenfl(x0)fl(x)andjfi:x0i=1gj>jfi:xi=1gj,whichcontradictstheassumptionthatxisanoptimalsolutionwiththemaximumnumberof1's.Thus,Dlisanl-dominatingsetwithcardinalityrm=g(G)andthereforel(G)g(G).Thisconcludestheproofofthetheorem. 2.2 follows. Proof.

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Thus,(G)=nmin0xi1;i=1;:::;nnXi=10@xi+(1xi)(1Y(i;j)2Exj)1A=max0xi1;i=1;:::;nnXi=1(1xi)Y(i;j)2Exj:

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Inthischapter,wepresentseveralheuristicsforthemaximumindependentsetproblemusingcontinuousformulations.Theseresultshavepreviouslyappearedinseveralpublications[ 3 54 ]. 3.1 HeuristicBasedonFormulation(P1) Inthissection,wediscussanalgorithmforndingamaximalindependentsetbasedonformulation(P1)presentedinSection 2.3 .Aspointedoutbefore,thefunctionF(x)islinearwithrespecttoeachvariable,soAi(x)andBi(x)canbecomputedforanyi2f1;2;:::;ng.ToproduceamaximalindependentsetusingF(x),rstletx02[0;1]nbeanystartingpoint.Theproceduredescribedbelowproducesasequenceofnpointsx1;x2;:::;xnsuchthatxncorrespondstoamaximalindependentset.LetV=f1;2;:::;ngandconsidersomei2V.From( 2.9 { 2.12 )itfollowsthatifwesetx1i=8>><>>:0;ifAi(x0)>Bi(x0);1;otherwise. andx1j=x0j,ifj6=i,weobtainforthepointx1=(x11;x12;:::;x1n)thatF(x1)F(x0). IfweupdateV=Vnfig,wecanconstructthenextpointx2fromx1inthesamemanner.Runningthisprocedurentimes,weobtainapointxnwhichsatisestheinequalityF(xn)F(x0):

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Fromthediscussionabove,wehavethefollowingalgorithmtondamaximalindependentset. INPUT:x02[0;1]n 1. 2. 3. Theorem3.2. Proof.

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Thismeansthatneitherinoranynodefromtheneighborhoodofiisincludedintheindependentsetthatweobtainedafterstep1.Thus,wecanincreasethecardinalityofthissetincludingiinitbysettingvi=0. ThetimecomplexityoftheproposedalgorithmisO(2n),sinceAi(v)andBi(v)canbecalculatedinO(2)time. 3.2 HeuristicBasedonFormulation(P2) Wenowfocusourattentiononanalgorithm,similartoAlgorithm1,basedonformulation(P2). INPUT:x02[0;1]n 1. 2. 3. Theorem3.3. Proof. ThetimecomplexityofthisalgorithmisO(n).

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3.3 Examples Thealgorithmspresentedbuildamaximalindependentsetfromanygivenpointx02[0;1]ninpolynomialtime.Theoutput,however,dependsonthechoiceofx0.Aninterestingquestionthatarisesishowtochoosesuchinputpointx0,sothatamaximumindependentsetcanbefound?Theproblemofndingsuchapointcannotbesolvedinpolynomialtime,unlessP=NP.However,wecanshowonsimpleexamplesthatinsomecasesevenstartingwitha\bad"startingpointx0(F(x0)1orH(x0)1),weobtainamaximumindependentsetastheoutput. 3.3.1 Example1 ConsiderthegraphinFigure 3{1 .Forthisexamplex=(x1;x2;x3;x4)2[0;1]4;F(x)=(1x1)x2x3x4+(1x2)x1+(1x3)x1+(1x4)x1;A1(x)=x2x3x4;B1(x)=(1x2)+(1x3)+(1x4);A2(x)=A3(x)=A4(x)=x1;B2(x)=(1x1)x3x4;B3(x)=(1x1)x2x4;B4(x)=(1x1)x2x3;H(x)=x1+x2+x3+x4x1x2x1x3x1x4: 2;1 2;1 2;1 2).SinceA1(x0)=1 8,B1(x0)=3 2,andsince1 8<3 2,thenextpointisx1=(1;1 2;1 2;1 2). Next,A2(x1)=1,B2(x1)=0,andx2=(1;0;1 2;1 2).Aftertwomoreiterationswegetx4=(1;0;0;0)withI=f2;3;4g,whichisthemaximumindependentsetofthegivengraph.WehavejIj=3,F(x0)=13 16,andtheobjectivefunctionincreaseisjIjF(x0)=35 16.

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IllustrationtoExample1 ForAlgorithm2,startingwithx0=(1;1;1;1),forwhichH(x0)=1,weobtainthemaximumindependentsetafterstep1.Note,thattheCaro{Weiboundforthisgraphis7 4. 3.3.2 Example2 IllustrationtoExample2 ForthegraphinFigure 3{2 ,wehavex=(x1;x2;x3;x4;x5)2[0;1]5andF(x)=(1x1)x2x3+(1x2)x1x3x4+(1x3)x1x2x5+(1x4)x2+(1x5)x3:

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ApplyingAlgorithm1forthisgraphwithinitialpointx0=(1 2;1 2;1 2;1 2;1 2),weobtain,attheendofstep1,thesolutionx5=(1;1;1;0;0),whichcorrespondstotheindependentsetI=f4;5g.Attheendofstep2,thesolutionis(0;1;1;0;0),whichcorrespondstothemaximumindependentsetI=f1;4;5g.ForthiscasewehavejIj=3,F(x0)=7 8,andtheobjectivefunctionimprovementis17 8.Withtheinitialpointx0=(0;0;0;0;0),H(x0)=0,andAlgorithm2ndsthemaximumindependentsetafterstep1.Forthisexample,theCaro{Weiboundequals11 6. 3.3.3 Example3 Thisexampleshows,thattheoutputofAlgorithm1andAlgorithm2dependsnotonlyoninitialpointx0,butalsoontheorderinwhichweexaminevariablesinsteps1and2.Forexample,ifweconsiderthegraphfromFigure 3{1 withadierentorderofnodes(asinFigure 3{3 ),andrunAlgorithm1andAlgorithm2forthisgraphwithinitialpointx0=(1;1;1;1),weobtainI=f4gasoutputforbothalgorithms.Notethat,forthegraphfromFigure 3{1 ,bothoutputswouldbethemaximumindependentsetofthegraph. IllustrationtoExample3 AsExample3shows,wemaybeabletoimprovebothalgorithmsbyincludingtwoprocedures(oneforeachstep)which,givenasetofremainingnodes,chooseanodetobeexaminednext.ConsiderAlgorithm2.Letindex1()andindex2()be

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proceduresfordeterminingtheorderofexaminingnodesonstep1andstep2ofthealgorithm,respectively.Thenwehavethefollowingalgorithm. INPUT:x02[0;1]n 1. (b) (c) 2. (b) (c) 3. 3.3.4 ComputationalExperiments Thissectionpresentscomputationalresultsofthealgorithmsdescribedintheprevioussection.WehavetestedthealgorithmsonsomeoftheDIMACScliqueinstanceswhichcanbedownloadedfromtheURL

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Table3{1: Resultsonbenchmarkinstances:Algorithm1,randomx0. NamenDensity!(G)Sol.FoundAverageSol.Time(sec.) a9450.927161513.050.01MANN a273780.99012611368.168.14MANN a4510350.996345283198.76243.94 0.9 12000.900705339.080.14san200 0.9 22000.900603429.410.14san200 0.9 32000.900443127.580.23san400 0.9 14000.9001005346.184.54 First,eachalgorithmwasexecuted100timeswithrandominitialsolutionsuniformlydistributedintheunithypercube.TheresultsoftheseexperimentsaresummarizedinTables 3{1 3{2 ,and 3{3 .Thecolumns\Name,"\n,"\Density,"and\!(G)"representthenameofthegraph,thenumberofitsvertices,itsdensity,anditscliquenumber,respectively.ThisinformationisavailablefromtheDIMACSwebsite.Thecolumn\Sol.Found"containsthesizeofthelargestcliquefoundafter100runs.Thecolumns\AverageSol."and\Time(sec.)"containaveragesolutionandaverageCPUtime(inseconds)takenover100runsofanalgorithm,respectively.

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Table3{2: Resultsonbenchmarkinstances:Algorithm2,randomx0. NamenDensity!(G)Sol.FoundAverageSol.Time(sec.) a9450.927161614.450.01MANN a273780.990126120119.161.89MANN a4510350.996345334331.6636.20 0.9 12000.900706138.860.14san200 0.9 22000.900603229.090.13san200 0.9 32000.900443026.930.18san400 0.9 14000.9001005444.202.68 Table 3{4 containstheresultsofcomputationsforallthealgorithmswiththeinitialsolutionx0,suchthatx0i=0;i=1;:::;n.Inthistable,\A3"standsforAlgorithm3.Ascanbeseenfromthetables,thebestsolutionsforalmostallinstancesobtainedduringtheexperimentscanbefoundamongtheresultsforAlgorithm3withx0i=0;i=1;:::;n(seeTable 3{4 ). InTable 3{5 wecomparetheseresultswithresultsforsomeothercontinuousbasedheuristicsforthemaximumcliqueproblemtakenfromthepaperofBomzeetal.[ 39 ].Thecolumns\ARH",\PRD(1 2)",\PRD(0)"and\CBH"containthesizeofacliquefoundusingtheannealedreplicationheuristicofBomzeetal.[ 39 ],theplainreplicatordynamicsappliedfortwodierentparameterizations(withparameters1 2and0)oftheMotzkin{Strausformulation[ 40 ],andthecontinuous-basedheuristic

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Table3{3: Resultsonbenchmarkinstances:Algorithm3,randomx0. NamenDens.!(G)Sol.FoundAverageSol.Time(sec.) a9450.927161614.980.01MANN a273780.990126121119.214.32MANN a4510350.996345334331.5787.78 0.9 12000.900704645.030.37san200 0.9 22000.900603734.940.38san200 0.9 32000.900443226.860.37san400 0.9 14000.9001005150.012.54 ofGibbonsetal.[ 97 ],respectively.Thecolumn\A3(0)"representstheresultsforAlgorithm3withx0i=0;i=1;:::;n. 3.4 HeuristicBasedonOptimizationofaQuadraticOveraSphere RecalltheMotzkin-Straussformulationofthemaximumcliqueproblem.LetAGbetheadjacencymatrixofGandletebethen-dimensionalvectorwithallcomponentsequalto1. 2xTAGx;

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Table3{4: Resultsonbenchmarkinstances:Algorithms1{3,x0i=0,fori=1;:::;n. NamenDens.!(G)Sol.FoundTime(sec.) A1A2A3A1A2A3 a9450.92716169160.010.010.01MANN a273780.990126125271258.172.013.72MANN a4510350.99634534045342170.3136.4279.15 0.9 12000.900704243473.190.270.56san200 0.9 22000.900602936403.210.270.54san200 0.9 32000.900442921343.051.040.48san400 0.9 14000.90010052357588.473.286.55 Table3{5: Resultsonbenchmarkinstances:comparisonwithothercontinuousbasedapproaches.x0i=0;i=1;:::;n. NamenDens.!(G)Sol.Found ARHPRD(1 2)PRD(0)CBHA3(0) a9450.927161612121616MANN a273780.990126117117117121125 0.9 12000.900704545454647san200 0.9 22000.900603936353640san200 0.9 32000.900443132333034san400 0.9 14000.9001005040555075 3.1 a)x0: 3.1 b)

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211 98 ]extendedthisresultbyprovidingacharacterizationofmaximalcliquesintermsoflocalsolutions.Moreover,theystudiedoptimalityconditionsoftheMotzkin-Strausprogramandinvestigatedthepropertiesofanewlyintroducedparametrizationofthecorrespondingquadraticprogram. Gibbonsetal.[ 97 ]proposedanother,moretractablecontinuousformulationofthemaximumindependentsetproblem. 2xTAGx+(nXi=1xi1)2(3.2) 3.2 a)x0: 3.2 b) 3.2 )thenV(k)=0ithereexistsanindependentsetIinGsuchthatjIjk. 3.2 )-( 3.2 a)toextractamaximalindependentsetfromthegraph.TheapproachthatweproposeinthissectionisbasedonthesameformulationasinTheorem 3.5 .Moreover,wealsoadopttheideaofusingtheproblemofoptimizationofthesamequadraticoveraspheretondalargeindependentset,butinourapproachweuseinformationaboutallstationarypointsoftheproblem( 3.2 )-( 3.2 a).WewillusethecomputationalresultsofGibbonsetal.[ 97 ]toestimatetheeciencyofourapproachinsubsection 3.4.3

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3.4.1 OptimizationofaQuadraticFunctionOveraSphere AlthoughoptimizationofaquadraticfunctionsubjecttolinearconstraintsisNP-hard,quadraticprogrammingsubjecttoanellipsoidconstraintispolynomiallysolvable.Infact,Ye[ 203 ]provedthatitcanbesolvedinO(log(log(1=)))iterations,whereistheerrortolerance.Duetoitsapplicationintrustregionmethodsofnonlinearoptimization,theproblemofoptimizationofaquadraticfunctionoverasphereisawellstudiedone[ 89 110 153 164 ].Inthissectionwewilldiscusssomeresultsrelatedtothisproblem. Considerthefollowingproblem: minf(x)=2cTx+xTQx;(3.3) subjecttojjxjj2=r; 3.3 a) wherex2IRnisthevectorofvariables,Qisasymmetricnnmatrix,c2IRnisanarbitraryxedvector,r>0issomenumber,andjjjj2isthesecondnorminIRn,i.e.,foranyx2IRn:jjxjj2=nPi=1x2i1 2. Weareinterestedinallstationarypointsoftheproblem( 3.3 ),whichcanbefoundfromtheLagrangianL(x;)off(x).Wehave whereistheLagrangemultiplieroftheconstraintof( 3.3 ),andthestationarityconditions:

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Thesetofconditions( 3.5 )isequivalenttothesystem (QI)x=c;(3.7) whereIstandsforthennidentitymatrix.Thecondition( 3.6 )issimplytheconstraint( 3.3 a).Therefore,xisastationarypointofproblem( 3.3 )-( 3.3 a)ifandonlyifthereexistsarealnumber=(x0)suchthat (QI)x=c;(3.8) Thesetofallrealforwhichthereexistsanxsuchthat( 3.8 )and( 3.9 )aresatised,iscalledthespectrumoftheproblem( 3.3 )-( 3.3 a). ApplyingtheeigenvaluedecompositionforthematrixQwehaveQ=Rdiag(1;2;:::;n)RT; 3.7 )canberewrittenas (i)xi=ci;i=1;:::;n:(3.10) Assumingthat6=i,wehave xi=ci

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Rewritingexpression( 3.9 )inthebasisassociatedwiththerowsofmatrixRandsubstituting( 3.11 )intoit,weobtaintheso-calledsecularequation Theleft-handsideofthelastexpressionisaunivariatefunctionconsistingofn+1pieceseachofwhichiscontinuousandconvex.Thisimpliesthatitcanhaveatmosttworootsatanyinterval(i;i+1);i=0;:::;ncorrespondingtotheithpiece,whereby0andn+1wemeanand+1respectively.Moreover,thespectrumincludestheset0ofallsolutionstothesecularequation( 3.12 )andn0=fi:ci=0g(so,=0iffi:ci=0g=;). ForsytheandGolub[ 89 ]haveshownthattheglobalminimumofproblem( 3.3 )correspondstothesmallestelementofthespectrum=min.Itimpliesthatifci6=0then=min0. Wementiononemoreresult,whichrelatesthemaximumindependentsetproblemtotheproblemofoptimizationofaquadraticoverasphere.Recently,deAngelisetal.[ 70 ]haveshownthatintheproblemofoptimizingaquadraticoverasphereofradiusraglobalsolutionx(r)canbechosen,whichdependscontinuouslyuponr.Thelastfactgivesusanewpointofviewonthemaximumindependentsetproblem.Namely,itcanbeformulatedasaproblemofoptimizationofaquadraticfunctionoveraspherewithanunknownradius: 2;p 2ejj2=r; 2)andS2(ofradiusp

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theunithypercuberespectively.Wehavemaxx2S1f(x)(G)=maxx2[0;1]nf(x)maxx2S2f(x) andbytheIntermediateValueTheoremthelemmaiscorrect. 3.4.2 TheHeuristic Inthissubsection,wepresenttheheuristicforthemaximumindependentsetproblemwhichemploystheformulationmentionedinTheorem 3.5 ,andutilizesthemethodssimilartothoseintheso-calledQUALEXheuristic[ 53 ].InQUALEX,thefeasibleregionoftheformulation(P2)ischangedfromtheunithypercubetothespherewiththecenterx0=0andacertainradiusr,dependingontheinputgraph,resultinginthefollowingproblem: maxf(x)=nXi=1xiX(i;j)2Exixj(3.14) subjecttojjxx0jj22=r2: 3.14 a) Inshort,QUALEXcanberegardedasasophisticatedgreedyheuristic,whichrstndsalocalsolutionusingastraightforwardgreedyapproach,andthenattemptstondabettersolutionusinginformationprovidedbythestationarypointsoftheaboveproblem. Weapplysimilartechniquesfortheproblemwhichisobtainedfrom( 3.2 )-( 3.2 b)bychangingtheinequalitytotheequalityin( 3.2 a)andrelaxingnonnegativityconstraints( 3.2 b): min1 2xTAGx+(nXi=1xi1)2(3.15)

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subjecttonXi=1x2i=1 3.15 a) Asthevalueofkweusethecardinalityofamaximalindependentsetfoundbyasimplegreedyheuristic(numericalexperimentssuggestthatvaryingkdoesnotmakeresultsmuchbetter,but,naturally,requiresmorecomputationaltime).Foreachstationarypointxoftheproblem( 3.15 )-( 3.15 a)weapplyagreedyheuristictondamaximalindependentsetIx,whichusesorderoftheverticesinthegraphcorrespondingtothenonincreasingorderofthecomponentsofx.ThelargestfoundIxisreportedastheoutputmaximalindependentset. Belowwepresentanoutlineofouralgorithm. INPUT:GraphGwithadjacencymatrixAG;=0 Applythefollowinggreedyheuristic: (a) (b) InG0recursivelyremovetheneighborhoodofavertexwiththeminimumdegreeinthecurrentgraphuntilanemptygraphwithvertexsetIisobtained; (c) 1. Foreachinterval(i;i+1)begin ApplythetechniquesdiscussedinSection 3.4.1 fortheproblem( 3.15 )-( 3.15 a)tondthesetofallstationarypointsScorrespondingto(i;i+1); (b) Foreachx2SndIxbythefollowingprocedure:

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i. Orderthecomponentsofxinnonincreasingorderxi1xi2:::xin; ii. iii. (d) IfjIj>thenI=Iand=jIj; end 3.4.3 ComputationalExperiments Thissectionpresentsresultsofcomputationalexperimentsfortheproposedheuristic.ThealgorithmhasbeentestedonthecomplementgraphsofsomeoftheDIMACScliqueinstanceswhichcanbedownloadedfromtheURLhttp://dimacs.rutgers.edu/Challenges/.Theperformanceofthealgorithm(denotedbyQSH)onthebenchmarksissummarizedinTables 3{6 and 3{7 .Thecolumns\Name,"\n,"\Density,"and\!(G)"representthenameofthegraph,thenumberofitsvertices,itsdensity,anditscliquenumber,respectively.ThisinformationisavailablefromtheDIMACSwebsite.Recallthatweareworkingwithcomplementsoftheconsideredgraphs,andthedensitiesarespeciedfortheoriginal(maximumclique)instances.Thecolumn\Sol.Found"containsthesizeofthecliquefoundbyeachalgorithm.InTable 3{6 ,sub-columns\CBH"and\QSH"representthecontinuousbasedheuristic[ 97 ]andtheheuristicproposedinthissection,correspondingly.Finally,thecolumn\Time"reportscomputationaltimesforQSHinsecondsobtainedusingtheCfunctiontime.TheresultsforCBHweretakenfrom

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Table3{6: Resultsonbenchmarkinstances,partI. NamenDensity!(G)Sol.FoundTime CBHQSH(sec.) a9450.9271616160MANN a273780.9901261211258 12000.7452120211brock200 22000.4961212121brock200 32000.6051514151brock200 42000.6581716172brock400 14000.74827232721brock400 24000.74929242921brock400 34000.74831233120brock400 44000.74933243321brock800 18000.649232017245brock800 28000.651241924243brock800 38000.649252025228brock800 48000.650261926233 Gibbonsetal.[ 97 ].AmodicationoftheC++QUALEXcodeavailableon-linefromBusygin'swebpage[ 53 ]wasusedtoobtaintheresultsforQSH.ThecodewascompiledandexecutedonanIBMcomputerwith2PowerPCprocessors,333Mhzeach. InTable 3{8 wecomparetheperformanceofCBHandQSHondierenttypesofgraphs.Ineachrowofthistabletherstcolumncontainsanameofthefamilyof

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Table3{7: Resultsonbenchmarkinstances,partII. NamenDensity!(G)Sol.FoundTimeCBHQSH(sec.) hat300-13000.2448876p hat300-23000.4892525246p hat300-33000.7443636336p hat500-15000.25399948p hat500-25000.50536353349p hat500-35000.75249494648p hat700-17000.24911118143p hat700-27000.498444442143p hat700-37000.748626059143 0.7 12000.7003015301san200 0.7 22000.7001812181san200 0.9 12000.9007046701san200 0.9 22000.9006036601san200 0.9 32000.9004430351san400 0.5 14000.500138920san400 0.7 14000.70040204021san400 0.7 24000.70030153019san400 0.7 34000.70022141621san400 0.9 14000.9001005010023 0.72000.7001818151sanr200 0.92000.9004241371sanr400 0.54000.50013121120sanr400 0.7 14000.70021201819 graphs,whereeachfamilyconsistsofagroupofgraphswiththesameletternames(inTables 3{6 and 3{7 suchfamiliesareseparatedbyhorizontallines).Inthenextthreecolumns\+"representsthenumberofinstancesfromeachfamilyforwhichCBHfoundabettersolutionthanQSH;\{"representsthenumberofinstancesforwhichQSHfoundabettersolutionthanCBH;nally,\="standsforthenumberofinstancesforwhichcardinalitiesoftheindependentsetsfoundbythetwoalgorithmswereequal.

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Table3{8: Comparisonoftheresultsonbenchmarkinstances. GraphsfamilyCBHvsQSHname+={ a011brock1110c-fat070hamming040johnson040keller002p hat810san0010sanr400 AsonecanseeQSHperformedbetteronfourfamilies(MANN a,brock,keller,san)anddidworseontwo(p hatandsanr).Analyzingthedierenceinqualityoftheresults,onecannoticeahugegapbetweenthesolutionsobtainedforsangroup.TheresultsofnumericalexperimentssuggestthatQSHissuperiortoCBHandmanyotherheuristicspresentedintheliterature. 3.5 ConcludingRemarks Inthischapter,wediscussedseveralheuristicsforthemaximumindependentsetproblembasedoncontinuousformulations.InSections 3.1 and 3.2 ,weoerthreesyntacticallyrelatedalgorithmsforndinglargemaximalindependentsets.Acomputationalinvestigationofthesealgorithmsshowstheircompetitiveness. InSection 3.4 ,wepresentaheuristicforthemaximumindependentsetproblembasedontechniquesusedinoptimizationofaquadraticoverasphere.Weshowthevalidityofthisapproachbydemonstratingtheresultsofnumericalexperiments.Wewanttostressthattheinformationprovidedbythestationarypointsotherthanpointsofglobaloptimalitymayhelpustondbetterindependentsetsinthecorrespondinggraph.

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InLemma 3.1 weformulatedthemaximumindependentsetproblemasaproblemofoptimizationofaquadraticoveraspherewithanunknownradius.Infutureresearch,whenusingsimilartechniquesforcombinatorialoptimizationproblems,oneshouldtrytoaddressthefollowingissues: Toanswerthesequestionsboththeoreticalandempiricalstudiesareneeded.

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Massivedatasetsariseinabroadspectrumofscientic,engineeringandcommercialapplications.Theseincludegovernmentandmilitarysystems,telecommunications,medicineandbiotechnology,astrophysics,nance,ecology,geographicalinformationsystems,etc[ 5 ].Someofthewiderangeofproblemsassociatedwithmassivedatasetsaredatawarehousing,compressionandvisualization,informationretrieval,clusteringandpatternrecognition,andnearestneighborsearch.Handlingtheseproblemsrequiresspecialinterdisciplinaryeortsindevelopingnovelsophisticatedtechniques.Thepervasivenessandcomplexityoftheproblemsbroughtbymassivedatasetsmakeitoneofthemostchallengingandexcitingareasofresearchforyearstocome. Inmanycases,amassivedatasetcanberepresentedasaverylargegraphwithcertainattributesassociatedwithitsverticesandedges.Theseattributesmaycontainspecicinformationcharacterizingthegivenapplication.Studyingthestructureofthisgraphisimportantforunderstandingthestructuralpropertiesoftheapplicationitrepresents,aswellasforimprovingstorageorganizationandinformationretrieval. Inthischapter,werstreviewcurrentdevelopmentinstudyingmassivegraphs.Then,inSection 4.2 wepresentourresearchconcerningthemarketgraphrepresentingtheU.S.stockmarkets.WewillconcludewithsomeremarksinSection 4.4 .ThischapterisbasedonjointpublicationswithBoginskiandPardalos[ 32 33 ]. 4.1 ModelingandOptimizationinMassiveGraphs Inthissectionwediscussrecentadvancesinmodelingandoptimizationformassivegraphs.Asexamples,call,Internet,andWebgraphswillbeused. 59

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Asbefore,byG=(V;E)wewilldenoteasimpleundirectedgraphwiththesetofnverticesVandthesetofedgesE.Amulti-graphisanundirectedgraphwithmultipleedges. Thedistancebetweentwoverticesisthenumberofedgesintheshortestpathbetweenthem(itisequaltoinnityforverticesrepresentingdierentconnectedcomponents).ThediameterofagraphGisusuallydenedasthemaximaldistancebetweenpairsofverticesofG.Note,thatinthecaseofadisconnectedgraphtheusualdenitionofthediameterwouldresultintheinnitediameter,thereforethefollowingdenitionisinorder.Bythediameterofadisconnectedgraphwewillmeanthemaximumniteshortestpathlengthinthegraph(whichisthesameasthelargestofdiametersofthegraph'sconnectedcomponents). 4.1.1 ExamplesofMassiveGraphs Thecallgraph HerewediscussanexampleofamassivegraphrepresentingtelecommunicationstracdatapresentedbyAbello,PardalosandResende[ 4 ].Inthiscallgraphtheverticesaretelephonenumbers,andtwoverticesareconnectedbyanedgeifacallwasmadefromonenumbertoanother. Abelloetal.[ 4 ]experimentedwithdatafromAT&Ttelephonebillingrecords.Togiveanideaofhowlargeacallgraphcanbewementionthatagraphbasedonone20-dayperiodhad290millionverticesand4billionedges.Theanalyzedone-daycallgraphhad53,767,087verticesandover170millionsofedges.Thisgraphappearedtohave3,667,448connectedcomponents,mostofthemtiny;only302,468(or8%)componentshadmorethan3vertices.Agiantconnectedcomponentwith44,989,297verticeswascomputed.ItwasobservedthattheexistenceofagiantcomponentresemblesabehaviorsuggestedbytherandomgraphstheoryofErdosandRenyi,butbythepatternofconnectionsthecallgraphobviouslydoesnottintothistheory.ThiswillbediscussedinmoredetailinSubsection 4.1.3 .The

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maximumcliqueproblemandproblemofndinglargequasi-cliqueswithprespecieddensitywereconsideredinthisgiantcomponent.Theseproblemswereattackedusingagreedyrandomizedadaptivesearchprocedure(GRASP)[ 85 86 ].Inshort,GRASPisaniterativemethodthatateachiterationconstructs,usingagreedyfunction,arandomizedsolutionandthenndsalocallyoptimalsolutionbysearchingtheneighborhoodoftheconstructedsolution.Thisisaheuristicapproachwhichgivesnoguaranteeaboutqualityofthesolutionsfound,butprovedtobepracticallyecientformanycombinatorialoptimizationproblems.Tomakeapplicationofoptimizationalgorithmsintheconsideredlargecomponentpossible,theauthorsusesomesuitablegraphdecompositiontechniquesemployingexternalmemoryalgorithms(seeSubsection 4.1.2 ). Figure4{1: FrequenciesofcliquesizesinthecallgraphfoundbyAbelloetal.[ 4 ]. Abelloetal.ran100,000GRASPiterationstaking10parallelprocessorsaboutoneandahalfdaystonish.Ofthe100,000cliquesgenerated,14,141appearedtobedistinct,althoughmanyofthemhadverticesincommon.Theauthorssuggestedthatthegraphcontainsnocliqueofasizegreaterthan32.Figure 4{1 showsthenumberofdetectedcliquesofvarioussizes.Finally,largequasi-cliqueswithdensityparameters

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Figure4{2: Numberofverticeswithvariousout-degrees(a)andin-degrees(b);thenumberofconnectedcomponentsofvarioussizes(c)inthecallgraph,duetoAielloetal.[ 11 ]. Aielloetal.[ 11 ]usedthesamedataasAbelloetal.[ 4 ]toshowthattheconsideredcallgraphtstotheirrandomgraphmodelwhichwillbediscussedinSubsection 4.1.3 .TheplotsinFigure 4{2 demonstratesomeconnectivitypropertiesofthecallgraph. Figure 4{2 (a,b)showsplotsofthenumberofverticesforeveryout-degreeandin-degree,respectively.FrequenciesofthesizesofconnectedcomponentsarerepresentedinFigure 4{2 (c).

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TheInternetandWebgraphs TheroleoftheInternetinthemodernworldisdiculttooverestimate;itsinventionchangedthewaypeopleinteract,learn,andcommunicatelikenothingbefore.Alongsidewithincreasingsignicance,theInternetitselfcontinuestogrowatanoverwhelmingrate.Figure 4{3 showsthedynamicsofgrowthofthenumberofInternethostsforthelast13years.AsofJanuary2002thisnumberwasestimatedtobecloseto150million.Thenumberofwebpagesindexedbylargesearchenginesexceeds2billion,andthenumberofwebsitesisgrowingbythousandsdaily. Figure4{3: NumberofInternethostsfortheperiod01/1991-01/2002.DatabyInternetSoftwareConsortium[ 126 ]. ThehighlydynamicandseeminglyunpredictablestructureoftheWorldWideWebattractsmoreandmoreattentionofscientistsrepresentingmanydiversedisciplines,includinggraphtheory.InagraphrepresentationoftheWorldWideWeb,theverticesaredocumentsandtheedgesarehyperlinkspointingfromonedocumenttoanother.Similarlytothecallgraph,theWebisadirectedmultigraph,althoughoftenitistreatedasanundirectedgraphtosimplifytheanalysis.AnothergraphisassociatedwiththephysicalnetworkoftheInternet,wheretheverticesare

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Figure4{4: AsampleofpathsofthephysicalnetworkofInternetcablescreatedbyW.CheswickandH.Burch[ 60 ].CourtesyofLumetaCorporation.Patent(s)Pending.CopyrightcLumetaCorporation2003.AllRightsReserved. 4{4 illustratesthetremendouscomplexityofthisnetwork.ThisandmanyothermapsaimingtovisualizetheInternettopologyareproductsofthe\InternetMappingProject"[ 60 ].TheyarecreatedfromdataobtainedbytracingroutesfromoneterminaltoasetofotherInternetdomains. Graphtheoryhasbeenappliedforwebsearch[ 42 59 141 ],webmining[ 160 161 ]andotherproblemsarisingintheInternetandWorldWideWeb.Inseveralrecentstudies,therewereattemptstounderstandsomestructuralpropertiesoftheWebgraphbyinvestigatinglargeWebcrawls.AdamicandHuberman[ 8 124 ]used

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Figure4{5: Numberofverticeswithvariousout-degrees(left)anddistributionofsizesofstronglyconnectedcomponents(right)inWebgraph[ 43 ]. crawlswhichcoveredalmost260,000pagesintheirstudies.BarabasiandAlbert[ 26 ]analyzedasubgraphoftheWebgraphapproximately325,000nodesrepresentingnd.edupages.Inanotherexperiment,Kumaretal.[ 146 ]examinedadatasetcontainingabout40millionpages.Inarecentstudy,Broderetal.[ 43 ]usedtwoAltavistacrawls,eachwithabout200millionpagesand1.5billionlinks,thussignicantlyexceedingthescaleoftheprecedingexperiments.ThisworkyieldedseveralremarkableobservationsaboutlocalandglobalpropertiesoftheWebgraph.Allofthepropertiesobservedinoneofthetwocrawlswerevalidatedfortheotheraswell.Below,bytheWebgraphwewillmeanoneofthecrawls,whichhas203,549,046nodesand2130millionarcs. TherstobservationmadebyBroderetal.conrmsapropertyoftheWebgraphsuggestedinearlierworks[ 26 146 ]claimingthatthedistributionofdegreesfollowsapowerlaw.Thatis,thenumberofverticesofdegreekisproportionaltokforsome>1.Interestingly,thedegreedistributionoftheWebgraphresemblesthepower-lawrelationshipoftheInternetgraphtopology,whichwasrstdiscoveredbyFaloutsosetal.[ 82 ].Broderetal.[ 43 ]computedthein-andout-degreedistributionsforbothconsideredcrawlsandshowedthatthesedistributionsagreewithpowerlaws.Moreover,theyobservedthatinthecaseofin-degreestheconstant2:1isthe

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sameastheexponentofpowerlawsdiscoveredinearlierstudies[ 26 146 ].InanothersetofexperimentsconductedbyBroderetal.,directedandundirectedconnectedcomponentswereinvestigated.Itwasnoticedthatthedistributionofsizesoftheseconnectedcomponentsalsoobeysapowerlaw.Figure 4{5 illustratestheexperimentswithdistributionsofout-degreesandconnectedcomponentsizes. ThelastseriesofexperimentsdiscussedbyBroderetal.[ 43 ]aimedtoexploretheglobalconnectivitystructureoftheWeb.Thisledtothediscoveryoftheso-calledBow-TiemodeloftheWeb[ 44 ].Similarlytothecallgraph,theconsideredWebgraphappearedtohaveagiantconnectedcomponent,containing186,771,290nodes,orover90%ofthetotalnumberofnodes.Takingintoaccountthedirectednatureoftheedges,thisconnectedcomponentcanbesubdividedintofourpieces:stronglyconnectedcomponent(SCC),InandOutcomponents,and\Tendrils".Overall,theWebgraphintheBow-Tiemodelisdividedintothefollowingpieces: Figure 4{6 showstheconnectivitystructureoftheWeb,aswellassizesoftheconsideredcomponents.Asonecanseefromthegure,thesizesofSCC,In,OutandTendrilscomponentsareroughlyequal,andtheDisconnectedcomponentissignicantlysmaller.

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Figure4{6: ConnectivityoftheWebduetoBroderetal.[ 43 ]. Broderetal.[ 43 ]havealsocomputedthediametersoftheSCCandofthewholegraph.ItwasshownthatthediameteroftheSCCisatleast28,andthediameterofthewholegraphisatleast503.Theaverageconnecteddistanceisdenedasthepairwisedistanceaveragedoverthosedirectedpairs(i;j)ofnodesforwhichthereexistsapathfromitoj.Theaverageconnecteddistanceofthewholegraphwasestimatedas16.12forin-links,16.18forout-links,and6.83forundirectedlinks.Interestingly,itwasalsofoundthatforarandomlychosendirectedpairofnodes,thechancethatthereisadirectedpathbetweenthemisonlyabout24%.TheseresultsarenotinagreementwiththepredictionofAlbertetal.[ 12 ],whosuggestedthattheaveragedistancebetweentworandomlychosendocumentsoftheWebis18.59.Letusmentionthatthepropertyofalargenetworktohaveasmalldiameterhasbeenobservedinmanyreal-lifenetworksandisfrequentlyreferredtoasthesmallworldphenomenon[ 15 198 199 ].

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4.1.2 ExternalMemoryAlgorithms Inmanycases,thedataassociatedwithmassivegraphsistoolargetotentirelyinsidethefastcomputer'sinternalmemory,thereforeaslowerexternalmemory(forexampledisks)needstobeused.Theinput/outputcommunication(I/O)betweenthesememoriescanresultinanalgorithm'sslowperformance.Externalmemory(EM)algorithmsanddatastructuresaredesignedwithaimtoreducetheI/Ocostbyexploitingthelocality.Recently,externalmemoryalgorithmshavebeensuccessfullyappliedforsolvingbatchedproblemsinvolvinggraphs,includingconnectedcomponents,topologicalsorting,andshortestpaths. TherstEMgraphalgorithmwasdevelopedbyUllmanandYannakakis[ 194 ]in1991anddealtwiththeproblemoftransitiveclosure.Manyotherresearcherscontributedtotheprogressinthisareaeversince[ 2 18 19 49 61 147 197 ].Chiangetal.[ 61 ]proposedseveralnewtechniquesfordesignandanalysisofecientEMgraphalgorithmsanddiscussedapplicationsofthesetechniquestospecicproblems,includingminimumspanningtreeverication,connectedandbiconnectedcomponents,graphdrawing,andvisibilityrepresentation.Abelloetal.[ 2 ]proposedafunctionalapproachforEMgraphalgorithmsandusedtheirmethodologytodevelopdeterministicandrandomizedalgorithmsforcomputingconnectedcomponents,maximalindependentsets,maximalmatchings,andotherstructuresinthegraph.Inthisapproacheachalgorithmisdenedasasequenceoffunctions,andthecomputationcontinuesinaseriesofscanoperationsoverthedata.Iftheproducedoutputdata,oncewritten,cannotbechanged,thenthefunctionissaidtohavenosideeects.Thelackofsideeectsenablestheapplicationofstandardcheckpointingtechniques,thusincreasingthereliability.Abelloetal.presentedasemi-externalmodelforgraphproblems,whichassumesthatonlytheverticestinthecomputer'sinternalmemory.Thisisquitecommoninpractice,andinfactthiswasthecaseforthecallgraphdescribedinSubsection 4.1.1 ,forwhichecientEM

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algorithmsdevelopedbyAbelloetal.[ 2 ]wereusedinordertocomputeitsconnectedcomponents[ 4 ]. Formoredetailonexternalmemoryalgorithmsseethebook[ 6 ]andtheextensivereviewbyVitter[ 197 ]ofEMalgorithmsanddatastructures. 4.1.3 ModelingMassiveGraphs Thesizeofreal-lifemassivegraphs,manyofwhichcannotbeheldevenbyacomputerwithseveralgigabytesofmainmemory,vanishesthepowerofclassicalalgorithmsandmakesonelookfornovelapproaches.Externalmemoryalgorithmsanddatastructuresdiscussedintheprevioussubsectionrepresentoneoftheresearchdirectionsaimingtoovercomedicultiescreatedbydatasizes.Butinsomecasesnotonlyistheamountofdatahuge,butthedataitselfisnotcompletelyavailable.Forinstance,onecanhardlyexpecttocollectcompleteinformationabouttheWebgraph;infact,thelargestsearchenginesareestimatedtocoveronly38%oftheWeb[ 148 ]. Someapproachesweredevelopedforstudyingthepropertiesofreal-lifemassivegraphsusingonlytheinformationaboutasmallpartofthegraph.Forinstance,Goldreich[ 105 ]proposestworandomizedalgorithmsfortestingifagivenmassivegraphhassomepredenedproperty.Thesealgorithmsanalyzeapartofthegraphandwithsomeprobabilitygivetheanswerifthisgraphhasagivenpropertyornot,basedonacertaincriterion. Anothermethodologyofinvestigatingreal-lifemassivegraphsistousetheavailableinformationinordertoconstructpropertheoreticalmodelsofthesegraphs.Oneoftheearliestattemptstomodelrealnetworkstheoreticallygoesbacktothelate1950's,whenthefoundationsofrandomgraphtheoryhadbeendeveloped.Inthissubsectionwewillpresentsomeoftheresultsproducedbythisandother(morerealistic)graphmodels.

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Uniformrandomgraphs TheclassicaltheoryofrandomgraphsfoundedbyErdosandRenyi[ 78 79 ]dealswithseveralstandardmodelsoftheso-calleduniformrandomgraphs.TwoofsuchmodelsareG(n;m)andG(n;p)[ 35 ].Therstmodelassignsthesameprobabilitytoallgraphswithnverticesandmedges,whileinthesecondmodeleachpairofverticesischosentobelinkedbyanedgerandomlyandindependentlywithprobabilityp. Inmostcasesforeachnaturalnaprobabilityspaceconsistingofgraphswithexactlynverticesisconsidered,andthepropertiesofthisspaceasn!1arestudied.Itissaidthatatypicalelementofthespaceoralmostevery(a.e.)graphhaspropertyQwhentheprobabilitythatarandomgraphonnverticeshasthispropertytendsto1asn!1.WewillalsosaythatthepropertyQholdsasymptoticallyalmostsurely(a.a.s.).ErdosandRenyidiscoveredthatinmanycaseseitheralmosteverygraphhaspropertyQoralmosteverygraphdoesnothavethisproperty. Manypropertiesofuniformrandomgraphshavebeenwellstudied[ 34 35 129 143 ].Belowwewillsummarizesomeknownresultsinthiseld. Probablythesimplestpropertytobeconsideredinanygraphisitsconnectivity.ItwasshownthatforauniformrandomgraphG(n;p)2G(n;p)thereisa\threshold"valueofpthatdetermineswhetheragraphisalmostsurelyconnectedornot.Morespecically,agraphG(n;p)isa.a.s.disconnectedifp
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dependsontheinterrelationshipoftheparametersofthemodelnandp.However,thisdependencyturnsouttoberathercomplicated.Itwasdiscussedinmanypapers,andthecorrespondingresultsaresummarizedbelow. ItwasprovedbyKleeandLarman[ 140 ]thatarandomgraphasymptoticallyalmostsurelyhasthediameterd,wheredisacertainintegervalue,ifthefollowingconditionsaresatised 35 ]provedthatifnplogn!1thenthediameterofarandomgraphisa.a.s.concentratedonnomorethanfourvalues. Luczak[ 151 ]consideredthecasenp<1,whenauniformrandomgrapha.a.s.isdisconnectedandhasnogiantconnectedcomponent.LetdiamT(G)denotethemaximumdiameterofallconnectedcomponentsofG(n;p)whicharetrees.Thenif(1np)n1=3!1thediameterofG(n;p)isa.a.s.equaltodiamT(G). ChungandLu[ 62 ]investigatedanotherextremecase:np!1.TheyshowedthatinthiscasethediameterofarandomgraphG(n;p)isa.a.s.equalto (1+o(1))logn (1+o(1))logn np+1:

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log(np)diam(G(n;p))2666log33c20 62 ]thatitisa.a.s.trueonlyifnp>3:5128. Potentialdrawbacksoftheuniformrandomgraphmodel Thereweresomeattemptstomodelthereal-lifemassivegraphsbytheuniformrandomgraphsandtocomparetheirbehavior.However,theresultsoftheseexperimentsdemonstratedasignicantdiscrepancybetweenthepropertiesofrealgraphsandcorrespondinguniformrandomgraphs. Thefurtherdiscussionanalyzesthepotentialdrawbacksofapplyingtheuniformrandomgraphmodeltothereal-lifemassivegraphs. Thoughtheuniformrandomgraphsdemonstratesomepropertiessimilartothereal-lifemassivegraphs,manyproblemsarisewhenonetriestodescribetherealgraphsusingtheuniformrandomgraphmodel.Asitwasmentionedabove,agiantconnectedcomponenta.a.s.emergesinauniformrandomgraphatacertainthreshold.ItlooksverysimilartothepropertiesoftherealmassivegraphsdiscussedinSubsection 4.1.3 .However,afterdeeperinsight,itcanbeseenthatthegiant

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connectedcomponentsintheuniformrandomgraphsandthereal-lifemassivegraphshavedierentstructures.Thefundamentaldierencebetweenthemisasfollows:itwasnoticedthatinalmostalltherealmassivegraphsthepropertyofso-calledclusteringtakesplace[ 198 199 ].Itmeansthattheprobabilityoftheeventthattwogivenverticesareconnectedbyanedgeishigheriftheseverticeshaveacommonneighbor(i.e.,avertexwhichisconnectedbyanedgewithbothofthesevertices).Theprobabilitythattwoneighborsofagivenvertexareconnectedbyanedgeiscalledtheclusteringcoecient.Itcanbeeasilyseenthatinthecaseoftheuniformrandomgraphs,theclusteringcoecientisequaltotheparameterp,sincetheprobabilitythateachpairofverticesisconnectedbyanedgeisindependentofallothervertices.Inreal-lifemassivegraphs,thevalueoftheclusteringcoecientturnsouttobemuchhigherthanthevalueoftheparameterpoftheuniformrandomgraphswiththesamenumberofverticesandedges.Adamic[ 7 ]foundthatthevalueoftheclusteringcoecientforsomepartoftheWebgraphwasapproximately0.1078,whiletheclusteringcoecientforthecorrespondinguniformrandomgraphwas0.00023.Pastor-Satorrasetal.[ 172 ]gotsimilarresultsforthepartoftheInternetgraph.Thevaluesoftheclusteringcoecientsfortherealgraphandthecorrespondinguniformrandomgraphwere0.24and0.0006respectively. Anothersignicantproblemarisinginmodelingmassivegraphsusingtheuniformrandomgraphmodelisthedierenceindegreedistributions.Itcanbeshownthatasthenumberofverticesinauniformrandomgraphincreases,thedistributionofthedegreesoftheverticestendstothewell-knownPoissondistributionwiththeparameternpwhichrepresentstheaveragedegreeofavertex.However,asitwaspointedoutinSubsection 4.1.3 ,theexperimentsshowthatintherealmassivegraphsdegreedistributionsobeyapowerlaw.Thesefactsdemonstratethatsomeothermodelsareneededtobetterdescribethepropertiesofrealmassivegraphs.Next,

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wediscusstwoofsuchmodels;namely,therandomgraphmodelwithagivendegreesequenceanditsmostimportantspecialcase-thepower-lawmodel. Randomgraphswithagivendegreesequence Besidestheuniformrandomgraphs,therearemoregeneralwaysofmodelingmassivegraphs.Thesemodelsdealwithrandomgraphswithagivendegreesequence.Themainideaofhowtoconstructthesegraphsisasfollows.Foralltheverticesi=1:::nthesetofthedegreesfkigisspecied.Thissetischosensothatthefractionofverticesthathavedegreektendstothedesireddegreedistributionpkasnincreases. Itturnsoutthatsomepropertiesoftheuniformrandomgraphscanbegeneralizedforthemodelofarandomgraphwithagivendegreesequence. Recallthenotationofso-called\phasetransition"(i.e.,thephenomenonwhenatacertainpointagiantconnectedcomponentemergesinarandomgraph)whichhappensintheuniformrandomgraphs.Itturnsoutthatasimilarthingtakesplaceinthecaseofarandomgraphwithagivendegreesequence.ThisresultwasobtainedbyMolloyandReed[ 162 ].Theessenceoftheirndingsisasfollows. Considerasequenceofnon-negativerealnumbersp0,p1,...,suchthatPkpk=1.AssumethatagraphGwithnverticeshasapproximatelypknverticesofdegreek.IfwedeneQ=Pk1k(k2)pkthenitcanbeprovedthatGa.a.s.hasagiantconnectedcomponentifQ>0andthereisa.a.s.nogiantconnectedcomponentifQ<0. Asadevelopmentoftheanalysisofrandomgraphswithagivendegreesequence,theworkofCooperandFrieze[ 66 ]shouldbementioned.Theyconsideredasparsedirectedrandomgraphwithagivendegreesequenceandanalyzeditsstrongconnectivity.Inthestudy,thesizeofthegiantstronglyconnectedcomponent,aswellastheconditionsofitsexistence,werediscussed.

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Theresultsobtainedforthemodelofrandomgraphswithagivendegreesequenceareespeciallyusefulbecausetheycanbeimplementedforsomeimportantspecialcasesofthismodel.Forinstance,theclassicalresultsonthesizeofaconnectedcomponentinuniformrandomgraphsfollowfromtheaforementionedfactpresentedbyMolloyandReed.Next,wepresentanotherexampleofapplyingthisgeneralresulttooneofthemostpracticallyusedrandomgraphmodels-thepower-lawmodel. Power-lawrandomgraphs Oneofthemostimportantspecialcasesofthemodelofrandomgraphswithagivendegreesequenceisthepower-lawrandomgraphmodel.Thepower-lawrandomgraphsarealsosometimesreferredtoas(;)-graphs.Thismodelwasrecentlyappliedtodescribesomereal-lifemassivegraphssuchasthecallgraph,theInternetgraphandtheWebgraphmentionedabove.SomefundamentalresultsforthismodelwereobtainedbyAiello,ChungandLu[ 10 11 ]. Thebasicideaofthepower-lawrandomgraphmodelP(,)isasfollows.Ifwedeneytobethenumberofnodeswithdegreex,thenaccordingtothismodel Equivalently,wecanwrite logy=logx:(4.2) Thisrepresentationismoreconvenientinthesensethattherelationshipbetweenyandxcanbeplottedasastraightlineonalog-logscale,sothat(-)istheslope,andistheintercept.Thisimpliesthefollowingpropertiesofagraphdescribedbythepowerlawmodel[ 11 ]:

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Xx=1e where(t)=1Pn=11 2e Xx=1xe 2(1)e;>2;1 4e;=2;1 2e2=.(2);0<<2:(4.4) Sincethepower-lawrandomgraphmodelisaspecialcaseofthemodelofarandomgraphwithagivendegreesequence,theresultsdiscussedabovecanbeappliedtothepowerlawgraphs.Weneedtondthethresholdvalueofinwhichthe\phasetransition"(i.e.,theemergenceofagiantconnectedcomponent)occurs.InthiscaseQ=Px1x(x2)pxisdenedas Px=1x(x2)e Px=1e Px=1e Hence,thethresholdvalue0canbefoundfromtheequation Theresultsonthesizeoftheconnectedcomponentofapower-lawgraphwerepresentedbyAiello,ChungandLu[ 11 ].Theseresultsaresummarizedbelow.

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Thepower-lawrandomgraphmodelwasdevelopedfordescribingreal-lifemassivegraphs.Sothenaturalquestionishowwellitreectsthepropertiesofthesegraphs. Thoughthismodelcertainlydoesnotreectallthepropertiesofrealmassivegraphs,itturnsoutthatthemassivegraphssuchasthecallgraphortheInternetgraphcanbefairlywelldescribedbythepower-lawmodel.Thefollowingexampledemonstratesit. Aiello,ChungandLu[ 11 ]investigatedthesamecallgraphthatwasanalyzedbyAbelloetal.[ 4 ].ThismassivegraphwasalreadydiscussedinSubsection 4.1.3 ,soitisinterestingtocomparetheexperimentalresultspresentedbyAbelloetal.[ 4 ]withthetheoreticalresultsobtainedin[ 11 ]usingthepower-lawrandomgraphmodel. Figure 4{2 showsthenumberofverticesinthecallgraphwithcertainin-degreesandout-degrees.Recallthataccordingtothepower-lawmodelthedependencybetweenthenumberofverticesandthecorrespondingdegreescanbeplottedasastraightlineonalog-logscale,soonecanapproximatetherealdatashowninFigure 4{2 byastraightlineandevaluatetheparameterandusingthevaluesoftheinterceptandtheslopeoftheline.Thevalueofforthein-degreedatawasestimatedtobeapproximately2.1,andthevalueofewasapproximately30106.Thetotalnumberofnodescanbeestimatedusingformula( 4.3 )as(2:1)e=1:56e47106(comparewithSubsection 4.1.3 ). Accordingtotheresultsforthesizeofthelargestconnectedcomponentpresentedabove,apower-lawgraphwith1<3:47875a.a.s.hasagiantconnected

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component.Since2:1fallsinthisrange,thisresultexactlycoincideswiththerealobservationsforthecallgraph(seeSubsection 4.1.3 ). Anotheraspectthatisworthmentioningishowtogeneratepower-lawgraphs.Themethodologyfordoingitwasdiscussedindetailintheliterature[ 10 48 65 ].Thesepapersuseasimilarapproach,whichisreferredtoasarandomgraphevolutionprocess.Themainideaistoconstructapower-lawmassivegraph\step-by-step":ateachtimestep,anodeandanedgeareaddedtoagraphinaccordancewithcertainrulesinordertoobtainagraphwithaspeciedin-degreeandout-degreepower-lawdistribution.Thein-degreeandout-degreeparametersoftheresultingpower-lawgrapharefunctionsoftheinputparametersofthemodel.AsimpleevolutionmodelwaspresentedbyKumaretal.[ 145 ].Aiello,ChungandLu[ 10 ]developedfourmoreadvancedmodelsforgeneratingbothdirectedandundirectedpower-lawgraphswithdierentdistributionsofin-degreesandout-degrees.Asanexample,wewillbrieydescribeoneoftheirmodels.Itwasthebasicmodeldevelopedinthepaper,andtheotherthreemodelsactuallywereimprovementsandgeneralizationsofthismodel. Themainideaoftheconsideredmodelisasfollows.Atthersttimemomentavertexisaddedtothegraph,anditisassignedtwoparameters-thein-weightandtheout-weight,bothequalto1.Thenateachtimestept+1anewvertexwithin-weight1andout-weight1isaddedtothegraphwithprobability1,andanewdirectededgeisaddedtothegraphwithprobability.Theoriginanddestinationverticesarechosenaccordingtothecurrentvaluesofthein-weightsandout-weights.Morespecically,avertexuischosenastheoriginofthisedgewiththeprobabilityproportionaltoitscurrentout-weightwhichisdenedaswoutu;t=1+outu;twhereoutu;tistheout-degreeofthevertexuattimet.Similarly,avertexvischosenasthedestinationwiththeprobabilityproportionaltoitscurrentin-weightwinv;t=1+inv;twhereinv;tisthein-degreeofvattimet.Fromtheabovedescriptionitcanbeseenthatattimetthetotalin-weightandthetotalout-weightarebothequaltot.Sofor

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eachparticularpairofverticesuandv,theprobabilitythatanedgegoingfromutovisaddedtothegraphattimetisequalto Thenotionoftheso-calledscaleinvariance[ 26 27 ]mustalsobementioned.Thisconceptarisesfromthefollowingconsiderations.Theevolutionofmassivegraphscanbetreatedastheprocessofgrowingthegraphatatimeunit.Now,ifwereplaceallthenodesthatwereaddedtothegraphatthesameunitoftimebyonlyonenode,thenwewillgetanothergraphofasmallersize.Thebiggerthetimeunitis,thesmallerthenewgraphsizewillbe.Theevolutionmodeliscalledscale-free(scale-invariant)ifwithhighprobabilitythenew(scaled)graphhasthesamepower-lawdistributionofin-degreesandout-degreesastheoriginalgraph,foranychoiceofthetimeunitlength.Itturnsoutthatmostoftherandomevolutionmodelshavethisproperty.Forinstance,themodelsofAielloetal.[ 10 ]wereprovedtobescale-invariant. 4.1.4 OptimizationinRandomMassiveGraphs Recentrandomgraphmodelsofreal-lifemassivenetworks,someofwhichwerementionedinSubsection 4.1.3 increasedinterestinvariouspropertiesofrandomgraphsandmethodsusedtodiscovertheseproperties.Indeed,numericalcharacteristicsofgraphs,suchascliqueandchromaticnumbers,couldbeusedasoneofthestepsinvalidationoftheproposedmodels.Inthisregard,theexpectedcliquenumberofpower-lawrandomgraphsisofspecialinterestduetotheresultsbyAbelloetal.[ 4 ]andAielloetal.[ 10 ]mentionedinSubsections 4.1.1 and 4.1.3 .Ifcomputed,

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itcouldbeusedasoneofthepointsinverifyingthevalidityofthemodelforthecallgraphproposedbyAielloetal.[ 10 ]. Inthissubsectionwepresentsomewell-knownfactsregardingthecliqueandchromaticnumbersinuniformrandomgraphs. Cliquenumber TheearliestresultsdescribingthepropertiesofcliquesinuniformrandomgraphsareduetoMatula[ 159 ],whonoticedthatforaxedpalmostallgraphsG2G(n;p)haveaboutthesamecliquenumber,ifnissucientlylarge.BollobasandErdos[ 37 ]furtherdevelopedtheseremarkableresultsbyprovingsomemorespecicfactsaboutthecliquenumberofarandomgraph.Letusdiscusstheseresultsinmoredetailbypresentingnotonlythefactsbutalsosomereasoningbehindthem.FormoredetailseebooksbyBollobas[ 34 35 ]andJansonetal.[ 129 ]. Assumethat00g:

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Theexpectationofkl(Gn)canbecalculatedasE(kl(Gn))=nlp(l2): Usingthisobservationandthesecondmomentmethod,BollobasandErdos[ 37 ]provedthatifp=p(n)satisesn0thereexistsaconstantc,suchthatforc

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GrimmettandMcDiarmid[ 106 ]werethersttostudytheproblemofcoloringrandomgraphs.Manyotherresearcherscontributedtosolvingthisproblem[ 13 36 ].Wewillmentionsomefactsemergedfromthesestudies. Luczak[ 150 ]improvedtheresultsabouttheconcentrationof(G(n;p))previouslyprovedbyShamirandSpencer[ 182 ],provingthatforeverysequencep=p(n)suchthatpn6=7thereisafunctionch(n)suchthata:a:s:ch(n)(G(n;p))ch(n)+1: 13 ]provedthatforanypositiveconstantthechromaticnumberofauniformrandomgraphG(n;p),wherep=n1 2,isa.a.s.concentratedintwoconsecutivevalues.Moreover,theyprovedthataproperchoiceofp(n)mayresultinaone-pointdistribution.Thefunctionch(n)isdiculttond,butinsomecasesitcanbecharacterized.Forexample,Jansonetal.[ 129 ]provedthatthereexistsaconstantc0suchthatforanyp=p(n)satisfyingc0 35 ]yieldsthefollowingestimate:(G(n;p))=n 4.1.5 Remarks Wediscussedadvancesinseveralresearchdirectionsdealingwithmassivegraphs,suchasexternalmemoryalgorithmsandmodelingofmassivenetworksasrandom

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graphswithpower-lawdegreedistributions.Despitetheevidencethatuniformrandomgraphsarehardlysuitableformodelingtheconsideredreal-lifegraphs,theclassicalrandomgraphstheorystillmayserveasagreatsourceofideasinstudyingpropertiesofmassivegraphsandtheirmodels.Werecalledsomewell-knownresultsproducedbytheclassicalrandomgraphstheory.Theseincluderesultsforconcentrationofcliquenumberandchromaticnumberofrandomgraphs,whichwouldbeinterestingtoextendtomorecomplicatedrandomgraphmodels(i.e.,power-lawgraphsandgraphswitharbitrarydegreedistributions).Externalmemoryalgorithmsandnumericaloptimizationtechniquescouldbeappliedtondanapproximatevalueofthecliquenumber(asitwasdiscussedinSubsection 4.1.1 ).Ontheotherhand,probabilisticmethodssimilartothosediscussedinSubsection 4.1.4 couldbeutilizedinordertondtheasymptoticaldistributionofthecliquenumberinthesamenetwork'srandomgraphmodel,andthereforeverifythismodel. 4.2 TheMarketGraph AlthoughnotsoobviouslyasintheexamplesinSubsection 4.1.1 ,nancialmarketscanalsoberepresentedasgraphs.Forastockmarketonenaturalrepresentationisbasedonthecrosscorrelationsofstockpriceuctuations.Amarketgraphcanbeconstructedasfollows:eachstockisrepresentedbyavertex,andtwoverticesareconnectedbyanedgeifthecorrelationcoecientofthecorrespondingpairofstocks(calculatedforacertainperiodoftime)isaboveaprespeciedthreshold;11. 4.2.1 ConstructingtheMarketGraph ThemarketgraphthatwestudyinthischapterrepresentsthesetofnancialinstrumentstradedintheU.S.stockmarkets.Morespecically,weconsider6546instrumentsandanalyzedailychangesoftheirpricesoveraperiodof500consecutivetradingdaysin2000-2002.Basedonthisinformation,wecalculatethecross-correlationsbetweeneachpairofstocksusingthefollowingformula[ 156 ]:

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Figure4{7: Distributionofcorrelationcoecientsinthestockmarket ThecorrelationcoecientsCijcanvaryfrom-1to1.Figure 4{7 showsthedistributionofthecorrelationcoecientsbasedonthepricesdatafortheyears2000-2002.Itcanbeseenthatthisdistributionisnearlysymmetricaroundthemean,whichisapproximatelyequalto0.05. Themainideaofconstructingamarketgraphisasfollows.Letthesetofnancialinstrumentsrepresentthesetofverticesofthegraph.Also,wespecifyacertainthresholdvalue;11andaddanundirectededgeconnectingtheverticesiandjifthecorrespondingcorrelationcoecientCijisgreaterthanorequalto.Obviously,dierentvaluesofdenethemarketgraphswiththesamesetofvertices,butdierentsetsofedges.

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Figure4{8: Edgedensityofthemarketgraphfordierentvaluesofthecorrelationthreshold. Itiseasytoseethatthenumberofedgesinthemarketgraphdecreasesasthethresholdvalueincreases.Infact,ourexperimentsshowthattheedgedensityofthemarketgraphdecreasesexponentiallyw.r.t..ThecorrespondinggraphispresentedonFigure 4{8 4.2.2 ConnectivityoftheMarketGraph InSubsection 4.1.3 wementionedtheconnectivitythresholdsinrandomgraphs.Themainideaofthisconceptisndingathresholdvalueoftheparameterofthemodel(pinthecaseofuniformrandomgraphs,andinthecaseofpower-lawgraphs)thatwilldeneifthegraphisconnectedornot.Moreover,ifthegraphisdisconnected,anotherthresholdvaluecanbedenedtodetermineifthegraphhasagiantconnectedcomponentorallofitsconnectedcomponentshaveasmallsize. Forinstance,inthecaseofthepower-lawmodel=1isathresholdvaluethatdeterminestheconnectivityofthepower-lawgraph,i.e.,thegraphisa.a.s.connected

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Figure4{9: Plotofthesizeofthelargestconnectedcomponentinthemarketgraphasafunctionofcorrelationthreshold. if<1,anditisa.a.s.disconnectedotherwise.Similarly,3:47875denestheexistenceofagiantconnectedcomponentinthepower-lawgraph. Nowanaturalquestionarises:whatistheconnectivitythresholdforthemarketgraph?Sincethenumberofedgesinthemarketgraphdependsonthechosencorrelationthreshold,weshouldndavalue0thatdeterminestheconnectivityofthegraph.Asitwasmentionedabove,thesmallervalueofwechoose,themoreedgesthemarketgraphwillhave.So,ifwedecrease,afteracertainpoint,thegraphwillbecomeconnected.Wehaveconductedaseriesofcomputationalexperimentsforcheckingtheconnectivityofthemarketgraphusingthebreadth-rstsearchtechnique,andweobtainedarelativelyaccurateapproximationoftheconnectivitythreshold:0'0:14382.Moreover,weinvestigatedthedependencyofthesizeofthelargestconnectedcomponentinthemarketgraphw.r.t..ThecorrespondingplotisshownonFigure 4{9

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4.2.3 DegreeDistributionsintheMarketGraph Asitwasshownintheprevioussection,thepower-lawmodelfairlywelldescribessomeofthereal-lifemassivegraphs,suchastheWebgraphandthecallgraph.Inthissubsection,wewillshowthatthemarketgraphalsoobeysthepower-lawmodel. Itshouldbenotedthatsinceweconsiderasetofmarketgraphs,whereeachgraphcorrespondstoacertainvalueof,thedegreedistributionswillbedierentforeach. Theresultsofourexperimentsturnedouttoberatherinteresting. Ifwespecifyasmallvalueofthecorrelationthreshold,suchas=0,=0:05,=0:1,=0:15,thedistributionofthedegreesoftheverticesisvery\noisy"anddoesnothaveanywell-denedstructure.Notethatforthesevaluesofthemarketgraphisconnectedandhasahighedgedensity.Themarketgraphstructureseemstobeverydiculttoanalyzeinthesecases. However,thesituationchangesdrasticallyifahighercorrelationthresholdischosen.Astheedgedensityofthegraphdecreases,thedegreedistributionmoreandmoreresemblesapowerlaw.Infact,for0:2thisdistributionisapproximatelyastraightlineinthelog-logscale,whichisexactlythepowerlawdistribution,asitwasshowninSection 4.1 .Figure 4{10 demonstratesthedegreedistributionsofthemarketgraphsforsomevaluesofthecorrelationthreshold. Aninterestingobservationisthattheslopeofthelines(whichisequaltotheparameterofthepower-lawmodel)israthersmall.Itcanbeseenfromformula( 4.1 )thatinthiscasethegraphwillcontainmanyverticeswithahighdegree.Thisfactisimportantforthenextsubjectofourinterest-ndingmaximumcliquesinthemarketgraph.Intuitively,onecanexpectalargecliqueinagraphwithasmallvalueoftheparameter.Aswewillseenext,thisassumptionistrueforthemarketgraph. Anothercombinatorialoptimizationproblemassociatedwiththemarketgraphisndingmaximumindependentsetsinthegraphswithanegativecorrelationthreshold

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Figure4{10: Degreedistributionofthemarketgraphfor(a)=0:2;(b)=0:3;(c)=0:4;(d)=0:5 However,wecanconsideracomplementarygraphforamarketgraphwithanegativevalueof.Inthisgraph,anedgewillconnectinstrumentsiandjifthecorrelationbetweenthemCij<.Recallthatamaximumindependentsetintheinitialgraphisamaximumcliqueinthecomplementarygraph,sothemaximumindependentsetproblemcanbereducedtothemaximumcliqueprobleminthecomplementarygraph. Therefore,itisalsousefultoinvestigatethedegreedistributionsofthesecomplementarygraphs.AsitcanbeseenfromFigure 4{7 ,thedistributionofthecorrelationcoecientsisalmostsymmetricaround=0:05,soforthevaluesof

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Figure4{11: Degreedistributionofthecomplementarymarketgraphfor(a)=0:15;(b)=0:2;(c)=0:25 closeto0theedgedensityofboththeinitialandthecomplementarygraphishighenough.So,forthesevaluesofthedegreedistributionofacomplementarygraphisalso\noisy"asinthecaseofthecorrespondinginitialgraph. Asdecreases(i.e.,increasesintheabsolutevalue),thedegreedistributionofacomplementarygraphtendstothepowerlaw.ThecorrespondinggraphsareshownonFigure 4{11 .However,inthiscase,theslopeofthelineinthelog-logscale(thevalueoftheparameter)ishigherthaninthecaseofpositivevaluesof.Itmeansthattherearenotmanyverticeswithahighdegreeinthesegraphs,sothesizeofamaximumcliqueshouldbesignicantlysmallerthaninthecaseofthemarketgraphswithapositivecorrelationthreshold.

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Table4{1: Clusteringcoecientsofthemarketgraph(-complementarygraph) -0.152:64105-0.10.00120.30.48850.40.44580.50.45220.60.48720.70.4886 Itisalsointerestingtocomparethedierenceinclusteringcoecientsofamarketgraph(withpositivevaluesof)oritscomplement(withnegativevaluesof)(seeTable 4{1 ). Intuitively,thelargeclusteringcoecientsshouldcorrespondtographswithlargercliques,therefore,fromTable 4{1 oneshouldexpectthatthecliquesinmarketgraphwithpositivearemuchlargerthantheindependentsetsinmarketgraphwithnegative.Thispredictionwillbeconrmedinthenextsubsection,wherewepresentthecomputationalresultsofsolvingthemaximumcliqueandmaximumindependentsetproblemsinthemarketgraph. 4.2.4 CliquesandIndependentSetsintheMarketGraph Asitwasmentionedabove,themaximumcliqueandthemaximumindependentsetproblemsareNP-hard.Itmakestheseproblemsespeciallychallenginginlargegraphs.Themaximumcliqueproblemadmitsanintegerprogrammingformulation,however,inthecaseofthegraphwith6546verticesthisintegerprogrammingproblemcannotbesolvedinareasonabletime.Therefore,weusedagreedyheuristicforndingalowerboundofthecliquenumber,andaspecialpreprocessingtechniquewhichreducesaproblemsize. Tondalargeclique,weapplythefollowinggreedyalgorithm.Startingwithanemptyset,werecursivelyaddtothecliqueavertexfromtheneighborhoodofthecliqueadjacenttothemostverticesintheneighborhoodoftheclique.Ifwedenoteby

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Afterrunningthisalgorithm,weappliedthefollowingpreprocessingprocedure[ 4 ].WerecursivelyremovefromthegraphalloftheverticeswhicharenotinCandwhosedegreeislessthanjCj,whereCisthecliquefoundbytheabovealgorithm.Thissimpleprocedureenabledustosignicantlyreducethesizeofthemaximumcliquesearchspace.LetusdenotebyG0(V0;E0)thegraphinducedbyremainingvertices.Table 4{2 presentsthesizesofthecliquesfoundusingthegreedyalgorithm,andsizesofthegraphsremainingafterapplyingthepreprocessingprocedure. Table4{2: Sizesofcliquesfoundusingthegreedyalgorithmandsizesofgraphsremainingafterapplyingthepreprocessingtechnique InordertondthemaximumcliqueofG0(whichisalsothemaximumcliqueintheoriginalgraphG),weusedthefollowingintegerprogrammingformulationofthemaximumcliqueproblem(seeSection 2.1 ): maximizeXxi

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Table4{3: Sizesofthemaximumcliquesinthemarketgraphwithdierentvaluesofthecorrelationthreshold 0.350.00901930.40.00471440.450.00241090.50.0013850.550.0007630.60.0004450.650.0002270.70.000122 s.t.xi+xj1;(i;j)=2E0;xi2f0;1g;i=1;:::;n: 125 ]tosolvethisintegerprogramforsomeoftheconsideredinstances. Table 4{3 summarizesthesizesofthemaximumcliquesfoundinthegraphfordierentvaluesof.Itturnsoutthatthesecliquesareratherlarge.Infact,evenfor=0:6,whichisaveryhighcorrelationthreshold,thecliqueofsize45wasfound. Theseresultsareinagreementwiththeabovediscussion,whereweanalyzedthedegreedistributionsofthemarketgraphswithpositivevaluesofandcametotheconclusionthatthecliquesinthesegraphsshouldbelarge. Thenancialinterpretationofthecliqueinthemarketgraphisthatitdenesthesetofstockswhosepriceuctuationsexhibitasimilarbehavior.Ourresultsshowthatinthemodernstockmarkettherearelargegroupsofinstrumentsthatarecorrelatedwitheachother. Next,weconsiderthemaximumindependentsetprobleminthemarketgraphswithnonpositivevaluesofthecorrelationthreshold.Thisproblemcanbeeasilyrepresentedasamaximumcliqueprobleminacomplementarygraph.Interestingly,thepreprocessingprocedurethatwasveryhelpfulforndingmaximumcliquesin

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Table4{4: Sizesofindependentsetsfoundusingthegreedyalgorithm 0.050.4794360.00.200112-0.050.04315-0.10.0053-0.150.00052 originalgraphswasabsolutelyuselessinthecasewiththeircomplements.Therefore,weconcludethatthemaximumindependentsetappearstobemorediculttocomputethanthemaximumcliqueinthemarketgraph.Table 4{4 presentstheresultsobtainedusingthegreedyalgorithmdescribedabove. Asonecansee,thesizesofthecomputedindependentsetsareverysmall,whichcoincideswiththepredictionthatwasmadebeforebasedontheanalysisofthedegreedistributions. Fromthenancialpointofview,theindependentsetinthemarketgraphrepresents\completelydiversied"portfolio,whereallinstrumentsarenegativelycorrelatedwitheachother.Itturnsoutthatchoosingsuchaportfolioisnotaneasytask,andonecannotexpecttoeasilyndalargegroupofnegativelycorrelatedinstruments. 4.2.5 InstrumentsCorrespondingtoHigh-DegreeVertices Uptothispoint,westudiedthepropertiesofthemarketgraphasonebigsystem,anddidnotconsiderthecharacteristicsofeveryvertexinthisgraph.However,averyimportantpracticalissueistoinvestigatethedegreeofeachvertexinthemarketgraphandtondtheverticeswithhighdegrees,i.e.,theinstrumentsthatarehighlycorrelatedwithmanyotherinstrumentsinthemarket.Clearly,thisinformationwillhelpustoansweraveryimportantquestion:whichinstrumentsmostaccuratelyreectthebehaviorofthemarket?

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Forthispurpose,wechosethemarketgraphwithahighcorrelationthreshold(=0:6),calculatedthedegreesofeachvertexinthisgraphandsortedtheverticesinthedecreasingorderoftheirdegrees. Interestingly,eventhoughtheedgedensityoftheconsideredgraphisonly0.04%(onlyhighlycorrelatedinstrumentsareconnectedbyanedge),therearemanyverticeswithdegreesgreaterthan100. Accordingtoourcalculations,thevertexwiththehighestdegreeinthismarketgraphcorrespondstotheNASDAQ100IndexTrackingStock.Thedegreeofthisvertexis216,whichmeansthatthereare216instrumentsthatarehighlycorrelatedwithit.AninterestingobservationisthatthedegreeofthisvertexistwicehigherthanthenumberofcompanieswhosestockpricestheNASDAQindexreects,whichmeansthatthese100companiesgreatlyinuencethemarket. InTable 4{5 wepresentthe\top25"instrumentsintheU.S.stockmarket,accordingtotheirdegreesintheconsideredmarketgraph.Thecorrespondingsymbolsdenitionscanbefoundonseveralwebsites,forexamplehttp://www.nasdaq.com.Notethatmostofthemareindicesthatincorporateanumberofdierentstocksofcompaniesindierentindustries.Althoughthisresultisnotsurprisingfromthenancialpointofview,itisimportantasapracticaljusticationofthemarketgraphmodel. 4.3 EvolutionoftheMarketGraph Inordertoinvestigatethedynamicsofthemarketgraphstructure,wechosetheperiodof1000tradingdaysin1998-2002andconsideredeleven500-dayshiftswithinthisperiod.Thestartingpointsofeverytwoconsecutiveshiftsareseparatedbytheintervalof50days.Therefore,everypairofconsecutiveshiftshad450daysincommonand50daysdierent(seeFigure 4{12 ).DatescorrespondingtoeachshiftaresummarizedinTable 4{6

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Table4{5: Top25instrumentswithhighestdegrees(=0:6). symbolvertexdegree QQQ216IWF193IWO193IYW193XLK181IVV175MDY171SPY162IJH159IWV158IVW156IAH155IYY154IWB153IYV150BDH144MKH143IWM142IJR134SMH130STM118IIH116IVE113DIA106IWD106 Theadvantageofthisprocedureisthatitallowsustoaccuratelyreectthestructuralchangesofthemarketgraphusingrelativelysmallintervalsbetweenshifts,butatthesametimewecanmaintainlargesamplesizesofthestockpricesdataforcalculatingcross-correlationsforeachshift.Weshouldnotethatinouranalysisweconsideredonlystockswhichwereamongthosetradedasofthelastofthe1000tradingdays.Forpracticalreasonswedidnottakeintoaccountstockswhichhadbeenwithdrawnfromthemarket.However,thesecouldbeincludedinamoredetailedanalysistoobtainabetterglobalpictureofthemarketevolution. Therstsubjectofourconsiderationisthedistributionofcorrelationcoecientsbetweenallpairsofstocksinthemarket.Recallthatforthemarketgraphconsidered

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Figure4{12: Timeshiftsusedforstudyingtheevolutionofthemarketgraphstructure. intheprevioussectionthisdistributionwasnearlysymmetricaround0.05andlikeanormaldistributionitwasconcentratedarounditsmean.Aninterpretationofthisfactisthatthecorrelationofmostpairsofstocksisclosetozero,therefore,thestructureofthestockmarketissubstantiallyrandom,andonecanmakeareasonableassumptionthatthepricesofmoststockschangeindependently.Asweconsidertheevolutionofthecorrelationdistributionovertime,itturnsoutthatthisdistributionremainsalmostunchangedforalltimeintervals,whichisillustratedbyFigure 4{13 Thestabilityofthecorrelationcoecientsdistributionofthemarketgraphintuitivelymotivatesthehypothesisthatthedegreedistributionshouldalsoremainstable.Toverifythisassumption,wehavecalculatedthedegreedistributionofthegraphsconstructedforallconsideredtimeshifts.Thecorrelationthreshold=0:5waschosen.Ourexperimentsshowthatthedegreedistributionissimilarforallintervals,andinallcasesitiswelldescribedbyapowerlaw.Figure 4{14 shows

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Table4{6: Datescorrespondingtoeach500-dayshift. Period#StartingdateEndingdate 109/24/199809/15/2000212/04/199811/27/2000302/18/199902/08/2001404/30/199904/23/2001507/13/199907/03/2001609/22/199909/19/2001712/02/199911/29/2001802/14/200002/12/2002904/26/200004/25/20021007/07/200007/08/20021109/18/200009/17/2002 Figure4{13: Distributionofthecorrelationcoecientsbetweenallconsideredpairsofstocksinthemarket,forodd-numberedtimeshifts. thedegreedistributions(inthelogarithmicscale)forsomeinstancesofthemarketgraphcorrespondingtodierentintervals.Asonecansee,alltheseplotscanbewellapproximatedbystraightlines,whichmeansthattheyrepresentthepower-lawdistribution. Thecross-correlationdistributionandthedegreedistributionofthemarketgraphrepresenttheglobalcharacteristicsofthemarket,andtheaforementionedresultsleadustotheconclusionthatthegeneralstructureofthemarketisstableovertime.However,aswewillseenow,someglobalchangesinthestockmarketstructuredo

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Figure4{14: Degreedistributionofthemarketgraphforperiods(fromlefttoright,fromtoptobottom)1,4,7,and11(logarithmicscale). takeplace.Inordertodemonstrateit,welookatanothercharacteristicofthemarketgraph,itsedgedensity. Forstudyingtheedgedensityofthemarketgraph,wechosearelativelyhighcorrelationthreshold=0:5thatwouldensurethatweconsideronlytheedgescorrespondingtothepairsofstocks,whicharesignicantlycorrelatedwitheachother.Inthiscase,theedgedensityofthemarketgraphwouldrepresenttheproportionofthosepairsofstocksinthemarket,whosepriceuctuationsaresomewhatsimilarandcorrespondinglyinuenceeachother.Thesubjectofourinterestistostudyhowthisproportionchangesduringtheconsideredperiodoftime.Table 4{7 summarizestheobtainedresults.

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Table4{7: Numberofverticesandnumberofedgesinthemarketgraph(=0:5)fordierentperiods. PeriodjVjjEjEdgedensity 1543022580.015%2550726140.017%3559337720.024%4566652760.033%5576868410.041%6586677700.045%76013104280.058%86104124570.067%96262129110.066%106399197070.096%116556278850.130% Asitcanbeseenfromthistable,boththenumberofverticesandthenumberofedgesinthemarketgraphincreaseastimegoes.Obviously,thenumberofverticesgrowssincenewstocksappearinthemarket,andwedon'tconsiderthosestockswhichceasedtoexistbythelastof1000tradingdaysusedinouranalysis,sothemaximumpossiblenumberofedgesinthegraphincreasesaswell.However,itturnsoutthatthenumberofedgesgrowsfaster;therefore,theedgedensityofthemarketgraphincreasesfromperiodtoperiod.AsonecanseefromFigure 4{15 ,thegreatestincreaseoftheedgedensitycorrespondstothelasttwoperiods.Infact,theedgedensityforthelatestintervalisapproximately8.5timeshigherthanfortherstinterval! Figure4{15: Growthdynamicsoftheedgedensityofthemarketgraphovertime.

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Thisdramaticjumpsuggeststhatthereisatrendtothe\globalization"ofthemodernstockmarket,whichmeansthatnowadaysmoreandmorestockssignicantlyaectthebehavioroftheothers,andthestructureofthemarketbecomesnotpurelyrandom. Itshouldbenotedthattheincreaseoftheedgedensitycouldbepredictedfromtheanalysisofthedistributionofthecross-correlationsbetweenallpairsofstocks,whichwasmentionedabove.FromFigure 4{13 ,onecanobservethateventhoughthedistributionscorrespondingtodierentperiodshaveasimilarshapeandthesamemean,the\tail"ofthedistributioncorrespondingtothelatestperiod(period11)issignicantly\heavier"thanfortheearlierperiods,whichmeansthattherearemorepairsofstockswithhighervaluesofthecorrelationcoecient. Next,wendthestockwiththehighestdegreeinthemarketgraph.Table 4{8 presentsthestockswiththehighestdegreesinthemarketgraphwiththecorrelationthreshold=0:5,fordierenttimeintervals.Asonewouldexpect,thesestocksareexchangetradedfunds:NASDAQ-100indextrackingstock(QQQ),MidCapSPDRTrustSeriesI(MDY),andSPDRTrustSeriesI(SPY).Note,thatthevaluesofthehighestdegreeinthemarketgraphincreasefromperiodtoperiod,whichisanotherconrmationofthefactthatthe\globalization"ofthestockmarketactuallytakesplace. 4.4 Conclusion Agreatnumberofproblemsarisinginmassivegraphsexhibitachallengingandattractiveeldofresearchforrepresentativesofavarietyofdiverseeldsofscienceandengineering,includingprobability,graphtheory,andcomputerscienceamongmanyothers. Inthischapterwehavegivenabriefreviewoftherecentprogressinselectedtopicsassociatedwithmassivegraphs.Althoughinexampleswerestrictedourattentiontographsarisingintelecommunications,theInternetandnanceonly,

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Table4{8: Verticeswiththehighestdegreesinthemarketgraphfordierentperiods(=0:5). PeriodStockwiththehighestdegreeDegree 1SPY1062QQQ1303QQQ1814QQQ2245QQQ2526QQQ2677QQQ3188QQQ3389QQQ33710MDY40211MDY514 therearemanyotherreal-lifenetworkswithsimilarproblemswhichhaveattractedconsiderableresearcheortsinthepastfewyears[ 117 130 131 173 ]. Wepresentedadetailedstudyofthepropertiesofthemarketgraph,includingtheanalysisofitsdegreedistribution,connectivity,aswellascomputingofcliquesandindependentsets.Findingcliquesandindependentsetsinthemarketgraphgivesusanewtooloftheanalysisofthemarketstructurebyclassifyingthestocksintodierentgroups. Moreover,weprovedthatthepower-lawmodel,whichwelldescribesthemassivegraphsarisingintelecommunicationsandtheInternet,isalsoapplicableinnance.Itconrmsanamazingobservationthatmanyreal-lifemassivegraphshaveasimilarpower-lawstructure. InSection 4.3 ,wehavestudiedtheevolutionofdierentcharacteristicsofthemarketgraphovertimeandcametoseveralinterestingconclusionsbasedonouranalysis.Itturnsoutthatthepower-lawstructureofthemarketgraphisquitestableovertheconsideredtimeintervals.Anotherinterestingresultisthefactthattheedgedensityofthemarketgraphandthehighestdegreesoftheverticessteadily

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increaseduringthelastseveralyears,whichsupportsthewell-knownideaabouttheglobalizationofeconomywhichhasbeenwidelydiscussedrecently. Althoughweaddressedmanyissuesinouranalysisofthemarketgraph,therearestillalotofopenproblems.Forinstance,sincetheindependentsetsinthemarketgraphturnedouttobeverysmall,thereisapossibilitytoconsiderquasi-cliquesinsteadofcliquesinthecomplementarygraph.Thiswillallowustondlargerdiversiedportfolioswhichisimportantfromthepracticalpointofview.Also,onecanconsideranothertypeofthemarketgraphbasedonthedataoftheliquidityofdierentinstruments,insteadofconsideringthereturns.Itwouldbeveryinterestingtostudythepropertiesofthisgraphandcompareitwiththemarketgraphconsideredinthischapter.Therefore,thisresearchdirectionisverypromisingandimportantforadeeperunderstandingofmarketbehavior.

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Errorcorrectingcodeslieintheheartofdigitaltechnology,makingcellphones,compactdiskplayersandmodemspossible.Theyarealsoofspecialsignicanceduetoincreasingimportanceofreliabilityissuesininternettransmissions.Afundamentalproblemofinterestistosendamessageacrossanoisychannelwithamaximumpossiblereliability.Thisproblemisrelatedtotheproblemofndingthelargesterrorcorrectingcodesforcertaintypesoferrors.Computingestimatesofthesizeofcorrectingcodesisimportantfromboththeoreticalandpracticalperspectives. Inthischapterweusetheobservationthattheproblemofndingthelargestcorrectingcodesandtheirestimatescanbeformulatedintermsofmaximumindependentsetandgraphcoloringproblemsinspeciallyconstructedgraphs.Wesolvetheproblemofndingthelargestcorrectingcodesusingecientalgorithmsforoptimizationproblemsingraphs.Wereportnewexactsolutionsandestimates.ThischapterisbasedonpublicationsbyButenkoetal.[ 55 56 ]. 5.1 Introduction Letapositiveintegernbegiven.Forabinaryvectoru2BndenotebyFe(u)thesetofallvectors(notnecessaryofdimensionn)whichcanbeobtainedfromuasaconsequenceofcertainerrore,suchasdeletionortranspositionofbits.AsubsetCBnissaidtobeane-correctingcodeifFe(u)TFe(v)=;forallu;v2C;u6=v.Inthispaperweconsiderthefollowingcasesfortheerrore. 188 ]haspublishedasurveyofsingle-deletion-correctingcodes. 103

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Theproblemofourinterestistondthelargestcorrectingcodes.Itappearsthatthisproblemcanbeformulatedintermsofextremalgraphproblemsasfollows. ConsideragraphGnhavingavertexforeveryvectoru2Bn,withanedgejoiningtheverticescorrespondingtou;v2Bn;u6=vifandonlyifFe(u)TFe(v)6=;.ThenacorrectingcodecorrespondstoanindependentsetinGn.Hence,thelargeste-correctingcodecanbefoundbysolvingthemaximumindependentsetproblemintheconsideredgraph.Note,thatthisproblemcouldbeequivalentlyformulatedasthemaximumcliqueprobleminthecomplementgraphofG. Inthischapter,usingecientapproachestothemaximumindependentsetandgraphcoloringproblemsweimprovedsomeofthepreviouslyknownlowerboundsforasymmetriccodesandfoundtheexactsolutionsforsomeoftheinstances. Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wendlowerboundsandexactsolutionsforthelargestcodesusingecientalgorithmsforthemaximumindependentsetproblem.InSection 5.3 ,agraphcoloringheuristic

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andthepartitioningmethodareutilizedinordertoobtainbetterlowerboundsforsomeasymmetriccodes.Finally,concludingremarksaremadeinSection 5.4 5.2 FindingLowerBoundsandExactSizesoftheLargestCodes Inthissection,wesummarizesomepreviouslyobtainedresults[ 56 181 ].Westartwiththefollowingglobaloptimizationformulationforthemaximumindependentsetproblem,whichisverysimilartotheformulation(P1)fromChapter 2 Thisformulationisvalidifinsteadof[0;1]nwewillusef0;1gnasthefeasibleregion,thusobtaininganinteger0-1programmingproblem.Inproblem( 5.1 ),toeachvertexicorrespondsaBooleanexpression:i!ri=xi^24^(i;j)2E Toapplylocalsearchtechniquesfortheaboveproblemoneneedstodeneaproperneighborhood.Wedenetheneighborhoodonthesetofallmaximalindependentsets:

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IfI(G(Listjq))isanarbitrarymaximalindependentsetinG(Listjq),thenthesetsoftheform(Ifxjqg)[I(G(Listjq));q=1;:::;jIj 1. GivenarandomlygeneratedBooleanvectorx,ndanappropriateinitialmaximalindependentsetI. 2. Findamaximalindependentsetfromtheneighborhood(denedformaximalindependentsets)ofI,whichhasthelargestcardinality. Wetestedtheproposedalgorithmwiththefollowinggraphsarisingfromcodingtheory.ThesegraphsareconstructedasdiscussedinSection 5.1 andcanbedownloadedfromNeilSloane'swebsite[ 187 ]. TheresultsoftheexperimentsaresummarizedinTable 5{1 .Inthistable,thecolumns\Graph",\n",and\jEj"representthenameofthegraph,thenumberofitsvertices,anditsnumberofedges.Thisinformationisavailablefromthewebsite[ 187 ].Thecolumn\Solutionfound"containsthesizeoflargestindependentsetsfoundby

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Table5{1: Lowerboundsobtained. 1dc1281281471161dc2562563839301dc5125129727521dc1024102424063941dc20482048583671722dc128128517352dc2562561718372dc51251254895112dc10241024169162161tc6464192201tc128128512381tc2562561312631tc51251232641101tc1024102479361961tc20482048189443521et6464264181et128128672281et2562561664501et51251240321001et1024102496001711et204820482205283161zc1281281120181zc2562562816361zc5125126912621zc10241024161401121zc20482048394241981zc4096409692160379 thealgorithmover10runs.Asonecansee,theresultsareveryencouraging.Infact,foralloftheconsideredinstances,theywereatleastasgoodasthebestpreviouslyknownestimates. 5.2.1 FindingtheLargestCorrectingCodes Theproposedexactalgorithmconsistsofthefollowingsteps.

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Belowwegivemoredetailoneachofthesesteps. 0. 1. 2. 3.

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s.t.Pi2Cjxi1;j=1;:::;m; whereCj2Cisamaximalclique,C-asetofmaximalcliques,jCj=m.Forageneralgraphthelastconstraintshouldread0xi1;i=1;:::;n:But,sinceanisolatedvertexisanisolatedcliqueaswell,afterthepreprocessingstepourgraphdoesnotcontainisolatedverticesandtheaboveinequalitiesareimpliedbythesetofcliqueconstraints( 5.4 )alongwithnonnegativityconstraints( 5.5 ).WecallOC(G)thelinearcliqueestimate. InordertondatightboundOC(G)onenormallyneedstoconsideralargenumberofcliqueconstraints.Therefore,onedealswithlinearprogramsinwhichthenumberofconstraintsmaybemuchlargerthanthenumberofvariables.Inthiscaseitmakessensetoconsiderthelinearprogramwhichisdualtoproblem( 5.3 )-( 5.5 ).Thedualproblemcanbewrittenasfollows: s.t.mPj=1aijyj1;i=1;:::;n; whereaij=8><>:1;ifi2Cj;0;otherwise: 5.3 )-( 5.5 ).Ifm>nthedualproblemismoresuitableforsolvingwithsimplexmethodandinteriorpointmethods.Increasingthenumberofcliqueconstraintsinproblem( 5.3 )-( 5.5 )leadsonlytoincreasingthenumberofvariablesin

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problem( 5.6 )-( 5.8 ).Thisprovidesaconvenient\restart"scheme(startfromanoptimalsolutiontothepreviousproblem)whenadditionalcliqueconstraintsaregenerated. Tosolveproblem( 5.6 )-( 5.8 )weusedavariationofinteriorpointmethodproposedbyDikin[ 75 76 ].WewillcallthisversionofinteriorpointmethodDikin'sInteriorPointMethod,orDIPM.WepresentacomputationalschemeofDIPMforanLPprobleminthefollowingform: miny2Rm+nm+nXi=1ciyi;(5.9) s.t.Ay=e; HereAis(m+n)nmatrixinwhichrstmcolumnsaredeterminedbycoecientsaijandcolumnsam+i=eifori=1;:::;n,whereeiistheithorth.Vectorc2Rm+nhastherstmcomponentsequaltooneandtherestncomponentsequaltozero;e2Rnisthevectorofallones.Problem( 5.6 )-( 5.8 )canbereducedtothisformifinequalityconstraintsin( 5.7 )arereplacedbyequalityconstraints.AstheinitialpointfortheDIPMmethodwechoosey0,suchthaty0i=8><>:2;fori=1;:::;m;2mPj=1aij1;fori=m+1;:::;m+n: 5.9 )-( 5.11 ).IntheDIPMmethodthenextpointyk+1isobtainedbythefollowingscheme:

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Asthestoppingcriterionweusedtheconditionm+nXj=1cjykjm+nXj=1cjyk+1j<";where"=103: 77 ]forsolvingthefollowingsystemoflinearequations:AD2kATuk=AD2kc;uk2Rn: Thevaluesofvectoruk=(AD2kAT)1AD2kcfoundfromthelastsystemdenedualvariablesinproblem( 5.6 )-( 5.8 )(Lagrangemultipliersforconstraints( 5.7 )).Theoptimalvaluesofdualvariableswerethenusedasweightcoecientsforndingadditionalcliqueconstraints,whichhelptoreducethelinearcliqueestimateOC(G).Theproblemofndingweightedcliqueswassolvedusinganapproximationalgorithm;themaximumof1000cliqueswereaddedtotheconstraints. 4. Branching: Basedonthefactthatthenumberofverticesfromacliquethatcanbeincludedinanindependentsetisalwaysequalto0or1. (b) Bounding: WeusetheapproximatesolutionfoundasalowerboundandthelinearcliqueestimateOC(G)asanupperbound.

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Table5{2: Exactalgorithm:computationalresults. 1tc1281544.00022555.00023555.00024544.0001 1tc2561655.000221099.25013191313.750141099.50035655.0003 1tc51211077.00032181414.92213292223.68364292223.68115181414.923261077.0002 1dc5121755051.3167 2dc512116910.9674 1et1281323.00042645.00033977.00024977.00025645.00036323.0004 1et2561323.00022866.00063141012.00014221214.40025141012.00046866.00057323.0002 1et5121333.000021078.25023271818.00064292123.06265292123.10296271818.000971078.25018333.0000 5{2 and 5{3 containasummaryofthenumericalexperimentswiththeexactalgorithm.InTable 5{2 column\#"containsanumberassignedtoeachconnectedcomponentofagraphafterthepreprocessing.Columns\1",\2"and

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Table5{3: Exactsolutionsfound. GraphnjEj(G)Time(sec) 1dc51251297275221182dc512512548951126181tc1281285123871tc256256131263391tc51251232641101411et12812867228251et256256166450721et5125124032100143 \3"standfornumberofcliquesinthepartition,solutionfoundbytheapproximationalgorithmandthevalueoftheupperboundOC(G),respectively.InTable 5{3 column\(G)"containstheindependencenumberofthecorrespondinginstancefoundbytheexactalgorithm;column\Time"summarizesthetotaltimeneededtond(G). AmongtheexactsolutionspresentedinTable 5{3 ,onlytwowerepreviouslyknown,for2dc512and1et128.Therestwereeitherunknownorwerenotprovedtobeexact. 5.3 LowerBoundsforCodesCorrectingOneErrorontheZ-Channel Theerror-correctingcodesforZ-channelhaveveryimportantpracticalapplications.TheZ-channelshowninFigure 5{1 isanasymmetricbinarychannel,inwhichtheprobabilityoftransformationof1into0isp,andtheprobabilityoftransformationof0into1is0. Figure5{1: AschemeoftheZ-channel. TheproblemofourinterestistondgoodestimatesforthesizeofthelargestcodescorrectingoneerrorontheZ-channel.

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Letusintroducesomebackgroundinformationrelatedtoasymmetriccodes. TheasymmetricdistancedA(x;y)betweenvectorsx;y2Bnisdenedasfollows[ 177 ]: whereN(x;y)=jfi:(xi=0)^(yi=1)gj.ItisrelatedtotheHammingdistancedH(x;y)=nPi=1jxiyij=N(x;y)+N(y;x)bytheexpression 2dA(x;y)=dH(x;y)+jw(x)w(y)j;(5.13) wherew(x)=nPi=1xi=jfi:xi=1gjistheweightofx.LetusdenetheminimumasymmetricdistanceforacodeCBnas=minfdA(x;y)jx;y2C;x6=yg: 177 ]haveshownthatacodeCwiththeminimumasymmetricdistancecancorrectatmost(1)asymmetricerrors(transitionsof1to0).Inthissubsectionwepresentnewlowerboundsforcodeswiththeminimumasymmetricdistance=2. LetusdenethegraphG=(V(n);E(n)),wherethesetofverticesV(n)=Bnconsistsofallbinaryvectorsoflengthn,and(vi;vj)2E(n)ifandonlyifdA(vi;vj)<.Thentheproblemofndingthesizeofthecodewithminimalasymmetricdistanceisreducedtothemaximumindependentsetprobleminthisgraph.Table 5{4 containsthelowerboundsobtainedusingthealgorithmpresentedaboveinthissection(someofthemwerementionedinTable 5{1 ). 5.3.1 ThePartitioningMethod Thepartitioningmethod[ 47 81 195 ]usesindependentsetpartitionsoftheverticesofgraphGinordertoobtainalowerboundforthecodesize.Anindependentsetpartitionisapartitionofverticesintoindependentsetssuchthateachvertex

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Table5{4: Lowerboundsobtainedby:a-Varshamov[ 196 ];b-ConstantinandRao[ 64 ];c-DelsarteandPiret[ 71 ];d-EtzionandOstergard[ 81 ];e-Butenkoetal.([ 55 56 ],thischapter). nLowerboundUpperbound 44456a6612b12718c18836c36962c6210112d11711198d21012379e(378d)410 belongstoexactlyoneindependentset,thatis Recall,thattheproblemofndingthesmallestmforwhichapartitionoftheverticesintomdisjointindependentsetsexistsisthewellknowngraphcoloringproblem. Theindependentsetpartition( 5.14 )canbeidentiedbythevector(n)=(I1;I2;:::;Im): Intermsofthecodes,theindependentsetpartitionisapartitionofwords(binaryvectors)intoasetofcodes,whereeachcodecorrespondstoanindependentsetinthegraph.

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Similarly,forthesetofallbinaryvectorsofweightwwecanconstructagraphG(n;w),inwhichthesetofverticesisthesetofthenwvectors,andtwoverticesareadjacentitheHammingdistancebetweenthecorrespondingvectorsislessthan4.Thenanindependentsetpartition(n;w)=(Iw1;Iw2;:::;Iwm) canbeconsideredinwhicheachindependentsetwillcorrespondtoasubcodewithminimumHammingdistance4.Theindexvectoranditsnormaredenedinthesamewayasfor(n). Bythedirectproduct(n1)(n2;w)ofapartitionofasymmetriccodes(n1)=(I1;I2;:::;Im1)andapartitionofconstantweightcodes(n2;w)=(Iw1;Iw2;:::;Iwm2)wewillmeanthesetofvectorsC=f(u;v):u2Ii;v2Iwi;1img; 81 ]. InordertondacodeCoflengthnandminimumasymmetricdistance2bythepartitioningmethod,wecanusethefollowingconstructionprocedure. 1. Choosen1andn2suchthatn1+n2=n. 2. Choose=0or1. 3. Set 5.3.2 ThePartitioningAlgorithm Oneofthepopularheuristicapproachestotheindependentsetpartitioning(graphcoloring)problemisthefollowing.SupposethatagraphG=(V;E)isgiven.

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INPUT:G=(V;E);OUTPUT:I1;I2;:::;Im. 0. i=1; 1. 184 185 ]proposedtoimprovethisapproachbyndingateachstepaspeciednumberofmaximalindependentsets.ThenanewgraphGisconstructed,inwhichavertexcorrespondstoamaximalindependentset,andtwoverticesareadjacentithecorrespondingindependentsetshavecommonvertices.InthegraphG,afewmaximalindependentsetsarefound,andthebestofthem(say,theonewiththeleastnumberofadjacentedgesinthecorrespondingindependentsetsofG)ischosen.ThisapproachisformallydescribedinFigure 5{2 INPUT:G=(V;E);OUTPUT:I1;I2;:::;Im. 0. i=0; 1. end FindamaximalindependentsetMIS=fISi1;:::;ISipgofG; Algorithmforndingindependentsetpartitions. 5.3.3 ImprovedLowerBoundsforCodeSizes ThepartitionsobtainedusingthedescribedpartitionalgorithmaregiveninTables 5{5 and 5{6 .Thesepartitions,togetherwiththefactsthat[ 80 ]

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Table5{5: Partitionsofasymmetriccodesfound. 9162,62,62,61,58,56,53,46,29,18,52786811262,62,62,62,58,56,53,43,32,16,62785011362,62,62,61,58,56,52,46,31,17,52784811462,62,62,62,58,56,52,43,33,17,52783211562,62,62,62,58,56,54,42,31,15,82780611662,62,62,60,57,55,52,45,31,18,82779411762,62,62,60,58,55,51,45,37,16,42778811862,62,62,60,58,56,53,45,32,16,62778211962,62,62,62,58,56,52,43,32,17,627778111062,62,62,60,58,56,53,45,31,18,527776111162,62,62,62,58,56,50,45,32,18,527774111262,62,62,61,58,56,51,45,30,22,327772111362,62,62,62,58,56,50,44,34,16,627760111462,62,62,62,58,55,51,44,32,20,42774211 101112,110,110,109,105,100,99,88,75,59,37,16,497942132112,110,110,109,105,101,96,87,77,60,38,15,497850133112,110,110,108,106,99,95,89,76,60,43,15,197842134112,110,110,108,105,100,96,88,74,65,38,17,197828135112,110,110,108,106,103,95,85,76,60,40,15,497720136112,110,110,108,106,101,95,87,75,61,40,17,297678137112,110,109,108,105,101,96,86,78,63,36,17,39767413 (n;0)consistsofone(zero)codeword, (n;1)consistsofncodesofsize1, (n;2)consistsofn1codesofsizen=2forevenn, Indexvectorsof(n;w)and(n;nw)areequal; wereusedin( 5.15 ),with=0,toobtainnewlowerboundsforasymmetriccodespresentedinTable 5{7 .Toillustratehowthelowerboundswerecomputed,letusshowhowthecodeforn=18wasconstructed.Weusen1=8;n2=10.

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Table5{6: Partitionsofconstantweightcodesobtainedin:a-thischapter;b-Brouweretal.[ 47 ];c-EtzionandOstergard[ 81 ]. kw#Partitionindex-vectorNormm Table5{7: Newlowerbounds.Previouslowerboundswerefoundby:a-Etzion[ 80 ];b-EtzionandOstergard[ 81 ]. Lowerboundnnewprevious 181579215762a192947829334b205619656144b21107862107648b22202130201508b24678860678098b Thetotalis2(36+1280+6580)=15792codewords. 5.4 Conclusion Inthischapterwedealwithbinarycodesofgivenlengthcorrectingcertaintypesoferrors.Forsuchcodes,agraphcanbeconstructed,inwhicheachvertexcorrespondstoabinaryvectorandtheedgesarebuiltintheway,thateachindependentsetcorrespondstoacorrectingcode.Theproblemofndingthelargestcodeisthusreducedtothemaximumindependentsetprobleminthecorrespondinggraph.Forasymmetriccodes,wealsoappliedthepartitioningmethod,whichutilizes

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independentsetpartitions(orgraphcolorings)inordertoobtainlowerboundsforthemaximumcodesizes. Weuseecientapproachestothemaximumindependentsetandgraphcoloringproblemstodealwiththeproblemofestimatingthelargestcodesizes.Asaresult,someimprovedlowerboundsandexactsolutionsforthesizeoflargesterror-correctingcodeswereobtained.

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6.1 Introduction Existingroutingprotocolscanbeclassiedintothreecategories:proactive,reactiveandthecombinationofthetwo[ 132 134 175 176 ].Proactiveroutingprotocolsaskeachhosttomaintainglobaltopologyinformation,thusaroutecanbeprovidedimmediatelywhenrequested.Protocolsinthiscategorysuerfromlowerscalabilityandhighprotocoloverhead.Reactiveroutingprotocolshavethefeatureon-demand.Eachhostcomputesrouteforaspecicdestinationonlywhennecessary.Topologychangesthatdonotinuenceactiveroutesdonottriggeranyroutemaintenancefunction.Thus,communicationoverheadislowercomparedtoproactiveroutingprotocol.Thethirdcategorymaintainspartialtopologyinformationinsomehosts.Routingdecisionsaremadeeitherproactivelyorreactively.Oneimportantobservationontheseprotocolsisthatnoneofthemcanavoidtheinvolvementofooding.Forexample,proactiveprotocolsrelyonoodingforthedisseminationoftopologyupdatepackets,andreactiveprotocolsrelyonoodingforroutediscovery. Floodingsuersfromthenotoriousbroadcaststormproblem[ 166 ].Broadcaststormproblemreferstothefactthatoodingmayresultinexcessiveredundancy,contention,andcollision.Thiscauseshighprotocoloverheadandinterferenceto 121

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ongoingtrac.Ontheotherhand,oodingisveryunreliable[ 133 ],whichmeansthatnotallhostsgetallthebroadcastmessageswhenfreefromcollisions.Sinhaet.al.[ 186 ]claimedthat\inmoderatelysparsegraphstheexpectednumberofnodesinthenetworkthatwillreceiveabroadcastmessagewasshowntobeaslowas80%."Inreactiveprotocols,theunreliabilityofoodingmayobstructthedetectionoftheshortestpath,orsimplycannotdetectanypathatall,eventhoughthereexistsapath.Inproactiveprotocols,theunreliabilityofoodingmaycausetheglobaltopologyinformationtobecomeobsolete,thuscausingthenewly-computedpathobsolescence. RecentlyanapproachbasedonoverlayingavirtualinfrastructureonanadhocnetworkwasproposedbySinhaetal.[ 186 ].Routingprotocolsareoperatedoverthisinfrastructure,whichistermedcore.Allcorehostsformadominatingset.Thekeyfeatureinthisapproachisthenewcorebroadcastmechanismwhichusesunicasttoreplacetheoodingmechanismusedbymoston-demandroutingprotocols.Theunicastofrouterequestspacketstoberestrictedtocorenodesanda(small)subsetofnon-corenodes.SimulationresultswhenrunningDSR(DynamicSourceRouting[ 134 ])andAODV(AdhocOn-demandDistanceVectorrouting[ 176 ])overthecoreindicatethatthecorestructureiseectiveinenhancingtheperformanceoftheroutingprotocols.Priortothiswork,inspiredbythephysicalbackboneinawirednetwork,manyresearchersproposedtheconceptofvirtualbackboneforunicast,multicast/broadcastinadhocwirelessnetworks[ 17 67 202 ]. Inthischapter,wewillstudytheproblemofecientlyconstructingavirtualbackboneforadhocwirelessnetworks.Thenumberofhostsformingthevirtualbackbonemustbeassmallaspossibletodecreaseprotocoloverhead.Thealgorithmmustbetime/messageecientduetoresourcescarcity.Weuseaconnecteddominatingset(CDS)toapproximatethevirtualbackbone(seeFigure 6{1 ).Weassumeagivenadhocnetworkinstancecontainsnhosts.Eachhostisintheground

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Figure6{1: Approximatingthevirtualbackbonewithaconnecteddominatingsetinaunit-diskgraph andismountedbyanomni-directionalantenna.Thusthetransmissionrangeofahostisadisk.WefurtherassumethateachtransceiverhasthesamecommunicationrangeR.Thusthefootprintofanadhocwirelessnetworkisaunit-diskgraph.Ingraph-theoreticterminology,thenetworktopologyweareinterestedinisagraphG=(V;E)whereVcontainsallhostsandEisthesetoflinks.AlinkbetweenuandvexistsiftheyareseparatedbythedistanceofatmostR.Inarealworldadhocwirelessnetwork,sometimesevenwhenvislocatedinu'stransmissionrange,visnotreachablefromuduetohidden/exposedterminalproblems.Butinthischapterweonlyconsiderbidirectionallinks.Fromnowon,weusehostandnodeinterchangeablytorepresentawirelessmobile. ThereexistseveraldistributedalgorithmsforMCDScomputationinthecontextofadhocwirelessnetworking[ 14 67 202 ].ThealgorithmofAlzoubietal.[ 14 ]rstbuildsarootedtreeinadistributedfashion.Thenthestatus(insideoroutsideoftheCDS)isassignedforeachhostbasedonthelevelofthehostinthetree.DasandBharghavan[ 67 ]providethedistributedimplementationofthetwocentralizedalgorithmsgivenbyGuhaandKhuller[ 109 ].Bothimplementationssuerfromhighmessagecomplexities.TheonegivenbyWuandLiin[ 202 ]hasnoperformance

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analysis,butitneedsatleasttwo-hopneighborhoodinformation.Thestatusofeachhostisassignedbasedontheconnectivityofitsneighbors.Wewillcompareouralgorithmwiththeotherapproaches[ 14 67 202 ]inSections 6.2 and 6.3 Weadoptthefollowingdenitionsandnotations.Adominatingset(DS)DofGisasubsetofVsuchthatanynodenotinDhasatleastoneneighborinD.IftheinducedsubgraphofDisconnected,thenDisaconnecteddominatingset(CDS).AmongallCDSsofgraphG,theonewithminimumcardinalityiscalledaminimumconnecteddominatingset(MCDS).ComputinganMCDSinaunitgraphisNP-hard[ 63 ].NotethattheproblemofndinganMCDSinagraphisequivalenttotheproblemofndingaspanningtree(ST)withmaximumnumberofleaves.Allnon-leafnodesinthespanningtreeformtheMCDS.Itiseasytoprovethatanymaximalindependentset(MIS)isadominatingset. ForagraphG,ife=(u;v)2Eifflength(e)1,thenGiscalledaunit-diskgraph.Wewillonlyconsiderunit-diskgraphsinthischapter.Fromnowon,whenwesaya\graphG",wemeana\unit-diskgraphG".ThefollowinglemmawasprovedbyAlzoubietal.[ 14 ].ThislemmarelatesthesizeofanyMISofunit-diskgraphGtothesizeofitsoptimalCDS. 14 ]ThesizeofanyMISofGisatmost4opt+1,whereoptisthesizeofanyMCDSofG. Theremainderofthischapterisorganizedasfollows.OuralgorithmanditstheoreticperformanceanalysisarepresentedinSection 6.2 .ResultsofnumericalexperimentsaredemonstratedinSection 6.3 .Section 6.4 concludesthechapter.

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6.2 An8-ApproximateAlgorithmtoComputeCDS Inthissection,weproposeadistributedalgorithmtocomputeCDS.Thisalgorithmcontainstwophases.First,wecomputeamaximalindependentset(MIS);thenweuseatreetoconnectallverticesintheMIS.Wewillshowthatouralgorithmhasperformanceratioatmost8andismessageandtimeecient. 6.2.1 AlgorithmDescription Initiallyeachhostiscoloredwhite.Adominatoriscoloredblack,whileadominateeiscoloredgray.Theeectivedegreeofavertexisthetotalnumberofwhiteneighbors.Weassumethateachvertexknowsitsdistance-oneneighborsandtheireectivedegreesd.Thisinformationcanbecollectedbyperiodicorevent-drivenhellomessages. Wealsodesignateonehostastheleader.Thisisarealisticassumption.Forexample,theleadercanbethecommander'smobileforaplatoonofsoldiersinamission.Ifitisimpossibletodesignateanyleader,adistributedleader-electionalgorithmcanbeappliedtondoutaleader.Thisaddsmessageandtimecomplexity.Thebestleader-electionalgorithm[ 24 ]takestimeO(n)andmessageO(nlogn)andthesearethebest-achievableresults.Assumehostsistheleader.

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NUMOFBLACKNEIGHBORSwhenitdetectsthatnoneofitsneighborsiswhite.Phase1terminateswhennowhitevertexleft. 22 ]tocomputethetree. Ablackvertexwithoutanydominatorisactive.Initiallynoblackvertexhasadominatorandallhostsareunexplored.MessageMcontainsaeldnextwhichspeciesthenexthosttobeexplored.Agrayvertexwithatleastoneactiveblackneighboriseective.IfMisbuiltbyablackvertex,itsnexteldcontainstheidoftheunexploredgrayneighborwhichconnectstomaximumnumberofactiveblackhosts.IfMisbuiltbyagrayvertex,itsnexteldcontainstheidofanyunexploredblackneighbor.AnyblackhostureceivingaMmessagethersttimefromagrayhostvsetsitsdominatortovbybroadcastingmessagePARENT.WhenagrayhostureceivesmessageMfromvthatspeciesutobeexplorednext,uthencolorsitselfblack,setsitsdominatortovandbroadcastsitsownMmessage.AnygrayvertexreceivingmessagePARENTfromablackneighborwillbroadcastmessageNUMOFBLACKNEIGHBORS,whichcontainsthenumberofactiveblackneighbors.Ablackvertexbecomesinactiveafteritsdominatorisset.Agrayvertexbecomesineectiveifnoneofitsblackneighborsisactive.Agrayvertexwithoutactiveblackneighbor,orablackvertexwithouteectivegrayneighbor,willsendmessageDONEtothehostwhichactivatesitsexplorationortoitsdominator.WhensgetsmessageDONEandithasnoeectivegrayneighbors,thealgorithmterminates.

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Notethatphase1setsthedominatorsforallgrayvertices.Phase2maymodifythedominatorofsomegrayvertex.Themainjobforphase2istosetadominatorforeachblackvertex.AllblackverticesformaCDS. InPhase1,eachhostbroadcastseachofthemessagesDOMINATORandDOMINATEEatmostonce.ThemessagecomplexityisdominatedbymessageDEGREE,sinceitmaybebroadcastedtimesbyahost,whereisthemaximumdegree.ThusthemessagecomplexityofPhase1isO(n).ThetimecomplexityofPhase1isO(n). Inphase2,verticesareexploredonebyone.ThetotalnumberofverticesexploredisthesizeoftheoutputCDS.ThusthetimecomplexityisatmostO(n).ThemessagecomplexityisdominatedbymessageNUMOFBLACKNEIGHBORS,whichisbroadcastedatmost5timesbyeachgrayvertexbecauseagrayvertexhasatmost5blackneighborsinaunit-diskgraph.ThusthemessagecomplexityisalsoO(n). Fromtheaboveanalysis,wehave 6.2.2 PerformanceAnalysis Inthissubsection,westudytheperformanceofouralgorithm. Proof.

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iscoloredblack,allofitsneighborsarecoloredgray.Onceanodeiscoloredgray,itremainsincolorgrayduringPhase1. FromtheproofofLemma 6.2 ,itisclearthatif(id)insteadof(d;id)isused,westillgetanMIS.Intuitively,thiswouldyieldanMISofalargersize. Proof. Proof. Proof. 6.3 ,atleastonegrayvertexwithmaximumblackneighborswillbecoloredblackinphase2.Denotethisvertexbyu.Ifuiscoloredblack,thenallofitsblackneighborswillchooseuasitsdominator.Thus,theselectionofucausesmorethan1blackhoststobeconnected. Proof. 6.2 ,phase1computesanMIS.Wewillconsidertwocaseshere. Ifthereexistsagrayvertexwhichhasatleast3blackneighborsafterphase1,fromLemma 6.1 ,thesizeoftheMISisatmost4opt+1.Fromlemma 6.5 ,weknow

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thetotalnumberofblackverticesafterphase2isatmost4opt+1+((4opt+1)2)=8opt. Ifthemaximumnumberofblackneighborsagrayvertexhasis2,thenthesizeoftheMIScomputedinphase1isatmost2optsinceanyvertexinMCDSconnectstoatmost2verticesintheMIS.ThusfromLemma 6.4 ,thetotalnumberofblackhostswillbe2opt+2opt1<4opt. NotethatfromtheproofofTheorem 6.2 ,if(id)insteadof(d;id)isusedinphase1,ouralgorithmstillhasperformanceratioatmost8. Wecomparethetheoreticalperformanceofouralgorithmwithotheralgorithmsproposedintheliterature[ 14 67 202 ]inTable 6{1 .Theparametersusedforcomparisonincludethe(upperboundofthe)cardinalityofthegeneratedCDS(CDSC),themessageandtimecomplexities(MCandTC,respectively),themessagelength(ML)andneighborhoodinformation(NI). Table6{1: PerformancecomparisonofthealgorithmsofDasandBharghavan[ 67 ],WuandLi[ 202 ],andAlzoubietal.[ 14 ]andtheoneproposedinthischapter.HereoptisthesizeoftheMCDS;isthemaximumdegree;jCjisthesizeofthegeneratedconnecteddominatingset;misthenumberofedges;nisthenumberofhosts. 67 ]-I[ 67 ]-II[ 202 ][ 14 ]A 6{1 correspondstoouralgorithm.ItcanbeseenthatouralgorithmissuperiortothetwoalgorithmsofDasandBharghavan[ 67 ]withrespecttoallparameters.AlgorithmofWuandLi[ 202 ]takeslesstimethanouralgorithmbutithasmuchhighermessagecomplexityanditusesmorecomplicatedmessageinformation.ThealgorithmofAlzoubietal.[ 14 ]iscomparablewithouralgorithmsinmanyparameters.Butthesimulation

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resultsinSection 6.3 showthatouralgorithmcomputessmalleronaverageconnecteddominatingsetsforbothrandomandunformgraphs. 6.3 NumericalExperiments Table 6{1 intheprevioussectioncomparesouralgorithmswithotherstheoretically.Inthissection,wewillcomparethesizeoftheCDSscomputedbydierentalgorithms.Asmentionedearlier,thevirtualbackboneismainlyusedtodisseminatecontrolpackets.Thusthemostimportantparameteristhenumberofhostsinthevirtualbackboneafteritisconstructed.Thelargerthesizeofavirtualbackbone,thelargernumberoftransmissionstobroadcastamessagetothewholenetworkisneeded.NotethatthemessagecomplexitiesofthealgorithmsofDasandBharghavan[ 67 ]andWuandLi[ 202 ]aretoohighcomparedtootheralgorithmsandtheyneed2-hopneighborhoodinformation.Thuswewillnotconsidertheminthesimulationstudy.WewillcompareouralgorithmwiththeonegivenbyAlzoubietal.[ 14 ]. Wewillconsidertwokindsoftopologies:randomanduniform.WeassumethereareNhostsdistributedrandomlyoruniformlyina100100squareunits.TransmissionrangeRischosentobe15,25or50units.ForeachvalueofR,werunouralgorithms100timesfordierentvaluesofN.TheaveragedresultsarereportedinFigures 6{2 6{3 and 6{4 forrandomgraphs,andinFigures 6{5 6{6 and 6{7 foruniformgraphs.Fromtheseguresitisclearthatinallofoursimulationscenarios,ouralgorithmperformsbetterthanthealgorithmofAlzoubietal.[ 14 ]. 6.4 Conclusion Inthischapterweprovideadistributedalgorithmwhichcomputesaconnecteddominatingsetofasmallsize.Ouralgorithmhasperformanceratioatmost8.ThisalgorithmtakestimeO(n)andmessageO(n).Ourfutureworkistoinvestigatetheperformanceofvirtualbackboneroutingandtostudytheproblemofmaintainingtheconnecteddominatingsetinamobilityenvironment.

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Figure6{2: AveragedresultsforR=15inrandomgraphs. Figure6{3: AveragedresultsforR=25inrandomgraphs.

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Figure6{4: AveragedresultsforR=50inrandomgraphs. Figure6{5: AveragedresultsforR=15inuniformgraphs.

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Figure6{6: AveragedresultsforR=25inuniformgraphs. Figure6{7: AveragedresultsforR=50inuniformgraphs.

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Wepresentednewecientapproachestosolvingthemaximumindependentsetandrelatedcombinatorialoptimizationproblems.InChapter 2 weprovedseveralcontinuousoptimizationformulations.TheseformulationsandsomeclassicalresultsconcerningtheproblemofoptimizingaquadraticoveraspherewereusedtodevelopseveralheuristicsforthemaximumindependentsetprobleminChapter 3 .Inthefollowingsectionwewillshowhowtheseapproachescanbeextendedtoanotherimportantcombinatorialoptimizationproblem:theMAX-CUTproblem.Then,inSection 7.2 wepresentanewideaofusingtheso-calledcriticalsetsforthemaximumindependentsetproblemsolving.Finally,inSection 7.3 wediscussapplicationsconsideredinthisdissertationandthedirectionsoffutureworkconcerningtheapplications. 7.1 ExtensionstotheMAX-CUTProblem Givenanundirectedgraphwithedgeweights,theMAX-CUTproblemistondapartitionofthesetofverticesintotwosubsets,suchthatthesumoftheweightsoftheedgeshavingendpointsindierentsubsetsismaximized.Thiswell-knownNP-hardproblemhasapplicationsinVLSIdesign,statisticalphysics,andseveralotherelds[ 51 73 87 111 ].Similarlytothemaximumindependentsetproblem,theMAX-CUTproblemcanbeformulatedintermsofoptimizationofaquadraticovertheunithypercube.

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Proof. 7.1 )byf(x)=xTW(ex): 7.1 ),i.e.,maxx2[0;1]nxTW(ex)=maxx2f0;1gnxTW(ex): Alternatively,weconsideranypartitionofvertexsetintotwononoverlappingsets,V=V1[V2;V1\V2=;;

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Therefore,thereisaone-to-onecorrespondencebetweenbinaryvectorsoflengthnandcutsinG.So,maxx2[0;1]nxTW(ex)=maxx2f0;1gnxTW(ex)=maxV=V1SV2;V1TV2=;X(i;j)2V1V2wij 3 .Thus,itwouldbeinterestingtotrysimilarapproachesfortheMAX-CUTproblem.Infutureresearch,whenusingsimilartechniquesforcombinatorialoptimizationproblems,oneshouldtrytoaddressthefollowingissues: Toanswerthesequestionsboththeoreticalandempiricalstudiesarerequired. 7.2 CriticalSetsandtheMaximumIndependentSetProblem Inthissectionwediscussapossibilityofusingpropertiesofcriticalindependentsetsinstudyingthemaximumindependentsetproblem. LetG=(V;E)beasimpleundirectedgraphwiththevertexsetV=f1;2;:::;ngandthesetofedgesE.ForasetIV,letN(I)denotethesetofallverticesofG,whichareadjacenttoatleastonevertexofI.AnindependentsetIc(possiblyempty)iscalledcriticalifjIcjjN(Ic)j=maxfjIjjN(I)j:IisanindependentsetofGg: 9 204 ].Therefore,aninterestingquestiontoinvestigateis,howusefultheknowledgeaboutthecriticalindependent

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setsofagraphcanbeforsolvingthemaximumindependentsetprobleminthisgraph.Wewereabletoprovesomerelatedresults. 7.2.1 Results Proof. Proof. Proof.

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wehavejIcjjN(Ic)j=jIcnJj+jIc\JjjN(Ic)j<
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dierentinstruments,insteadofconsideringthereturns.Itwouldbeinterestingtostudythepropertiesofthisgraphandcompareitwiththemarketgraphconsideredinthiswork.Wealsoplantoinvestigateotherextensionsofthisnovelmethodology. InChapter 5 ,newexactvaluesandestimatesofsizeofthelargesterrorcorrectingcodeswerecomputedusingoptimizationinspeciallyconstructedgraphs.Errorcorrectingcodeslieintheheartofdigitaltechnology,makingcellphones,compactdiskplayersandmodemspossible.Theyarealsoofaspecialsignicanceduetoincreasingimportanceofreliabilityissuesininternettransmissions.Therearestillmanylarge-scaleproblemsinthisareathatneedtobesolved.Exactsolutionsandestimatesforthesizeoflargesterror-correctingcodesoflargerlengthcanbecomputedbydesigningandapplyingevenmoreecient,possiblyparallel,algorithmsforsolvingmaximumindependentsetproblemtooptimality. InChapter 6 ,ecientapproximatealgorithmsforconstructionofavirtualbackboneinadhocwirelessnetworksbymeansofsolvingtheminimumconnecteddominatingsetprobleminunit-diskgraphsweredevelopedandtested.Despiteexcellentperformanceoftheproposedapproach,oneofitsdisadvantagesisthatafeasiblesolutionisconstructedonlywhenthealgorithmterminates.Thisdrawbackcanbeovercomebydevelopingnewalgorithmswhichwouldmaintainafeasiblesolutionatanystageoftheirexecution.Anotherimportantproblemtoinvestigateismaintainingtheconnecteddominatingsetinamobilityenvironment. Infuturework,wewouldalsoliketoextendourworktoapplicationsinbiomedicalengineering.SomeofsuchpossibilitieswerebrieydiscussedinSection 1.6

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SergiyButenkowasbornonJune19,1977,inCrimea,Ukraine.In1994,hecompletedhishighschooleducationintheUkrainianPhysical-MathematicalLyceumofKievUniversityinKiev,Ukraine.Hereceivedhisbachelor'sandmaster'sdegreesinmathematicsfromKievUniversityinKiev,Ukraine,in1998and1999,respectively.InAugust1999,hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.HeearnedhisPh.D.inAugust2003. 155


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Permanent Link: http://ufdc.ufl.edu/UFE0001011/00001

Material Information

Title: Maximum independent set and related problems, with applications
Physical Description: Mixed Material
Creator: Butenko, Sergiy ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0001011:00001

Permanent Link: http://ufdc.ufl.edu/UFE0001011/00001

Material Information

Title: Maximum independent set and related problems, with applications
Physical Description: Mixed Material
Creator: Butenko, Sergiy ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0001011:00001


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MAXIMUM INDEPENDENT SET AND RELATED PROBLEMS, WITH
APPLICATIONS















By

SERGIY BUTENKO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003

















Dedicated to my family















ACKNOWLEDGMENTS

First of all, I would like to thank Panos Pardalos, who has been a great supervisor

and mentor throughout my four years at the University of Florida. His inspiring

enthusiasm and energy have been contagious, and his advice and constant support

have been extremely helpful.

Special thanks go to Stan Uryasev, who recruited me to the University of Florida

and was ahv--li- very supportive. I would also like to acknowledge my committee

members William Hager, Edwin Romeijn, and Max Shen for their time and guidance.

I am grateful to my collaborators James Abello, Vladimir Boginski, Stas Buii--in,

Xiuzhen C'!. i:- Ding-Zhu Du, Alexander Golodnikov, Carlos Oliveira, Mauricio G.

C. Resende, Ivan V. Sergienko, Vladimir Shylo, Oleg Prokopyev, Petro Stetsyuk, and

Vitally Yatsenko, who have been a pleasure to work with.

I would like to thank Donald Hearn for facilitating my graduate studies by

providing financial as well as moral support. I am also grateful to the faculty, staff,

and students of the Industrial and Systems Engineering Department at the University

of Florida for helping make my experience here unforgettable.

Finally, my utmost appreciation goes to my family members, and especially my

wife Joanna, whose love, understanding and faith in me made it a key ingredient of

this work.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iii

LIST OF TABLES ...................... ......... vii

LIST OF FIGURES ...................... ........ ix

ABSTRACT ....................... ........... xi

CHAPTER

1 INTRODUCTION ........................... 1

1.1 Definitions and Notations ........... ............ 4
1.2 Complexity Results .......................... 7
1.3 Lower and Upper Bounds ........... ............ 7
1.4 Exact Algorithm s ........................... 9
1.5 Heuristics ................... ....... 11
1.5.1 Simulated Annealing ......... ........ ... 11
1.5.2 Neural Networks .................. .. .. 13
1.5.3 Genetic Algorithms .................. ..... 14
1.5.4 Greedy Randomized Adaptive Search Procedures ..... 14
1.5.5 Tabu Search ................. ....... .. 15
1.5.6 Heuristics Based on Continuous Optimization . ... 15
1.6 Applications ................ ........... .. 16
1.6.1 Matching Molecular Structures by Clique Detection . 16
1.6.2 Macromolecular Docking .................. .. 17
1.6.3 Integration of Genome Mapping Data . . 17
1.6.4 Comparative Modeling of Protein Structure ........ 17
1.6.5 Covering Location Using Clique Partition . ... 18
1.7 Organization of the Dissertation .................. .. 19

2 MATHEMATICAL PROGRAMMING FORMULATIONS ........ 21

2.1 Integer Programming Formulations ................ .. 21
2.2 Continuous Formulations .................. .. 22
2.3 Polynomial Formulations Over the Unit Hypercube . ... 26
2.3.1 Degree (A + 1) Polynomial Formulation . .... 26
2.3.2 Quadratic Polynomial Formulation . . 29
2.3.3 Relation Between (P2) and Motzkin-Straus QP ...... ..31
2.4 A Generalization for Dominating Sets . . ...... 33









3 HEURISTICS FOR THE MAXIMUM INDEPENDENT SET PROBLEM 37

3.1 Heuristic Based on Formulation (P1) ...... ........ 37
3.2 Heuristic Based on Formulation (P2) ...... ........ 39
3.3 Examples ................... ....... 40
3.3.1 Example 1 ................... .... 40
3.3.2 Example 2 .................. ........ .. 41
3.3.3 Example 3 .................. ........ .. 42
3.3.4 Computational Experiments ..... . . . 43
3.4 Heuristic Based on Optimization of a Quadratic Over a Sphere .46
3.4.1 Optimization of a Quadratic Function Over a Sphere .. 49
3.4.2 The Heuristic .................. ..... .. 52
3.4.3 Computational Experiments ................ .. 54
3.5 Concluding Remarks .................. ..... .. 57

4 APPLICATIONS IN MASSIVE DATA SETS .............. .. 59

4.1 Modeling and Optimization in Massive Graphs . .... 59
4.1.1 Examples of Massive Graphs ................ .. 60
4.1.2 External Memory Algorithms ................ .. 68
4.1.3 Modeling Massive Graphs .................. .. 69
4.1.4 Optimization in Random Massive Graphs . ... 79
4.1.5 Rem arks .................. ......... .. 82
4.2 The M market Graph .................. .... 83
4.2.1 Constructing the Market Graph .... . . 83
4.2.2 Connectivity of the Market Graph . . 85
4.2.3 Degree Distributions in the Market Graph . ... 87
4.2.4 Cliques and Independent Sets in the Market Graph . 90
4.2.5 Instruments Corresponding to High-Degree Vertices . 93
4.3 Evolution of the Market Graph .............. .. .. 94
4.4 Conclusion ............... ......... .. 100

5 APPLICATIONS IN CODING THEORY ...... . . 103

5.1 Introduction ....... ... ... ... ............ 103
5.2 Finding Lower Bounds and Exact Sizes of the Largest Codes 105
5.2.1 Finding the Largest Correcting Codes . . ... 107
5.3 Lower Bounds for Codes Correcting One Error on the Z-C('! I 113
5.3.1 The Partitioning Method .................. .. 114
5.3.2 The Partitioning Algorithm ..... . . ..... 116
5.3.3 Improved Lower Bounds for Code Sizes . . ... 117
5.4 Conclusion .................. ............ .. 119

6 APPLICATIONS IN WIRELESS NETWORKS . . 121

6.1 Introduction .............. .. .......... .. 121
6.2 An 8-Approximate Algorithm to Compute CDS . .... 125









6.2.1 Algorithm Description ............... . 125
6.2.2 Performance Analysis .................. .. 127
6.3 Numerical Experiments .................. .. .. .. 130
6.4 Conclusion .................. ............ 130

7 CONCLUSIONS AND FUTURE RESEARCH . . ..... 134

7.1 Extensions to the MAX-CUT Problem . . . 134
7.2 Critical Sets and the Maximum Independent Set Problem ..... 136
7.2.1 R results . . . . . . .. .. 137
7.3 Applications .................. ........... 138

REFERENCES .................. ................ .. 140

BIOGRAPHICAL SKETCH .................. ......... 155















LIST OF TABLES
Table page

3-1 Results on benchmark instances: Algorithm 1, random x. ....... 44

3-2 Results on benchmark instances: Algorithm 2, random x. ....... ..45

3-3 Results on benchmark instances: Algorithm 3, random . . 46

3-4 Results on benchmark instances: Algorithms 1-3, x = 0, for i = 1,... n. 47

3-5 Results on benchmark instances: comparison with other continuous
based approaches. x 0, = 1,..., n. .. ..... ........... 47

3-6 Results on benchmark instances, part I. ............... 55

3-7 Results on benchmark instances, part II. ............... 56

3-8 Comparison of the results on benchmark instances. .......... ..57

4-1 C('l-I, ing coefficients of the market graph (* complementary graph) 90

4-2 Sizes of cliques found using the greedy algorithm and sizes of graphs
remaining after applying the preprocessing technique . ... 91

4-3 Sizes of the maximum cliques in the market graph with different values
of the correlation threshold .................. .. 92

4-4 Sizes of independent sets found using the greedy algorithm . 93

4-5 Top 25 instruments with highest degrees (0 = 0.6). ........ ..95

4-6 Dates corresponding to each 500-diw shift. ............. ..97

4-7 Number of vertices and number of edges in the market graph (0 = 0.5)
for different periods. .................. .... 99

4-8 Vertices with the highest degrees in the market graph for different
periods (0 = 0.5). ............... ..... .... 101

5-1 Lower bounds obtained. ............... .... .. 107

5-2 Exact algorithm: computational results. .... . .... 112

5-3 Exact solutions found. ............... ....... 113

5-4 Lower bounds ................ ........... .. 115









5-5 Partitions of .,-mmetric codes found. ................ 118

5-6 Partitions of constant weight codes. ................. 119

5-7 New lower bounds. .................. .......... .. 119

6-1 Performance comparison of the algorithms . . ..... 129















LIST OF FIGURES
Figure page

3-1 Illustration to Example 1 .................. ..... .. 41

3-2 Illustration to Example 2 .................. ... .. 41

3-3 Illustration to Example 3 .................. ... .. 42

4-1 Frequencies of clique sizes in the call graph found by Abello et al. .61

4-2 Number of vertices with various out-degrees (a) and in-degrees (b);
the number of connected components of various sizes (c) in the call
graph, due to Aiello et al. .................. .... 62

4-3 Number of Internet hosts for the period 01/1991-01/2002. Data by
Internet Software Consortium. .................. .... 63

4-4 A sample of paths of the physical network of Internet cables created
by W C('!i -.-. 1: and H. Burch ................ .. .. 64

4-5 Number of vertices with various out-degrees (left) and distribution of
sizes of strongly connected components (right) in Web graph . 65

4-6 Connectivity of the Web due to Broder et al ............... ..67

4-7 Distribution of correlation coefficients in the stock market ...... ..84

4-8 Edge density of the market graph for different values of the correlation
threshold. . . . . . .. . . 85

4-9 Plot of the size of the largest connected component in the market graph
as a function of correlation threshold 0. .............. ..86

4-10 Degree distribution of the market graph for (a) 0 = 0.2; (b) 0 = 0.3;
(c) 0 = 0.4; (d) 0 = 0.5 ............. .... .. ... 88

4-11 Degree distribution of the complementary market graph for (a) 0
-0.15; (b) 0 = -0.2; (c) 0 = -0.25 ................. 89

4-12 Time shifts used for studying the evolution of the market graph structure. 96

4-13 Distribution of the correlation coefficients between all considered pairs
of stocks in the market, for odd-numbered time shifts. . 97









4-14 Degree distribution of the market graph for periods (from left to right,
from top to bottom) 1, 4, 7, and 11 (logarithmic scale) .. .....

4-15 Growth dynamics of the edge density of the market graph over time..

5-1 A scheme of the Z-channel . ...................

5-2 Algorithm for finding independent set partitions .. .........

6-1 Approximating the virtual backbone with a connected dominating set
in a unit-disk graph . . . . . . .

6-2 Averaged results for R = 15 in random graphs .. ..........

6-3 Averaged results for R = 25 in random graphs .. ..........

6-4 Averaged results for R = 50 in random graphs .. ..........

6-5 Averaged results for R = 15 in uniform graphs .. ..........

6-6 Averaged results for R = 25 in uniform graphs .. ..........

6-7 Averaged results for R = 50 in uniform graphs .. ..........















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MAXIMUM INDEPENDENT SET AND RELATED PROBLEMS, WITH
APPLICATIONS

By

Sergiy Butenko

August 2003

C'!I ir: Panagote M. Pardalos
Major Department: Industrial and Systems Engineering

This dissertation develops novel approaches to solving computationally difficult

combinatorial optimization problems on graphs, namely maximum independent set,

maximum clique, graph (.1.ii:ir. minimum dominating sets and related problems.

The application areas of the considered problems include information retrieval,

classification theory, economics, scheduling, experimental design, and computer vision

among many others.

The maximum independent set and related problems are formulated as nonlinear

programs, and new methods for finding good quality approximate solutions in

reasonable computational times are introduced. All algorithms are implemented and

successfully tested on a number of examples from diverse application areas. The

proposed methods favorably compare with other competing approaches.

A large part of this dissertation is devoted to detailed studies of selected

applications of the problems of interest. Novel techniques for analyzing the structure

of financial markets based on their network representation are proposed and verified

using massive data sets generated by the U.S. stock markets. The network









representation of a market is based on cross-correlations of price fluctuations of the

financial instruments, and provides a valuable tool for classifying the instruments.

In another application, new exact values and estimates of size of the largest error

correcting codes are computed using optimization in specially constructed graphs.

Error correcting codes lie in the heart of digital technology, making cell phones,

compact disk pl i, rs and modems possible. They are also of a special significance

due to increasing importance of reliability issues in Internet transmissions.

Finally, efficient approximate algorithms for construction of virtual backbone

in wireless networks by means of solving the minimum connected dominating set

problem in unit-disk graphs are developed and tested.















CHAPTER 1
INTRODUCTION

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 [88, 169]. At the same time,

one of the most striking trends in optimization is the constantly increasing emphasis

on the interdisciplinary nature of the field. Optimization tod 'c is a basic research

tool in all areas of engineering, medicine and the sciences. The decision making tools

based on optimization procedures are successfully applied in a wide range of practical

problems arising in virtually any sphere of human activities, including biomedicine,

energy management, aerospace research, telecommunications and finance.

Having been applied in many diverse areas, the problems studied in this thesis

provide a perfect illustration of the interdisciplinary developments in optimization.

For instance, the maximum independent set problem put forward in the title of this

thesis has applications in numerous fields, including information retrieval, signal

transmission analysis, classification theory, economics, scheduling, and biomedical

engineering. New findings concerning some of these applications will be discussed in

a latter part of this thesis. But before we will present a formal description of results

obtained in this work, let us gently touch the historical grounds of optimization theory

and mathematical programming.

The problems of finding the "b. -I and the -. I-1I have .1.-- l,- been of great

interest. For example, given n sites, what is the fastest way to visit all of them

consecutively? In rn iu ,11 iuiii.r how should one cut plates of a material so that the

waste is minimized? Some of the first optimization problems were solved in ancient









Greece and are regarded among the most significant discoveries of that time. In

the first century A.D., the Alexandrian mathematician Heron solved the problem of

finding the shortest path between two points by way of the mirror. This result, also

known as the Heron's theorem of the light ray, can be viewed as the origin of the

theory of geometrical optics. The problem of finding extreme values gained a special

importance in the seventeenth century when it served as one of motivations in the

invention of differential calculus. The soon-after-developed calculus of variations and

the theory of stationary values lie in the foundation of the modern mathematical

theory of optimization.

The invention of the digital computer served as a powerful spur to the field

of numerical optimization. During World War II optimization algorithms were

used to solve military logistics and operations problems. The military applications

motivated the development of linear programming (LP), which studies optimization

problems with linear objective function and constraints. In 1947 George Dantzig

invented the simplex method for solving linear programs arising in U.S. Air Force

operations. Linear programming has become one of the most popular and well studied

optimization topics ever since.

Despite a fine performance of the simplex method on a wide variety of

practical instances, it has an exponential worst-case time complexity and therefore

is unacceptably slow in some large-scale cases. The question of existence

of a theoretically efficient algorithm for LP remained open until 1979, when

Leonid Khachian published his polynomial time ellipsoid algorithm for linear

programming [138]. This theoretical breakthrough was followed by the interior

point algorithm of Narendra Karmarkar [137] published in 1984. Not only does this

algorithm have a polynomial time theoretical complexity, it is also extremely efficient

practically, allowing for solving larger instances of linear programs. Nowad-, .









various versions of interior point methods are an integral part of the state-of-the-

art optimization solvers.

Linear programming can be considered as a special case of a broad optimization

area called combinatorial optimization, in which a the feasible region is a finite, but

usually very large, set. All of the problems studied in this work are essentially

combinatorial optimization problems. In nonlinear optimization, one deals with

optimizing a nonlinear function over a feasible domain described by a set of, in

general, nonlinear functions. It is fascinating to observe how naturally nonlinear

and combinatorial optimization are bridged with each other to yield new, better

optimization techniques. Combining the techniques for solving combinatorial

problems with nonlinear optimization approaches is especially promising since it

provides an alternative point of view and leads to new characterizations of the

considered problems. These ideas also give a fresh insight into the complexity issues

and frequently provide a guide to the discovery of nontrivial connections between

problems of seemingly different nature. For example, the ellipsoid and interior

point methods for linear programming mentioned above are based on nonlinear

optimization techniques. Let us also mention that an integrality constraint of the

form x E {0, 1} is equivalent to the nonconvex quadratic constraint x2 x = 0. This

straightforward fact -i-.-.- -1; that it is the presence of nonconvexity, not integrality

that makes an optimization problem difficult [123]. The remarkable association

between combinatorial and nonlinear optimization can also be observed throughout

this thesis. In C'! lpter 2 we prove several continuous formulations of the maximum

independent set problem. Consequently, in C!I Ipter 3 we derive efficient algorithms

for finding large independent sets based on these formulations.

As a result of ongoing enhancement of the optimization methodology and of

improvement of available computational facilities, the scale of the problems solvable

to optimality is continuously rising. However, many large-scale optimization problems









encountered in practice cannot be solved using traditional optimization techniques.

A variety of new computational approaches, called heuristics, have been proposed for

finding good sub-optimal solutions to difficult optimization problems. Etymologically,

the word "heuristic" comes from the Greek heuriskein (to find). Recall the famous

"Eureka! Eureka!" (I have found it! I have found it!) by Archimedes (287-212

B.C.). A heuristic in optimization is any method that finds an .. pltable" feasible

solution. Many classical heuristics are based on local search procedures, which

iteratively move to a better solution (if such solution exists) in a neighborhood of

the current solution. A procedure of this type usually terminates when the first local

optimum is obtained. Randomization and restarting approaches used to overcome

poor quality local solutions are often ineffective. More general strategies known as

metaheuristics usually combine some heuristic approaches and direct them towards

solutions of better quality than those found by local search heuristics. Heuristics and

metaheuristics phi. a key role in the solution of large difficult applied optimization

problems. New efficient heuristics for the maximum independent set and related

problems have been developed and successfully tested in this work.

In the remainder of this chapter we will first formally introduce definitions and

notations used in this thesis. Then we will state some fundamental facts concerning

the considered problems. Finally, in Section 1.7 we will outline the organization of

the remaining chapters of this thesis.

1.1 Definitions and Notations

Let G = (V,E) be a simple undirected graph with vertex set V = {1,..., n}

and set of edges E. The complement p lj''l of G is the graph G = (V, E), where E is

the complement of E. For a subset W C V let G(W) denote the subgraph induced

by W on G. N(i) will denote the set of neighbors of vertex i and di = N(i)| is the

degree of vertex i. We denote by A A(G) the maximum degree of G.









A subset I C V is called an independent set (stable set, vertex packing) if the edge

set of the subgraph induced by I is empty. An independent set is maximal if it is not a

subset of any larger independent set, and maximum if there are no larger independent

sets in the graph. The independence number a(G) (also called the ,Ibili, number)

is the cardinality of a maximum independent set in G.

Along with the maximum independent set problem, we consider some other

related problems for graphs, including the maximum clique, the minimum vertex

cover, and the maximum matching problems. A clique C is a subset of V such that

the subgraph G(C) induced by C on G is complete. The maximum clique problem is

to find a clique of maximum cardinality. The clique number u(G) is the cardinality

of a maximum clique in G. A vertex cover V' is a subset of V, such that every edge

(i,j) E E has at least one endpoint in V'. The minimum vertex cover problem is to

find a vertex cover of minimum cardinality. Two edges in a graph are incident if they

have an endpoint in common. A set of edges is independent if no two of them are

incident. A set M of independent edges in a graph G = (V, E) is called a matching.

The maximum matching problem is to find a matching of maximum cardinality.

It is easy to see that I is a maximum independent set of G if and only if I is a

maximum clique of G and if and only if V \ I is a minimum vertex cover of G. The

last fact yields Gallai's identity [92]


a(G)+ S =IV(G)|, (1.1)

where S is a minimum vertex cover of the graph G. Due to the close relation between

the maximum independent set and maximum clique problems, we will operate with

both problems while describing the properties and algorithms for the maximum

independent set problem. In this case, it is clear that a result holding for the maximum

clique problem in G will also be true for the maximum independent set problem in

G.









Sometimes, each vertex i e V is associated with a positive weight i' Then for

a subset S C V its weight W(S) is defined as the sum of weights of all vertices in S:


W(S) K"
iES

The maximum weight independent set problem seeks for independent sets of maximum

weight. The maximum weight clique problem is defined likewise.

The following definitions generalize the concept of clique. Namely, in some

applications, instead of cliques one is interested in dense subgraphs, or quasi-cliques.

A y-clique C,, also called a quasi-clique, is a subset of V such that G(C,) has at least

[Yq(q 1)/2] edges, where q is the cardinality of C,.
A 7 .iii (proper) coloring of G is an assignment of colors to its vertices so that no

pair of .Ii di:ent vertices has the same color. A coloring induces naturally a partition

of the vertex set such that the elements of each set in the partition are pairwise

1 n, i i :ent. These sets are precisely the subsets of vertices being assigned the same

color, which are independent sets by definition. If there exists a coloring of G that

uses no more than k colors, we -iv that G admits a k-coloring (G is k-colorable).

The minimal k for which G admits a k-coloring is called the chromatic number and is

denoted by X(G). The graph coloring problem is to find X(G) as well as the partition

of vertices induced by a X(G)-coloring.

Minimum clique partition problem, which is to partition vertices into minimum

number of cliques, is analogous to graph coloring problem. In fact, any proper coloring

in G is a clique partition in G.

For other standard definitions which are used in this paper the reader is referred

to a standard textbook in graph theory [31].

Konig's theorem (see page 30 in Diestel [74] and Rizzi [178] for a short proof)

states that the maximum cardinality of a matching in a bipartite graph G is equal to

the minimum cardinality of a vertex cover.









1.2 Complexity Results

The maximum clique and the graph coloring problems are NP-hard [96];

moreover, they are associated with a series of recent results about hardness of

approximations. The discovery of a remarkable connection between probabilistically

checkable proofs and approximability of combinatorial optimization problems [20, 21,

83] yielded new hardness of approximation results for many problems. Arora and

Safra [21] proved that for some positive c the approximation of the maximum clique

within a factor of n" is NP-hard. Recently, Hastad [116] has shown that in fact for any

6 > 0 the maximum clique is hard to approximate in polynomial time within a factor

n-6. Similar approximation complexity results hold for the graph coloring problem

as well. Garey and Johnson [95] have shown that obtaining colorings using sx(G)

colors, where s < 2, is NP-hard. It has been shown by Lund and Yannakakis [152]

that x(G) is hard to approximate within n" for some c > 0, and following Hastad [116],

Feige and Kilian [84] have shown that for any 6 > 0 the chromatic number is hard to

approximate within a factor of n1 -, unless NP C ZPP. All of the above facts together

with practical evidence [136] i-r-. -1 that the maximum clique and coloring problems

are hard to solve even in graphs of moderate sizes. The theoretical complexity and

the huge sizes of data make these problems especially hard in massive graphs.

Alternatively, the maximum matching problem can be solved in polynomial time

even for the weighted case (see, for instance, Papadimitriou and Steiglitz [168]).

1.3 Lower and Upper Bounds

In this section we briefly review some well-known lower and upper bounds on the

independence and clique numbers.

Perhaps the best known lower bound based on degrees of vertices is given by

Caro and Tuza [57], and Wei [200]:

1
ia( V di 1
iEV









In 1967, Wilf [201] showed that


w(G) < (AG) + 1,


where p(Ac) is the spectral radius of the .,i1] iency matrix of G (which is, by

definition, the largest eigenvalue of AG).

Denote by N-1 the number of eigenvalues of AG that do not exceed -1, and by

No the number of zero eigenvalues. Amin and Hakimi [16] proved that


w(G)

In the above, the equality holds if G is a complete multipartite graph.

One of the most remarkable methods for computing an upper bound on

independence number is based on semidefinite programming. In 1979, the Lov&sz

theta number 0(G) was introduced, which gives an upper bound on the stability

number a(G) [149]. This number is defined as the optimal value of the following

semidefinite program:
max eTXe

s.t. trace(X) 1,

Xi 0 if (i,j) E,

X > 0,

where X is a symmetric matrix of size n x n, the constraint X >- 0 requires X to

be positive semidefinite, and e is the unit vector of length n. The famous sandwich

theorem [142] states that the Lovasz number 0(G) of the complement of a graph is

sandwiched between the graph's clique number w(G) and the chromatic number X(G):


w(G) < 0(G) < X(G).









Since the above semidefinite program can be solved in polynomial time, the sandwich

theorem also shows that in a perfect graph G (such that w(G) = X(G) [154]), the

clique number can be computed in polynomial time.

1.4 Exact Algorithms

In 1957, Harary and Ross [114] published, allegedly, the first algorithm for

enumerating all cliques in a graph. Interestingly, their work was motivated by an

application in sociometry. The idea of their method is to reduce the problem on

general graphs to a special case with graphs having at most three cliques, and

then solve the problem for this special case. This work was followed by many

other algorithms. Paull and Unger [174], and Marcus [158] proposed algorithms

for minimizing the number of states in sequential switching functions. The work of

Bonner [41] was motivated by clustering problem. Bednarek and Taulbee [29] were

looking for all maximal chains in a set with a given binary relation. Although all

these approaches were designed to solve problems arising in different applications,

their idea is essentially to enumerate all cliques in some graph.

The development of computer technology in 1960's made it possible to test the

algorithms on graphs of larger sizes. As a result, in the early 1970's, many new

enumerative algorithms were proposed and tested. Perhaps the most notable of these

algorithms was the backtracking method by Bron and Kerbosch [45]. The advantages

of their approach include its polynomial storage requirement and exclusion of the

possibility of generating the same clique twice. The algorithm was successfully tested

on graphs with 10 to 50 vertices and with edge density in the range between 1(' and

95'. A modification of this approach considered by Tomita et al. [192] was claimed

to have the time complexity of O(3"/3), which is the best possible for enumerative

algorithms due to existence of graphs with 3"/3 maximal cliques [163]. Recently,

the Bron-Kerbosch algorithm was shown to be quite efficient with graphs arising in

matching 3-dimensional molecular structures [93].









All of the algorithms mentioned above in this section were designed for finding

all maximal cliques in a graph. However, to solve the maximum clique problem, one

needs to find only the clique number and the corresponding maximum clique. There

are many exact algorithms for the maximum clique and related problems available in

the literature. Most of them are variations of the branch and bound method, which

can be defined by different techniques for determining lower and upper bounds and

by proper branching strategies. Tarjan and Trojanowski [191] proposed a recursive

algorithm for the maximum independent set problem with the time complexity of

O(2't/). Later, this result was improved by Robson [179], who modified the algorithm

of Tarjan and Trojanowski to obtain the time complexity of 0(20.276n).

Another important algorithm was developed by Balas and Yu in 1986 [25].

Using an interesting new implementation of the implicit enumeration, they were

able to compute maximum cliques in graphs with up to 400 vertices and 30,000

edges. Carraghan and Pardalos [58] proposed yet another implicit enumerative

algorithm for the maximum clique problem based on examining vertices in the order

corresponding to the nondecreasing order of their degrees. Despite its simplicity, the

approach proved to be very efficient, especially for sparse graphs. A publicly available

implementation of this algorithm currently serves as a benchmark for comparing

different algorithms [136]. Recently, Ostergard [167] proposed a branch-and-bound

algorithm which analyzes vertices in order defined by their coloring and employs a

new pruning strategy. He compared the performance of this algorithm with several

other approaches on random graphs and DIMACS benchmark instances and claimed

that his algorithm is superior in many cases.

As most combinatorial optimization problems, the maximum independent set

problem can be formulated as an integer program. Several such formulations will be

mentioned in C'! lpter 2. The most powerful integer programming solvers used by

modern optimization packages such as CPLEX of ILOG [125] and Xpress of Dash









Optimization [68] usually combine branch-and-bound algorithms with cutting plane

methods, efficient preprocessing schemes, including fast heuristics, and sophisticated

decomposition techniques in order to find an exact solution.

1.5 Heuristics

Although exact approaches provide an optimal solution, they become impractical

(too slow) even on graphs with several hundreds of vertices. Therefore, when one deals

with maximum independent set problem on very large graphs, the exact approaches

cannot be applied, and heuristics provide the only available option.

Perhaps the simplest heuristics for the maximum independent set problem

are sequential greedy heuristics which repeatedly add a vertex to an intermediate

independent set ( "Best in") or remove a vertex from a set of vertices which is

not an independent set ( Worst ou, ), based on some criterion, -i- the degrees of

vertices [135, 144].

Sometimes in a search for efficient heuristics people turn to nature, which seems

to ah--,v- find the best solutions. In the recent decades, new types of optimization

algorithms have been developed and successfully tested, which essentially attempt

to imitate certain natural processes. The natural phenomena observed in annealing

processes, nervous systems and natural evolution were adopted by optimizers and

led to design of the simulated annealing [139], neural networks [121] and evolutionary

computation [119] methods in the area of optimization. Other popular metaheuristics

include greedy randomized adaptive search procedures or GRASP [86] and tabu

search [101]. Various versions of these heuristics have been successfully applied

to many important combinatorial optimization problems, including the maximum

independent set problem. Below we will briefly review some of such heuristics.

1.5.1 Simulated Annealing

The annealing process is used to obtain a pure lattice structure in physics. It

consists of first melting a solid by heating it up, and then solidifying it by slowly









cooling it down to a low-energy state. As a result of this process, the system's free

energy is minimized. This property of the annealing process is used for optimization

purposes in the simulated annealing method, where each feasible solution embodies

a state of the hypothetical physical system, and the objective function represents

the energy corresponding to the state. The basic idea of simulated annealing is the

following. Generate an initial feasible solution x(O). At iteration k+ 1, given a feasible

solution x(k), accept its neighbor x(k+l) as the next feasible solution with probability

f 1, if f(xk+l) < f(xk);
Pk+l f(+l)-f()
Pkexp f -f(k) otherwise,
t

where f(x) is the cost function, and t is a parameter representing the temperature,

which is modified as the optimization procedure is executed. A proper choice of

the cooling schedule describing the change in the temperature parameter is one of

the most important parts of the algorithm. A logarithmically slow cooling schedule

leads to the optimal solution in exponential time; however in practice, faster cooling

schedules yielding acceptable solutions are used.

An example of simulated annealing for the maximum independent set problem,

using a penalty function approach was described in the textbook by Aarts and

Korst [1]. They used the set of all possible subsets of vertices as the solution space,

and the cost function in the form f(V') = IV'I AIE'I, where V' C V is the given

subset of vertices, E' C V' x V' is the set of edges in G(V'). Homer and Peinado

[120] implemented a variation of the algorithm of Aarts and Korst with a simple

cooling schedule, and compared its performance with the performance of several other

heuristics for the maximum clique problem. Having run experiments on graphs with

up to 70,000 vertices, they concluded that the simulated annealing approach was

superior to the competing heuristics on the considered instances.









1.5.2 Neural Networks

Artificial neural networks (or simply, neural networks) represent an attempt

to imitate some of the useful properties of biological nervous systems, such as

adaptive biological learning. A neural network consists of a large number of

highly interconnected processing elements emulating neurons, which are tied together

with weighted connections analogous to synapses. Just like biological nervous

systems, neural networks are characterized by massive parallelism and a high

interconnectedness.

Although neural networks had been introduced as early as in the late 1950's,

they were not widely applied until the mid-1980's, when sufficiently advanced related

methodology has been developed. Hopfield and Tank [122] proved that certain neural

network models can be used to approximately solve some difficult combinatorial

optimization problems. N .--, .I vs, neural networks are applied to many complex

real-world problems. More detail on neural networks in optimization can be found,

for example, in Zhang [205].

Various versions of neural networks have been applied to the maximum

independent set problem by many researchers since the late 1980's. The efficiency

of early attempts is difficult to evaluate due to the lack of experimental results.

Therefore, we will mention some of the more recent contributions. Grossman [107]

considered a discrete version of the Hopfield model for the maximum clique. The

results of computational experiments with this approach on DIMACS benchmarks

were satisfactory but inferior to other competing methods, such as simulated

annealing. Jagota [127] proposed several different discrete and continuous versions

of the Hopfield model for the maximum clique problem. Later, Jagota et al. [128]

improved some of these algorithms to obtain considerably larger cliques than those

found by simpler heuristics, which ran only slightly faster.









1.5.3 Genetic Algorithms

Genetic algorithms in optimization were motivated by evolution processes in

natural systems. Optimization in a genetic algorithm is carried out on a population

of points, which are also called individuals or chromosomes. In the simplest version,

the chromosomes are represented by binary vectors. Each chromosome is associated

with a fitness value, which is the probability that the individual defined by this

chromosome in the next generation will survive to the adulthood. Individual fitness

values are used to compute values of the objective function representing the total

fitness of the population. The fundamental law of natural selection stating that

the total fitness of the population is non-decreasing from generation to generation,

can be used as a basis for an optimization procedure. In its simplest version, the

genetic algorithm starts with a population chosen randomly, and computes a new

population using one of the three basic mechanisms: reproduction, crossover, or

mutation [104]. Reproduction operator chooses the chromosomes used in the next

generation according to the probability given by their fitness. The crossover operator

is applied to produce new children from pairs of individuals. Finally, the mutation

operator reverses the value of each bit in a chromosome with a given probability.

Early attempts to apply genetic algorithms to the maximum independent set and

maximum clique problems were made in the beginning of 1990's. T ii successful

implementations have appeared in the literature ever since [50, 118, 157]. Most of the

genetic algorithms can be easily parallelized.

1.5.4 Greedy Randomized Adaptive Search Procedures

A greedy randomized adaptive search procedure (GRASP) is an iterative

randomized sampling technique in which each iteration provides an heuristic solution

to the problem at hand [86]. The best solution over all GRASP iterations is kept as the

final result. There are two phases within each GRASP iteration: the first constructs

a list of solutions called restricted candidate list (RCL) via an adaptive randomized









greedy function; the second applies a local search technique to the constructed solution

in hope of finding an improvement. GRASP has been applied successfully to a

v ii, I of combinatorial optimization problems including the maximum independent

set problem [85].

A recent addition to GRASP, the so-called path r. 1V. ..:,~1 [102] has been proposed

in order to enhance the performance of the heuristic by linking good quality solutions

with a path of intermediate feasible points. It appears that in many cases some of

these feasible points provide a better quality than the solutions used as the endpoints

in the path. GRASP procedure with path relinking can be applied for the maximum

independent set and the graph coloring problems.

1.5.5 Tabu Search

Tabu search [99, 100] is a variation of local search algorithms, which uses a tabu

strategy in order to avoid repetitions while searching the space of feasible solutions.

Such a strategy is implemented by maintaining a set of tabu lists, containing the

paths of feasible solutions previously examined by the algorithm. The next visited

solution is defined as the best 7.'/ (not prohibited by the tabu strategy) solution in

the neighborhood of the current solution, even if it is worse than the current solution.

Various versions of tabu search have been successfully applied to the maximum

independent set and maximum clique problems [28, 90, 155, 189].

1.5.6 Heuristics Based on Continuous Optimization

As it was discussed above, continuous approaches to combinatorial optimization

problems turn out to be especially attractive. Much of the recent efforts in developing

continuous-based heuristics for the maximum independent set and maximum clique

problems focused around the Motzkin-Straus [165] formulation which relates the

clique number of a graph to some quadratic program. This formulation will be proved

and discussed in more detail in C'!I Ilter 2 of this work. We will also prove some other









continuous formulations of the considered problems and develop and test efficient

heuristics based on these formulations.

Recently, we witnessed foundation and rapid development of semidefinite

programming techniques, which are essentially continuous. The remarkable result

by Goemans and Williamson [103] served as a 1i ii ,r step forward in development

of approximation algorithms and proved a special importance of semidefinite

programming for combinatorial optimization. Burer et al. [52] considered rank-one

and rank-two relaxations of the Lovdsz semidefinite program [108, 149] and derived

two continuous optimization formulations for the maximum independent set problem.

Based on these formulations, they developed and tested new heuristics for finding

large independent sets.

1.6 Applications

Practical applications of the considered optimization problems are abundant.

They appear in information retrieval, signal transmission analysis, classification

theory, economics, scheduling, experimental design, and computer vision [4, 23, 25,

30, 38, 72, 171, 193]. In this section, we will briefly describe selected applications. Our

work concerning applications in massive graphs, coding theory and wireless networks

will be presented in C'! lpters 4-6.

1.6.1 Matching Molecular Structures by Clique Detection

Two graphs G1 and G2 are called isomorphic if there exists a one-to-one

correspondence between their vertices, such that .,.li i:ent pairs of vertices in Gi are

mapped to .,.li i:ent pairs of vertices in G2. A common subgraph of two graphs G1

and G2 consists of subgraphs C' and G' of GC and G2, respectively, such that C' is

isomorphic to G2. The largest such common subgraph is the maximum common

subgraph (\!CS). For a pair of graphs, Gi = (Vi,Ei) and G2 = (V2,E2), their

correspondence graph C has all possible pairs (vl, v2), where, E V, i = 1,2, as









its set of vertices; two vertices (vl, v2) and (v', v2) are connected in C if the values of

the edges from v, to v' in G1 and from v2 to v' in G2 are the same.

For a pair of three-dimensional chemical molecules the MCS is defined as the

largest set of atoms that have matching distances between atoms (given user-defined

tolerance values). It can be shown that the problem of finding the MCS can be solved

efficiently using clique-detection algorithms applied to the correspondence graph [93].

1.6.2 Macromolecular Docking

Given two proteins, the protein docking problem is to find whether they interact

to form a stable complex, and, if they do, then how. This problem is fundamental

to all aspects of biological function, and development of reliable theoretical protein

docking techniques is an important goal. One of the approaches to the macromolecular

docking problem consists in representing each of two proteins as a set of potential

hydrogen bond donors and acceptors and using a clique-detection algorithm to find

maximally complementary sets of donor/acceptor pairs [94].

1.6.3 Integration of Genome Mapping Data

Due to differences in the methods used for analyzing overlap and probe data,

the integration of these data becomes an important problem. It appears that overlap

data can be effectively converted to probe-like data elements by finding maximal

sets of mutually overlapping clones [115]. Each set determines a site in the genome

corresponding to the region which is common among the clones of the set; therefore

these sets are called virtual probes. Finding the virtual probes is equivalent to finding

the maximal cliques of a graph.

1.6.4 Comparative Modeling of Protein Structure

The rapidly growing number of known protein structures requires the

construction of accurate comparative models. This can be done using a clique-

detection approach as follows. We construct a graph in which vertices correspond

to each possible conformation of a residue in an amino acid sequence, and edges









connect pairs of residue conformations (vertices) that are consistent with each other

(i.e., clash-free and satisfying geometrical constraints). Based on the interaction

between the atoms corresponding to the two vertices, weights are assigned to the

edges. Then the cliques with the largest weights in the constructed graph represent

the optimal combination of the various main-chain and side-chain possibilities, taking

the respective environments into account [180].

1.6.5 Covering Location Using Clique Partition

Covering problems arise in location theory. Given a set of demand points and a set

of potential sites for locating facilities, a demand point located within a prespecified

distance from a facility is said to be covered by that facility. Covering problems

are usually classified into two main classes [69]: mandatory coverage problems aim to

cover all demand points with the minimum number of facilities; and maximal coverage

problems intend to cover a maximum number of demand points with a given number

of facilities.

Location problems are also classified by the nature of sets of demand points and

potential sites, each of which can be discrete or continuous. In most applications both

sets are discrete, but in some cases at least one of them is continuous. For example,

Brotcorne et al. [46] consider an application of mandatory coverage problem arising

in cytological screening tests for cervical cancer, in which the set of demand points

is discrete and the set of potential sites is continuous. In this application, a servical

specimen on a glass slide has to be viewed by a screener, which is relocated on the

glass slide in order to explore the entire specimen. The goal is to maximize efficiency

by minimizing the number of viewing locations. The area covered by the screener can

be square or circular. To be specific, consider the case of the square screener, which

can move in any of four directions parallel to the sides of the rectangular glass slide.

In this case, we need to cover the viewing area of the slide by squares called tiles. To

locate a tile it is suffices to specify the position of its center on the tile.









Let us formulate the problem in mathematical terms. Given n demand points

inside a rectangle, we want to cover these points with a minimum number of squares

with side s (tiles).

Lemma 1.1. The following two statements are equivalent:

1. There exists a covering of n demand points in the r. .in.glll. using k tiles.

2. Given n tiles centered in the demand points, there exist k points in the r.. i,^,l,

such that each of the tiles contains at least one of them.

Let us introduce an undirected graph G = (V, E) associated with this problem.

The set of vertices V = {1, 2,..., n} corresponds to the set of demand points. The

set of edges E is constructed as follows. Consider the set T = {ti, t2,..., tn} of tiles,

each centered in a demand point. Then two vertices i and j are connected by an edge

if and only if ti n tj / 0. Then, in order to cover the demand points with minimum

number of tiles, or the same, minimize the number of viewing locations, it suffices

to solve minimum clique partition problem in the constructed graph (or the graph

coloring problem in its complement).

1.7 Organization of the Dissertation

Organizationally, this thesis is divided into seven chapters. ('!i Ipter 2

introduces mathematical programming formulations of the maximum independent

set and related problems. In particular, we prove several continuous optimization

formulations. These formulations and some classical results concerning the problem

of optimizing a quadratic over a sphere are utilized in ('!i lpter 3, where we propose

several heuristics for the maximum independent set problem. To demonstrate their

efficiency, we provide the results of numerical experiments with our approaches, as

well as comparison with other heuristics.

('!C lpter 4 discusses applications of the considered graph problems in studying

massive data sets. First, we review recent advances and challenges in modeling and

optimization of massive graphs. Then we introduce the notion of the so-called market









graph, in which financial instruments are represented by vertices, and edges are

constructed based on the values of pairwise cross-correlations of the price fluctuations

computed over a certain time period. We analyze structural properties of this graph

and -ii-:.- -1 some practical implications of the obtained results.

New exact solutions and improved lower bounds for the problem of finding the

largest correcting codes dealing with certain types of errors are reported in ('! Ilpter 5.

These results were obtained using efficient techniques for solving the maximum

independent set and graph coloring problems in specially constructed graphs. Yet

another application of the considered graph optimization problems is presented in

('!i lpter 6, in which we solve the problem of constructing a virtual backbone in ad

hoc wireless networks by approximating it with the minimum connected dominating

set problem in unit-disk graphs. The proposed distributed algorithm has a fixed

performance ratio and consists of two stages. Namely, it first computes a maximal

independent set and then connects it using some additional vertices. The results of

numerical simulation are also included. Finally, we conclude with some remarks and

ideas for the future research in ('! Ilpter 7.















CHAPTER 2
MATHEMATICAL PROGRAMMING FORMULATIONS

The maximum independent set problem has many equivalent formulations

as an integer programming problem and as a continuous nonconvex optimization

problem [38, 171]. In this chapter we will review some existing approaches and will

present proofs for some of the formulations. We will start with mentioning some

integer programming formulations in the next section. In Sections 2.2 and 2.3, we will

consider several continuous formulations of the maximum independent set problem

and will give proofs for some of them. This will follow by a generalization of one

of the formulations to dominating sets in Section 2.4. Most results presented in this

chapter previously appeared in Abello et al. [3].

2.1 Integer Programming Formulations

Given a vector w E IR of positive weights i, (associated with each vertex i, i

1,..., n), the maximum weight independent set problem asks for independent sets of

maximum weight. Obviously, it is a generalization of the maximum independent set

problem. One of the simplest formulations of the maximum weight independent set

problem is the following edge formulation:


max f(x) = (2.1)
i=1

subject to


Xi + Xj < 1,V(i,j) E, (2.1a)

x {0,1},i 1,...,n. (2.1b)









An alternative formulation of this problem is the following clique formulation

[108]:


max f(x) = (2.2)
i=1

subject to


x i < 1, VS E C {maximal cliques of G}, (2.2a)
iES

x e {0,1},i 1l,...,n. (2.2b)


The advantage of formulation (2.2) over (2.1) is a smaller gap between the optimal

values of (2.2) and its linear relaxation. However, since there is an exponential number

of constraints in (2.2a), finding an efficient solution of (2.2) is difficult.

Here we mention one more integer programming formulation. Let AG be the

.,1i ,i:ency matrix of a graph G, and let J denote the n x n identity matrix. The

maximum independent set problem is equivalent to the global quadratic zero-one

problem


max f(x) = xTAx, (2.3)


subject to


xi e {0,1},i= 1,...,n, (2.3a)


where A = J AG. If x* is a solution to (2.3), then the set I defined by I = {i

V : x = 1} is a maximum independent set of G with I| = f(x*). See Pardalos and

Rodgers [170] for details.

2.2 Continuous Formulations

Shor [183] considered an interesting formulation of the maximum weight

independent set problem by noticing that formulation (2.1) is equivalent to the









quadratically constrained global optimization problem


max f(x) = (2.4)
i=1

subject to


xixj = 0, V (i,j) e E, (2.4a)

Xi2 xi = 0, i 1,2,..., n. (2.4b)


Applying dual quadratic estimates, Shor reported good computational results and

presented a new way to compute the Lovdsz number of a graph [149].

Motzkin and Straus [165] established a remarkable connection between the

maximum clique problem and a certain standard quadratic programming problem.

The original proof of the Motzkin-Straus theorem was by induction. Below we present

a new proof. Let A0 be the .,1i ,i:ency matrix of G and let e be the n-dimensional

vector with all components equal to 1.

Theorem 2.1 (Motzkin-Straus). The jl,1l'.1l optimal value of the following

quadratic '*, ".iiii

1
max f(x) = xTAGx, (2.5)
2

subject to


eTx 1, (2.5a)

x > 0. (2.5b)


is given by


where (G is the clique number of .
2 l w(G)
where w(G) is the clique number of G.










Proof. We will use the following well known inequality for the proof:


n ni
a> > (2.6)
i= 1

where al, a2,... a are positive numbers. Equality takes place if and only if a1 =

a2 = ...= a.

Now consider the program (2.5). Let J denote the n x n identity matrix and let

O be the n x n matrix of all ones. Then


AG O J AG


and

yTAcy = yTOy yTjy yTAcy = 1 (yTJy + yTAcy),

where AG is the .,-i i,:ency matrix of the complement graph G.

Let R(x) = xJx + xTAGX. Program (2.7) is equivalent to (2.5).


min R(x) = xTJx + xTAGx, (2.7)


s.t. eTx = 1,

x > 0.

To check that there alv--, exists an optimal solution x* of (2.7) such that

*TAX* = 0, consider any optimal solution x of (2.7). Assume that xTAcx > 0.

Then there exists a pair (i,j) E E such that xij > 0. Consider the following

representation for R(x):

R(x) = Ri (x) + Rj (x),

where

Rij(x) x + x + 2xiX + 2xe k + 2xj xk;
(i,k)EE,k j (j,k)EE,k i

RW(x) = R(x) RW(x).









Without loss of generality, assume that Xk << Z Xk. Then, if we set
(i,k)EE,k i (j,k)EE,k7j

i + j, if k = i,

Xk = 0, if k = j,

Xk, otherwise,

we have:

R(x) = Rij(x) + R(x) =

(xi + j)2 + 2(xi + Xj). _k<
(i,k)EE,k i

x]+ _+x2xiXj + ik + 2X Xk = ).
(i,k)EE,k7j (j,k)EE,k i
If we denote by Z(x) = {(i,j) EE : xixj > 0}, then x is an optimal solution of

(2.7) with IZ(x)| < IZ(') Repeating this procedure a finite number of times we will

finally obtain an optimal solution x* for which |Z(x*) = 0 and thus x*TAox* = 0.

Note that x*TAox* 0 if and only if V(i, j) EE : x*x 0. This means that if

we consider the set C = {i : x* > 0} then C is a clique.

Without loss of generality, assume that x* > 0 for i 1, 2,..., m and x* = 0 for

m + 1 < i < n. Consider the objective function of (2.7),
m
R(x*) =X*TJx* xf2
i= 1

By inequality (2.6) and the feasibility of x* for (2.7),

nn f2
r^*2 > i _
m m
i 1

Since C is a clique of cardinality m, it follows m < w(G) and

1 1
R(*) > > --
m (G)









On the other hand, if we consider


1 k C*,
', ,if k e C*

0, otherwise,

where C* is a maximum clique of G, then x* is feasible and R(x*) = w(. Thus z* is

an optimal solution of (2.7). Returning back to the original quadratic program, the

result of the theorem follows. E

This result is extended by Gibbons et al. [98], who provided a characterization

of maximal cliques in terms of local solutions. Moreover, optimality conditions of

the Motzkin-Straus program have been studied and properties of a newly introduced

parametrization of the corresponding QP have been investigated. S6s and Straus [190]

further generalized the same theorem to hypergraphs.

2.3 Polynomial Formulations Over the Unit Hypercube

In this section we consider some of the continuous formulations originally proved

by Harant et al. [112, 113] using probabilistic methods. We prove deterministically

that the independence number of a graph G can be characterized as an optimization

problem based on these formulations. We consider two polynomial formulations, a

degree (A + 1) formulation and a quadratic formulation.

2.3.1 Degree (A + 1) Polynomial Formulation

Consider the degree (A + 1) polynomial of n variables

F(x) (1- xi) IJ xj, x [0,1t].
i 1 (i,j)EE

The following theorem characterizes the independence number of the graph G as

the maximization of F(x) over the n-dimensional hypercube.









Theorem 2.2. Let G = (V, E) be a simple graph on n nodes V = {1,..., n} and set

of edges E, and let a(G) denote the independence number of G. Then


a(G) max F(x)
0<_Xi<_l,i=1,...,n

max Y (1 ) xj, (P1)
O i=1 (i,j)EE

where each variable xi corresponds to node i E V.

Proof. Denote the optimal objective function value by f(G), i.e.,


f(G) = max F(x)=
OXil,i=1,...,n
n
max ( (- x1 ) T xj. (2.8)
O<_xi<_1,i=1,...,n
i=1 (i,j)EE

We want to show that ac(G) = f(G).

First we show that (2.8) ah--,v- has an optimal 0-1 solution. This is so because

F(x) is a continuous function and [0, 1]" {(xl, 2,... x) : 0 < i < ,i ..., n}

is a compact set. Hence, there ah-i-, exists x* E [0, 1]' such that F(x*) =

max F(x).
0,Xi<_l,i=1,...,n
Now, fix any i e V. We can rewrite F(x) in the form


F(x) = (1 xi)Ai(x) + xiBi(x) + C (x), (2.9)


where


Ai x) n xj, (2.10)
(i,j)EE

B,(x) = ( xk) xj, (2.11)
(i,k)EE (k,j)6E,j7i

C =(x) = ( Xk) Xj. (2.12)
(i,k) E (k,j)EE









Expressions (2.10 -2.12) can be interpreted in terms of neighborhoods. Ai(x)

and Bi(x) characterize the first- and the second-order neighborhoods of vertex i,

respectively, and Ci(x) is complementary to Bi(x) with respect to i in the sense that

it describes neighborhoods of all vertices, other than i, which are not characterized

by B,(x).

Notice that xi is absent in (2.10 2.12), and therefore F(x) is linear with respect

to each variable. It is also clear from the above representation that if x* is any optimal

solution of (2.8), then x* = 0 if A((x*) > Bi(x*), and x* = 1, if A,(x*) < Bi(x*).

Finally, if Ai(x*) = Bi(x*), we can set x* = 1. This shows that (2.8) ahv-- has an

optimal 0-1 solution.

To show that f(G) > a(G), assume that a(G) = m and let I be a maximum

independent set. Set

0, if i E I;
; = (2.13)
1, otherwise.

Then, f(G) max F(x) > F(x*) = m -a(G).
O Xi To complete the proof, we need to show that f(G) < a(G). Since the considered

problem ahv--, has an optimal 0-1 solution, it follows that f(G) must be integer.

Assume f(G) = m and take any optimal 0-1 solution of (2.8). Without loss of

generality we can assume that this solution is x* x ... x 0 ; x

x+ ... x* = 1, for some k. Then we have:


(I ,x) x + (1 x) x +... +
(1,j)EE (2,j)EE

(1 ) ]J = m. (2.14)
(k,j)EE







29

Each term in (2.14) is either 0 or 1. Therefore k > m and there exists a subset

I {1,..., k} such that I I = m and


Vi I: n =1.
(i,j)EE

Therefore, if (i,j) E E, then x = 1. Note that since x ... x* = 0,

it follows that V{i,j} C I we have (i,j) E and so I is an independent set by

definition. Thus, a(G) > |I = = f(G), which completes the proof. D

Corollary 2.1. The clique number uw(G) in a simple undirected 'r'l, G = (V, E)


n
w(G) max (1 x) x j.
0<_Xi<_l,i=1,...,n
i=1 (i,j)E,i j
Proof. The statement follows from Theorem 2.2 and the fact that any clique in G is

an independent set for G. D

Corollary 2.2. The size IS| of a minimum vertex cover S in a simple ,.'l'l, G

(V, E) .,/:.7,
n
IS n max (1 x~) J
O i=1 (i,j)EE
Proof. The result follows from Theorem 2.2 and Gallai's identity (1.1). D

Corollary 2.3. The size IMI of a maximum matching M in a bipartite g'jl'l, C

(V, E) -,/-7,

n
IMI = n max Y (1 x) J xj.
O i=1 (i,j)EE

Proof. The statement follows from Corollary 2.2 and Konig's theorem. D

2.3.2 Quadratic Polynomial Formulation

Consider now the quadratic polynomial

n
H(x) Z x, x x'j,
i 1 (i,j)EE










defined for x e [0, 1]".

The following theorem characterizes the independence number of the graph G as

the maximization of H(x) over the n-dimensional hypercube.

Theorem 2.3. Let G = (V, E) be a simple graph on n nodes V = {1,..., n} and set

of edges E, and let a(G) denote the independence number of G. Then


a(G)= max H(x)
0

max x, j xx (P2)
iO< i ( )1...,nE
-i=1 (ij)6E


where each variable xi corresponds to node i E V.

Proof. Denote the optimal objective function value by h(G), i.e.,


h(G) max x y- xx
05m^i1,i=1,...,n'. Y Y
i= 1 (i,j) EE

and let I be a maximum independent set. To prove that h(G) > a(G), let


1, if i I;

0, otherwise.


(2.15)






(2.16)


Since I is an independent set and x* = 0 for i I, then E x*x = 0. Furthermore,
(i,j)EE

Sx =I a(G). This yields h(G) > H(x*) = a(G).
i= 1
To complete the proof, we need to show that h(G) < a(G). Assume h(G) = m.

Since H(x) is linear with respect to each variable, problem (2.15) ahl--b has an

optimal 0-1 solution. Take any optimal 0-1 solution x of (2.15). Suppose, that there
n
exists (io,jo) E E such that xio = Xjo 1. ('!i ii, ii: xio to 0 decreases xi by 1
i= 1
and decreases Y xixj by at least 1. Thus, the objective function will not decrease.
(i,j)EE
Doing this for all such pairs (io,jo) will finally lead to an optimal solution x* such









that V(i,j) EE : x'x* = 0, and an independent set I = {i : x* = 1} of cardinality

h(G). This yields h(G) < a(G) and the theorem is proved. O

2.3.3 Relation Between (P2) and Motzkin-Straus QP

The next statement is a reformulation the Motzkin-Straus theorem for the

maximum independent set problem. We show how it can be obtained from the

formulation (P2).

Theorem 2.4. The 1l.. ll optimal value of the following QP

1
max f(x) = xTAGX, (2.17)
2
(2.18)


s.t. ex = 1,

x > 0.


is given by
1-
2( a (G)'

where a(G) is the independence number of G.

Proof. Consider formulation (P2):


a(G) = max xi xixj max (Tx -xTAG .
O
It is easy to see that changing the feasible region from [0, 1]" in the last QP to the

following

{x > 0 : eT a(G)}

does not change the optimal objective function value, thus, changing the variables to

(y x, we obtain:
Y/ a(G)


a(G)2YTAcy)
G m


a(G) ax (a(Gc T Y


(2.19)









s. t.

eTy = ,

S> 0.

Let I is a maximum independent set of G. Then

y 1, if iEI
f. a(G)'
S 0, otherwise,

is an optimal solution for the last program.

We have:


AG O J


and


yTAGy = yTOy yTjy yTAcy


1 yTJy yTAGy,


where AG is the .,i.i i:ency matrix of the complement graph G. Denote by F

0 : eTy = 1}. Then (2.19) is equivalent to the following:


a(G) -max a(G) + a(G)2(- + yTOy + yT Ay)


which yields

1 = max (yTJy + y AGy)

Since the maximum is reached in y*, we have:


= y*T AUy* < maxyTAGy.
a(G) yE


Now assume that for some y:


yTAGy= max yTAy > 1
yEE7


1
a(G)C


Then there exists y with I{(i,j) E : ygjy > 0}\

y we have:


1
a(G)'


1 yfJy > yTAcy > 1


0 such that yTAcy > TAgy. For









which yields the following:
1
yTjy< 7
a(G)'
Then, from the Lemma:

1 1
< y TJy< .
I{i :y > 0}1 (G)

Note that I {i : yi > 0} is an independent set and the last inequality implies I|| >

a(G), which contradicts the definition of a(G). Thus, max yTAGy 1 .


2.4 A Generalization for Dominating Sets

For a graph G = (V,E) with V {1,..., n}, let 1 (kl,... ,k,) be a vector of

integers such that 1 < ki < di for i E V, where di is the degree of vertex i E V. An

1-dominating set [113] is a set D, C V such that every vertex i E V\DI has at least ki

neighbors in D1. The 1-domination number 7l(G) of G is the cardinality of a smallest

1-dominating set of G.

For k, -. = kn 1, 1-domination corresponds to the usual definition of

domination. The domination number 7(G) of G is the cardinality of a smallest

dominating set of G. If ki = di for i = 1,..., n, then I = V \ Di is an independent

set and 7d(G) = n a(G) with d = (dl,... d).

The following theorem characterizes the domination number.

Theorem 2.5. The domination number can be expressed by

n n
nn(G) = mi f,(x) min Xi + (1 x) x
O i 1 i 1


Si xm (1 m ) (2.20)
SP-0 {i,... ip}CN(i) mE{ii,...,ip} mEN(i)\{i,...,ip}









Proof. Denote the objective function by g(G), i.e.,

n n
g(G) = min x + (1 xi) x
0 i=l i=l


S-1 nn (1 ) (1 (2.21)
P=0 {il,... ipCN(i) mE{ii,...,ip} mEN(i)\{ii,...,ip}

We want to show that 71(G) = g(G).

We first show that (2.21) ahv--, has an optimal 0-1 solution. Since fi(x) is a

continuous function and [0, 1]" {( (X1,2,... xn) : 0 < xi i < ,... ,n} is a

compact set, there ah--,~ exists x* c [0, 1]" such that fi(x*) min f (x).
0 The statement follows from linearity of fi(x) with respect to each variable.

Now we show that g(G) < 71(G). Assume 71(G) = m. Set


x t1, if i DI;

0, otherwise.

Then, g(G) min fi(x) < f,(x') = m= -7(G).
O Finally, we show that g(G) > 71(G). Since (2.21) ahb--i-s has an optimal 0-1

solution, then g(G) must be integer. Assume g(G) = m. Take an optimal 0-1 solution

x* of (2.20), such that the number of l's is maximum among all 0-1 optimal solutions.

Without loss of generality we can assume that this solution is x* x = x = 1;
1 = +2 = = 0, for some r. Let

ki-1
Qi(=) 5 n J Xm nl (1 -Xm);
P=0 {il,...ip}CN(i) mE{ii,...,ip} mEN(i)\{il,...,ip}



Q(x)= ( xi)x
i=r+l


=0 {il,... ip}CN(i) mE{il,...,ip} m N(i)\{il,...,ip}











D, = : x{ = 1}.


We have


r+ Q(x*) m.


From the last expression and nonnegativity of Q(x) it follows that ID1 = r < m.

We want to show that Di is an 1-dominating set. Assume that it is not. Let

S = e V \ DI : IN(i) n Di| < ki}. Then S / 0 and Vi E S : Q(x*) > 1. Note,

that changing x*, iE 5' from 0 to 1 will increase E x* by 1 and will decrease Q(x*)
i=
by at least 1, thus it will not increase the objective function.

Consider D* = Di U S and build x' as follows:


1, ifi E D;
0, otherwise.

Then fi(x') < fi(x*) and I{i : x' = 1}1 > I{i : x* = 1}1, which contradicts the

assumption that x* is an optimal solution with the maximum number of l's. Thus,

Di is an 1-dominating set with cardinality r < m= g(G) and therefore 71(G) < g(G).

This concludes the proof of the theorem. D

Corollary 2.4. For the case in which k = = kn 1, we have


n
7(G) min x' + (1 xi) 7 (1 -
O S i=1 (i,j)EE

Corollary 2.5. For the case ki di, i 1,..., n, the result of


Proof. It can be shown by induction for IN(i)|, that
di-,
E n Y0 H nXm (i- Xm) -
p-0 {i ...ip}cN(i) mE{ii,...,ip} mEN()\{ii ,...,ipl


Xj) .


Ti,.. I. ;,, 2.2 follows.


HI Xj.
jEN(i)











Thus,


n
u(G)= n in i + (1- x)(1
i= 1


(ij)EEI)


max (1
0xmi$1,i=1,...,n
i 1


Xj) f7 xj.
(ij)EE














CHAPTER 3
HEURISTICS FOR THE MAXIMUM INDEPENDENT SET PROBLEM

In this chapter, we present several heuristics for the maximum independent set

problem using continuous formulations. These results have previously appeared in

several publications [3, 54].

3.1 Heuristic Based on Formulation (P1)

In this section, we discuss an algorithm for finding a maximal independent set

based on formulation (P1) presented in Section 2.3. As pointed out before, the

function F(x) is linear with respect to each variable, so Ai(x) and Bi(x) can be

computed for any i E {1, 2,..., n}. To produce a maximal independent set using

F(x), first let x E [0, 1]" be any starting point. The procedure described below

produces a sequence of n points x1, x2,.., xn such that x" corresponds to a maximal

independent set. Let V = {1, 2,..., n} and consider some i E V. From (2.9-2.12) it

follows that if we set
1 0, if Ai(x) > Bi(x);

1, otherwise.

and xJ = x, ifj / i, we obtain for the point x' (x{, x1,..., x') that F(x1) > F(xo).

If we update V = V \ {i}, we can construct the next point x2 from x1 in the

same manner. Running this procedure n times, we obtain a point x" which satisfies

the inequality

F(x) > F(xo).

The following theorem states that x" has an independent set associated with it.

Theorem 3.1. If I = {i {1, 2,... n} : x = 0}, then I is an independent set.









Proof. Consider any (i,j) c E. We need to show, that {i,j} is not a subset of I.

Without loss of generality, assume that we check xi on the k-th iteration of the above

procedure. If x = 1, then i I. Alternatively, if xi 0, i.e., i E I, we need to show

that j I. Let 1 > k be an iteration on which we check xj. Then

Aj(Xl'-1 n[ 0,
(ij)EE

and therefore Aj(x"-1) < Bj(x-'1) and x 1, which implies that j ( I. O

From the discussion above, we have the following algorithm to find a maximal

independent set.

Algorithm 1:

INPUT: x e [0, 1]"

OUTPUT: Maximal independent set I

0. v := x;

1. for i 1,...,n do if A((v) > Bi(v) then : 0 else := 1;

2. for i 1,...,n do if A(v) = 1 then := 0;

3. 1 = { 1, 2,...,n} : 0};

END

Theorem 3.2. Algorithm 1 is correct.

Proof. We have already discussed steps 1 and 3 of the algorithm which guarantee

that an independent set is produced. We need to show that step 2 guarantees that

the independent set is maximal. Indeed, assume that after running step 1 we have,

for some index i, = 1 and


A(v) = Vij)
(ij)SE









This means that neither i nor any node from the neighborhood of i is included in the

independent set that we obtained after step 1. Thus, we can increase the cardinality

of this set including i in it by setting = 0. O

The time complexity of the proposed algorithm is O(A2n), since Ai(v) and Bi(v)

can be calculated in O(A2) time.

3.2 Heuristic Based on Formulation (P2)

We now focus our attention on an algorithm, similar to Algorithm 1, based on

formulation (P2).

Algorithm 2:

INPUT: x e [0, 1]"

OUTPUT: Maximal independent set I

0. v := x;

1. for i= 1,...,n do if E(ij)EvVj < 1 then : 1, else := 0;

2. for i= 1,..., n do if (ij)E vj = 0 then, : 1;

3. I= {ie {1,2,...,n}: 1};

END

Theorem 3.3. Algorithm 2 is correct.

Proof. Algorithm 2 is similar to Algorithm 1. In step 1 it finds an independent set

i {i {1, 2,..., n} : I = 1}. Set li is independent because after step 1 we have

that V(i,j) E E such that i < j, if = 1 then Z(j,k)EEV k > 1 and vj = 0. If I1

is not a maximal independent set, then there exists i such that + (ij))E vJ = 0.

We can increase the cardinality of I, by one, including i in it, by setting = 1 in

step 2. The resulting set is independent, because (i,j)E v =- 0, which requires that

Vj such that (i, j) EE : vj = 0. Thus no neighbors of i are included in the set. O


The time complexity of this algorithm is O(An).









3.3 Examples

The algorithms presented build a maximal independent set from any given point

x E [0, 1]" in polynomial time. The output, however, depends on the choice of

x. An interesting question that arises is how to choose such input point xO, so

that a maximum independent set can be found? The problem of finding such a point

cannot be solved in polynomial time, unless P = NP. However, we can show on simple

examples that in some cases even starting with a "bad" starting point xO (F(x) < 1

or H(x) < 1), we obtain a maximum independent set as the output.

3.3.1 Example 1

Consider the graph in Figure 3-1. For this example

X = (X1, X2, X3, 4) E [0, 1]4;

F(x) = (1 xI)2X3X4 + (1 x )x + (1 x3)x1 + (1 x)71;

Ai(x) = x2X3X4;

B,(x) (1 X2) + (1 X3) + (1 x );

A2(x) A3(x) = A4(x) = 21;

B2(x) =(1 XI)X3X4;

B3(x) =(1 Xi)x2X4;

B4(X) (1 x1)x2 3;

H(x) = x1 + x2 + X3 + x4 x1x2 71X3 x174.

Consider Algorithm 1 with .x = (, ). Since A,(x0) =, Bi(x0) = and

since i < 3, the next point is x1 (1, 1 1).

Next, A2(1) 1, B2(x1) = 0, and x2 = (1,0, ). After two more iterations

we get x4 = (1, 0, 0, 0) with I = {2, 3, 4}, which is the maximum independent set of

the given graph. We have III = 3, F(x0) = and the objective function increase is

III- F(x) = .
16"











2











Figure 3 1: Illustration to Example 1

For Algorithm 2, starting with xO = (1, 1, 1, 1), for which H(xo) = 1, we obtain

the maximum independent set after step 1. Note, that the Caro-Wei bound for this

graph is 7

3.3.2 Example 2








1






1 figure 3 2: i traction n to Example 2


For the graph in Figure 3-2, we have x = (x, x, 2,x3, x4,x5) E [0, 1]5 and


F(x) = (1 x)x )2X3 + (1 2)13X4 + ( -3)X125+


(1 X4)2 + (1 X5)x3.









Applying Algorithm 1 for this graph with initial point xO = (, 1, 1, 1, 1), we obtain, at

the end of step 1, the solution 5 = (1, 1, 1, 0, 0), which corresponds to the independent

set I = {4, 5}. At the end of step 2, the solution is (0, 1, 1, 0, 0), which corresponds to

the maximum independent set I {1, 4, 5}. For this case we have I|| = 3, F(xo) =

and the objective function improvement is ". With the initial point x = (0,0, 0, 0, 0),

H(xo) = 0, and Algorithm 2 finds the maximum independent set after step 1. For

this example, the Caro-Wei bound equals 1.

3.3.3 Example 3

This example shows, that the output of Algorithm 1 and Algorithm 2 depends

not only on initial point xO, but also on the order in which we examine variables

in steps 1 and 2. For example, if we consider the graph from Figure 3-1 with a

different order of nodes (as in Figure 3-3), and run Algorithm 1 and Algorithm 2 for

this graph with initial point xO = (1, 1, 1, 1), we obtain I = {4} as output for both

algorithms. Note that, for the graph from Figure 3-1, both outputs would be the

maximum independent set of the graph.











3

Figure 3-3: Illustration to Example 3


As Example 3 shows, we may be able to improve both algorithms by including

two procedures (one for each step) which, given a set of remaining nodes, choose a

node to be examined next. Consider Algorithm 2. Let index () and index2() be









procedures for determining the order of examining nodes on step 1 and step 2 of the

algorithm, respectively. Then we have the following algorithm.

Algorithm 3:

INPUT: x e [0, 1]"

OUTPUT: Maximal independent set I

0. v := x; V1 : V; V2 : V;

1. while V1 / 0 do

(a) k= indexl(Vi);

(b) if Y vj < 1 then := 1, else := 0;
(k,j)EE
(c) V1 : V1 \{,};
2. while V2 / 0 do

(a) k = index2(V2);

(b) if E vj 0 then := 1;
(k,j)EE
(c) V2 : V2\{,};
3. 1- {ie {1,2,...,n}: 1};

END

In general, procedures indexl) and index2() can be different. We propose the

same procedure index() for index () and index2():


index(Vo) argm-i-, V { vj}
(k,j)EE

breaking ties in favor of the node with the smallest neighborhood in V \ Vo and at

random if any nodes remain tied.

3.3.4 Computational Experiments

This section presents computational results of the algorithms described

in the previous section. We have tested the algorithms on some of

the DIMACS clique instances which can be downloaded from the URL









Table 3 1: Results on benchmark instances: A 1 1i:::: 1, random xz.


MANN_a9

MANN_a45
c-f..: --::" 1
c-fat200-2
c-f ..: I --
hamming6-2
1, I.' ',I I, I
hamming8-2

hamminglO-2
johnson8-2-4
johnson8-4-4
johnsonl6-2-4
johnson32-2-4
keller4
san200_0.9_1
s.- -^ _0.9_2
san200_0.9_3
san,-- _0.9_1


n
45
378
1035
200
200
200
64
64
256
256
1024

70

496
171
200
200
200
400


Density
0.927
0 -i

0.077
0.1 .
0.426

0.349


0 ::
0.556
0.768
0.765
0.879
0.649
0.900
0.900
0.900
0.900


u(G)
16
1 :
345
12

58
32
4
128
16
512
4
14
8
16
11
70

44
100


Sol. Found
15
113

12
24
58
32
4
128
16

4
14
8
S16
7

34
31
53


Average Sol.
13.05
68.16
1t: 76
10.. .
22.22
57.14
30.17
3.72
119.54
10.87
471.17
4.00
13.02

16.00
7.00
39.08
29.41
27.58
16.18


http://dimacs.rutgers.edu/Challenges/. All algorithms are programmed in C

and compiled and executed on an Intel Pentium III 600 Mhz PC under MS Windows

NT.

First, each algorithm was executed 100 times with random initial solutions

uniformly distributed in the unit hypercube. The results of these experiments are

summarized in Tables 3-1, 3-2, and 3-3. The columns X\ ii ," "n," "Density," and

"w(G)" represent the name of the graph, the number of its vertices, its density, and

its clique number, respectively. This information is available from the DIMACS web

site. The column "Sol. Found" contains the size of the largest clique found after 100

runs. The columns "Average Sol." and "Time(sec.)" contain average solution and

average CPU time (in seconds) taken over 100 runs of an algorithm, respectively.


Time(sec.)
0.01
8.14
243.94
33.72
31.43
17 ..
0 3
0.37
6.45

233.45
0.01
0.17
1.13
1; : 77
9.72
0.14
0.14
0.23
4.54










Table 3 2: Results on benchmark instances: A 1 i1:::: 2, random x.


MANN_a9
i i :7
MANN_a45
c-f.:: ::" 1
c-fat200-2
c-f.;,: I -
hamming6-2
,1 ,,1 ,, ,
hamming8-2

hamminglO-2
johnson8-2-4
johnson8-4-4
j ohnsonl6-2-4
johnson32-2-4
keller4
san200_0.9_1
s.- ,-. _0.9_2
san200_0.9_3
san,-- _0.9_1


n
45
378
1035
200
200
200
64
64
256
256
1024

70

496
171
200
200
200
400


Density
0.927
0 -i

0.077
0.1 .
0.426
0 '- I .

0 i3
0 : :
0 : 31

0.556
0.768
0.765
0.879
0.649
0.900
0.900
0.900
0.900


u(G)
16
1':
345
12
24
58
32
4
128
16
512
4
14
8
16
11
70

44
100


Sol. Found
16
1:i

12
24
58
32
4
121
16
14
4
14
8
S16
7
61
32

54


Average Sol.
14.45
119.16

1 : i
22.59
57.85
21.44



9.71
410.92
4.00
9.-

16.00
7.00
38.; :
29.09
"1. )3
44.20


Table 3-4 contains the results of computations for all the algorithms with the

initial solution x, such that x = 0, i = 1,..., n. In this table, "A3" stands for

Algorithm 3. As can be seen from the tables, the best solutions for almost all instances

obtained during the experiments can be found among the results for Algorithm 3 with

x 0, i = 1,..., n (see Table 3-4).

In Table 3-5 we compare these results with results for some other continuous

based heuristics for the maximum clique problem taken from the paper of Bomze et

al. [39]. The columns "ARH", "PRD(1)", "PRD(O)" and "CBH" contain the size of

a clique found using the annealed replication heuristic of Bomze et al. [39], the plain

replicator dynamics applied for two different parameterizations (with parameters 1

and 0) of the Motzkin-Straus formulation [40], and the continuous-based heuristic


Time(sec.)
0.01
1*

0.33
0.32
0
0.01
0.01
1.07
1.43
46.46
0.01
0.01
0.04
4.57
0.13
0.14
0.13
0.18
2.68










Table 3 3: Results on benchmark instances: A i1:::: 3, random xl .


Name
MANN_a9
7
MANN_a45
c-fat'-: --1
c-fat200-2
c-fat'-: --5
hamming6-2

2
. ........ 2
S 8-4
.. 1. 0-2
johnson8-2-4
johnson8-4-4
johnsonl6-2-4
johnson32-2-4
keller4
-_0.9_1
.. _0.9_2
_0.9_3
1 _0.9_1


n Dens.
45 0.927
10 0 '"7

0iA
O.A'
S0.163
: 0 I : .
I 0 ''
64

,- o0 .
1024 ( :
0.556
70 0.7"
0.
: 0.879
171 : :.

0.900

', ', :: '


w(G)
16
1'.
345
12
24
58

4


Sol. Found
16
121

12
24
58

4


16
512
4
14
8
16
11
70

44
100


Average Sol.
14 -:
119.21
331.57
11.64
22.47
57 -
27.49
4.00
1l-, 78
12.49
.53
4.00
11.22
8.00

7.54
45.03
34.94

:: ',1


Time(sec.)
0.01
4.32
87.78
0.47

0.42

0.02

1.13
:.11
0.01
0.02
0.09
10.20
0.28
0.37
0.38
0.37
2.54


of Gibbons et al. [97], respectively. The column "A3(0)" represents the results for

Algorithm 3 with x 0, i 1,..., n.

3.4 Heuristic Based on Optimization of a Quadratic Over a Sphere

Recall the Motzkin-Strauss formulation of the maximum clique problem. Let

AG be the .1li i:ency matrix of G and let e be the n-dimensional vector with all

components equal to 1.

Theorem 3.4 (Motzkin-Straus). The optimal value of the following quadratic

n i, ,, 1 111


1
max f(x) = xTAcx,
2


(3.1)










Table 3 4: Results on benchmark instances: Algorithms 1 3, x


Name

MANN_a9
MANN_a27
MANNa45
c-f z !- -' 1
c-fat200-2
c-fz ~ 55
2

2
!., :. ,. 2

!.; ,.... 2

hamming8-4
hamminglO-2
j, 2-4
johnson8-4-4
johnsonl6-2-4
johnson32-2-4
keller4
s; .'_0.9_1
sa,:: i_0.9_2
s;.:,: _0.9_3
san4i '_ 0.9 1


nr Dens. {u(G)


45 0
0.
1 0 "


0.
0.1.

0.9

64 0. .
0. -
0.
1 4 0 '"
0.556
0 : .
1 .: 0.765
0.879
171 0O ':
0
0
S0
Si 0 "


Al
16 16
i '. 125
345 i
12 12
24 24
58 58
32 32
4 2
i 128
16 16
512 512
4 4
14 14
8 8
16 16
11 7
70 42
60 29
44 29
100 52


Sol. Found


2
9
7
5
2
4
8

4


16
512
4
14
8
16
7
43

21
I .


A3 A
16 0.C
125 8.1
342 170.3
12 34.1
24 33.1
58 21.:
32 0
4 0.4
I 4.5
16 '*.0
512 508
4 0.0
14 1.3
8 1.5
16 191.2
11 7
47 3.1
:, 3.2
34 3.C
75 88.1


0, for i 1,...,


Time(sec.)
1 A2 A3
)1 : 11 1-: 11
7 2.01 3.72
1 ,- 42 79.15
3 0.48 1.59
5 0.79 ( : '
S. 0.48 12.02
0.03 0.05
2 0.03 0.06
.7 1' 1.32
17 1. 1.79
: 3" 82.19
ii 0.01 :-.
;3 1 i : 13
.6 0 0.19
4 5.97 9.81
' 0.18 0.
9 0.27 -.;
1 : ;7 1 ,4
5 1.04 0.48
7 3 6.55


Table 3 5: Results on benchmark instances: comparison with other c
approaches. x = 0, i = 1. ...., n.


*ontiniuous based


Name



MANN_a27
keller4
S---_0.9_1
san200_0.9_2
: _0.9_3
san400_0.9_1


n Dens. w(C(G


45 -
78
71 0.649
00 0.900
00 *
00 0.900
0111 .. i


Sol. Found


ARH
16 16
117
11 8
70 45
39
44 31
50


PRD(Y)
12
12
117
7


PRD(0)
12
117
7


eTx 1,


x > 0.


subject to


CBH
16
121
10
46

30
50


A3(0)
16
125
11
47

34
75


(3.la)

(3.lb)









is given by
-( 1-
2 l (G) '
where w(G) is the clique number of G.

Gibbons et al. [98] extended this result by providing a characterization of

maximal cliques in terms of local solutions. Moreover, they studied optimality

conditions of the Motzkin-Straus program and investigated the properties of a newly

introduced parametrization of the corresponding quadratic program.

Gibbons et al. [97] proposed another, more tractable continuous formulation of

the maximum independent set problem.

Theorem 3.5 (Gibbons et al.). Consider the following optimization problem:
n
V(k)= min xT AGX + ( x, 1-)2 (3.2)
i=

subject to


?< -1 (3.2a)
i= 1

x > 0. (3.2b)


If x is a solution to (3.2) then V(k) = 0 iff there exists an independent set I in G

such that II > k.

Based on this theorem the authors proposed a heuristic for the maximum clique

problem, which uses the global minimum of the problem (3.2)-(3.2a) to extract a

maximal independent set from the graph. The approach that we propose in this

section is based on the same formulation as in Theorem 3.5. Moreover, we also adopt

the idea of using the problem of optimization of the same quadratic over a sphere

to find a large independent set, but in our approach we use information about all

stationary points of the problem (3.2)-(3.2a). We will use the computational results

of Gibbons et al. [97] to estimate the efficiency of our approach in subsection 3.4.3.









3.4.1 Optimization of a Quadratic Function Over a Sphere

Although optimization of a quadratic function subject to linear constraints is

NP-hard, quadratic programming subject to an ellipsoid constraint is polynomially

solvable. In fact, Ye [203] proved that it can be solved in O(log(log(l/e))) iterations,

where c is the error tolerance. Due to its application in trust region methods of

nonlinear optimization, the problem of optimization of a quadratic function over a

sphere is a well studied one [89, 110, 153, 164]. In this section we will discuss some

results related to this problem.

Consider the following problem:

min f(x) = 2cTx + xTQx, (3.3)


subject to


IX12 = r, (3.3a)

where x E R is the vector of variables, Q is a symmetric n x n matrix, c E IR"

is an arbitrary fixed vector, r > 0 is some number, and I| 112 is the second norm in

IR, i.e., for any x IR" :I 2 (x- 2
i= 1
We are interested in all stationary points of the problem (3.3), which can be

found from the Lagrangian (x, A) of f(x). We have


L(x, A) f(x)- AIxll -2), (3.4)


where A is the Lagrange multiplier of the constraint of (3.3), and the stationarity

conditions:
a 0, i 1_ ,... ,n; (3.5)
Oxi

a (x, )
OxA) 0. (3.6)
OA









The set of conditions (3.5) is equivalent to the system


(Q AI)x = -c, (3.7)


where I stands for the n x n identity matrix. The condition (3.6) is simply the

constraint (3.3a). Therefore, x is a stationary point of problem (3.3)-(3.3a) if and

only if there exists a real number A = A(x') such that


(Q AI)x = -c; (3.8)



xf r2. (3.9)
i=1
The set A of all real A for which there exists an x such that (3.8) and (3.9) are

satisfied, is called the spectrum of the problem (3.3) (3.3a).

Applying the eigenvalue decomposition for the matrix Q we have


Q R diag(A, A2, ... )RT,


where A1 < A2 < ... < An are the eigenvalues of the matrix Q sorted in nondecreasing

order, R is the matrix having a corresponding orthonormal set of eigenvectors as its

columns. Transforming the vectors x and c to the basis R of the eigenvectors of Q,

we obtain

x = RTx, c =RTc.

Then (3.7) can be rewritten as


(A A)xi -ci, i= ,...,n. (3.10)


Assuming that A / Ai, we have


S Ci (3. 11)
Ai A'









Rewriting expression (3.9) in the basis associated with the rows of matrix R and

substituting (3.11) into it, we obtain the so-called secular equation

S2 2 0. (3.12)
A A>
The left-hand side of the last expression is a univariate function consisting of

n + 1 pieces each of which is continuous and convex. This implies that it can have

at most two roots at any interval (Ai, Ai+1), i = 0,... n corresponding to the ith

piece, where by Ao and A,+1 we mean -oo and +oc respectively. Moreover, the

spectrum A includes the set A of all solutions to the secular equation (3.12) and

A \ Ao {A, : c, 0} (so, A = A if {A : c, = 0 0}).

Forsythe and Golub [89] have shown that the global minimum of problem (3.3)

corresponds to the smallest element of the spectrum A* = minA. It implies that if

ci / 0 then A* min A0.

We mention one more result, which relates the maximum independent set

problem to the problem of optimization of a quadratic over a sphere. Recently, de

Angelis et al. [70] have shown that in the problem of optimizing a quadratic over a

sphere of radius r a global solution x*(r) can be chosen, which depends continuously

upon r. The last fact gives us a new point of view on the maximum independent set

problem. Namely, it can be formulated as a problem of optimization of a quadratic

function over a sphere with an unknown radius:

Lemma 3.1. There exists r E 2 such that


a(G) = max fa(x) = x- xj (3.13)
i=1 (i,j)EE

subject to IIx e 12 r,

To show the validity of the lemma, consider formulation (P2) and take two

spheres S1 (of radius -) and S2 (of radius v-) inscribed in and circumscribed over









the unit hypercube respectively. We have


max fo(x) < a(G) max f(x) < max ,(x)
XES1 x [0,1]n XES2

and by the Intermediate Value Theorem the lemma is correct.

3.4.2 The Heuristic

In this subsection, we present the heuristic for the maximum independent set

problem which employs the formulation mentioned in Theorem 3.5, and utilizes the

methods similar to those in the so-called QUALEX heuristic [53]. In QUALEX, the

feasible region of the formulation (P2) is changed from the unit hypercube to the

sphere with the center xO = 0 and a certain radius r, depending on the input graph,

resulting in the following problem:



max (x) = Xi- xXj (3.14)
i 1 (i,j)EE
subject to


IIx x0* 2 (3.14a)


In short, QUALEX can be regarded as a sophisticated greedy heuristic, which

first finds a local solution using a straightforward greedy approach, and then attempts

to find a better solution using information provided by the stationary points of the

above problem.

We apply similar techniques for the problem which is obtained from (3.2)-

(3.2b) by changing the inequality to the equality in (3.2a) and relaxing nonnegativity

constraints (3.2b):

min x AGx + ( 1)2 (3.15)
i= 1









subject to


xf =- (3. 15a)
S1i

As the value of k we use the cardinality of a maximal independent set found

by a simple greedy heuristic (numerical experiments -- .:: -1 that varying k does not

make results much better, but, naturally, requires more computational time). For

each stationary point x of the problem (3.15)-(3.15a) we apply a greedy heuristic

to find a maximal independent set I, which uses order of the vertices in the graph

corresponding to the nonincreasing order of the components of x. The largest found

Ix is reported as the output maximal independent set.

Below we present an outline of our algorithm.

Algorithm 4 [QSH]:

INPUT: Graph G with ..li i:'ency matrix AG; a = 0

OUTPUT: A maximal independent set I* and its cardinality a

0. Apply the following greedy heuristic:

(a) Go = G;

(b) In Go recursively remove the neighborhood of a vertex with the minimum

degree in the current graph until an empty graph with vertex set I is

obtained;

(c) If III > a then begin I* = I, a = I|I; end

(d) If Go \ I / 0 then begin Go Go \ I; go to step (b); end.

1. For k = a begin

For each interval (Ai, Ai+l) begin

(a) Apply the techniques discussed in Section 3.4.1 for the problem

(3.15)-(3.15a) to find the set of all stationary points S corresponding to

(Ab For Ei);
(b) For each x cE S find I, by the following procedure:









i. Order the components of x in nonincreasing order

Xil > Xi2 > .> Xi.;

ii. I = 0;

iii. For j = 1 to n begin

If N(iz) n Ix = 0 then I, I, UJi}

end

(c) x* = argmax{| I : x E S}, I Ix*;

(d) If III > a then I = I and a = |I|;

end

end

The results of numerical experiments with the algorithm are reported in the next

subsection.

3.4.3 Computational Experiments

This section presents results of computational experiments for the proposed

heuristic. The algorithm has been tested on the complement graphs of some

of the DIMACS clique instances which can be downloaded from the URL

http://dimacs.rutgers.edu/Challenges/. The performance of the algorithm

(denoted by QSH) on the benchmarks is summarized in Tables 3-6 and 3-7. The

columns \ ,ii ," "n," "Density," and "~(G)" represent the name of the graph,

the number of its vertices, its density, and its clique number, respectively. This

information is available from the DIMACS web site. Recall that we are working

with complements of the considered graphs, and the densities are specified for the

original (maximum clique) instances. The column "Sol. Found" contains the size of

the clique found by each algorithm. In Table 3-6, sub-columns "CBH" and "QSH"

represent the continuous based heuristic [97] and the heuristic proposed in this section,

correspondingly. Finally, the column "Time" reports computational times for QSH

in seconds obtained using the C function time. The results for CBH were taken from









Table 3 6: Results on benchmark instances, part I.


Name


MANN_a9
MANN_a27
brock200_1
brock200_2
brock200_3
bro(i .: ': _4
brock400_1
brock4: '_2
brock400_3
brock4i^ _4
b: i '_1
b: '-~,_2
b: i '_3
b: 2 -,._4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-f .- .:- --10
c-fat500-2
c-f.- ;*:: '-5
hamming6-2

S. 2
,-,., : **,8-4
johnson8-2-4
johnson8-4-4
johnsonl6-2-4
johnson32-2-4


37





4(
4i
4(
41









5(
{,',


n Density w(G) Sol. Found
CBH (' H
45 0.927 16 16 16
78 0. -' i '. 121
S 0.745 21 'i 21
S 0. ". 12 12 12
S 0. 15 14 15
0.658 17 16 17
)0 0.748 27 23 27
0.7'i 1 24 29
)0 0.748 31 23 31
0.71 2 24 33
0.6 23 I 17
0.651 24 19 24
0.6 25
0.650 ,. 19 26
0.077 12 12 1 2
0.1 24 24 24
0.4 '. 58 58 58
)0 0.1: ., 14 14 14
S0.374 > 126 126 126


0.073
A 0.186

64 0 '
6 0.-" '
O '

0.556
70 0.7.
S 0.765
S... 0.879


1' 1' 1


4

4


4
4

6


Gibbons et al. [97]. A modification of the C++ QUALEX code available on-line

from Bu,- ;in's webpage [53] was used to obtain the results for QSH. The code was

compiled and executed on an IBM computer with 2 PowerPC processors, 333 Mhz

each.

In Table 3-8 we compare the performance of CBH and QSH on different types

of graphs. In each row of this table the first column contains a name of the family of


Time
(sec.)
0
8

1


2t
1

2
21
21

21
245
243

233


1

33
20
31
23
0
0
2
1
0
0
0
24










Table 3 7: Results on benchmark instances, part II.


Name

keller4
keller5

p_hat300-2
p ..:- 3
phatE -1
plhatE: '-2
phatE -3
phat700-1
plhat71 --2
p_hat700-3
s;,:: -_0.7_1
san200_0.7_2
san200_0.9_1
san200_0.9_2
san200_0.9_3
san400_0.5_1
.: )0_0.7_1
san41 i_0.7_2
.: ;')0_0.7_3
san41 i_0.9_1
.... .. ').7
sanr -')*_0.9
sanr4 --,_0.5
sanr400 0.7_1


n Density

171 0.649
776 0.751
0.244
0.i -
0.744
51':1 0.253

5, :' 0.752
7:'! 0.249
0. 'i
.i: 0.748






: 0.
(: :: : : :



S : :



. i:: i! 0.70"'
i: : O '







0.700


wc(G) Sol. Found
CBH (C:"H
11 10 11
27 21 24
8 8 7
25 25 24
: 33
9 9 9
35 33
> 46
11 11 8
44 44 42
> 59
.. 15 ,
18 12 18
70 70
;' 60
44 35
13 8 9

15 30
22 14 16
1 ',' ,' I 100
18 18 15
> 42 41 37
13 12 11
> 21 '.: 18


graphs, where each family consists of a group of graphs with the same letter names

(in Tables 3-6 and 3-7 such families are separated by horizontal lines). In the next

three columns "+" represents the number of instances from each family for which

CBH found a better solution than QSH; "-" represents the number of instances for

which QSH found a better solution than CBH; finally, "=" stands for the number of

instances for which cardinalities of the independent sets found by the two algorithms

were equal.


Time
(sec.)
1
149
6
6
6
48

48
143
143











21
23
I
1
21






20
19
19









Table 3 8: Comparison of the results on benchmark instances.


( ..; .... iy CBH vs .'"H
name
'0 1 1
brock 1 1 10
c-fat 0 7 0
0 4 0
johnson 0 4 0
keller 0 0 2
p_hat 8 1 0
san 0 0 10
sanr 4 0 0
TOTAL 13 18


As one can see QSH performed better on four families (MANN_a, brock, keller,

san) and did worse on two (p_hat and sanr). Analyzing the difference in quality of

the results, one can notice a huge gap between the solutions obtained for san group.

The results of numerical experiments i -.i- -1 that QSH is superior to CBH and many

other heuristics presented in the literature.

3.5 Concluding Remarks

In this chapter, we discussed several heuristics for the maximum independent

set problem based on continuous formulations. In Sections 3.1 and 3.2, we offer

three syntactically related algorithms for finding large maximal independent sets. A

computational investigation of these algorithms shows their competitiveness.

In Section 3.4, we present a heuristic for the maximum independent set problem

based on techniques used in optimization of a quadratic over a sphere. We show the

validity of this approach by demonstrating the results of numerical experiments. We

want to stress that the information provided by the stationary points other than points

of global optimality may help us to find better independent sets in the corresponding

graph.









In Lemma 3.1 we formulated the maximum independent set problem as a

problem of optimization of a quadratic over a sphere with an unknown radius.

In future research, when using similar techniques for combinatorial optimization

problems, one should try to address the following issues:

How to optimally choose the parameters of the quadratic objective and of the
sphere used as the feasible set?

What is the optimal way to extract the sought combinatorial object from
the information provided by stationary (or optimal) points of the considered
quadratic problem?

To answer these questions both theoretical and empirical studies are needed.















CHAPTER 4
APPLICATIONS IN MASSIVE DATA SETS

Massive data sets arise in a broad spectrum of scientific, engineering

and commercial applications. These include government and military systems,

telecommunications, medicine and biotechnology, .i-1 i 111 i,-Bi- finance, ecology,

geographical information systems, etc [5]. Some of the wide range of problems

associated with massive data sets are data warehousing, compression and

visualization, information retrieval, clustering and pattern recognition, and nearest

neighbor search. Handling these problems requires special interdisciplinary efforts

in developing novel sophisticated techniques. The pervasiveness and complexity of

the problems brought by massive data sets make it one of the most challenging and

exciting areas of research for years to come.

In many cases, a massive data set can be represented as a very large graph with

certain attributes associated with its vertices and edges. These attributes may contain

specific information characterizing the given application. Studying the structure of

this graph is important for understanding the structural properties of the application

it represents, as well as for improving storage organization and information retrieval.

In this chapter, we first review current development in studying massive graphs.

Then, in Section 4.2 we present our research concerning the market graph representing

the U.S. stock markets. We will conclude with some remarks in Section 4.4. This

chapter is based on joint publications with Boginski and Pardalos [32, 33].

4.1 Modeling and Optimization in Massive Graphs

In this section we discuss recent advances in modeling and optimization for

massive graphs. As examples, call, Internet, and Web graphs will be used.









As before, by G = (V, E) we will denote a simple undirected graph with the set

of n vertices V and the set of edges E. A multi-graph is an undirected graph with

multiple edges.

The distance between two vertices is the number of edges in the shortest path

between them (it is equal to infinity for vertices representing different connected

components). The diameter of a graph G is usually defined as the maximal distance

between pairs of vertices of G. Note, that in the case of a disconnected graph the

usual definition of the diameter would result in the infinite diameter, therefore the

following definition is in order. By the diameter of a disconnected graph we will

mean the maximum finite shortest path length in the graph (which is the same as the

largest of diameters of the graph's connected components).

4.1.1 Examples of Massive Graphs

The call graph

Here we discuss an example of a massive graph representing telecommunications

traffic data presented by Abello, Pardalos and Resende [4]. In this call 'ji''l, the

vertices are telephone numbers, and two vertices are connected by an edge if a call

was made from one number to another.

Abello et al. [4] experimented with data from AT&T telephone billing records.

To give an idea of how large a call graph can be we mention that a graph based on

one 20-d-v period had 290 million vertices and 4 billion edges. The analyzed one-

d-4v call graph had 53,767,087 vertices and over 170 millions of edges. This graph

appeared to have 3,667,448 connected components, most of them tiny; only 302,468

(or '.) components had more than 3 vertices. A giant connected component with

44,989,297 vertices was computed. It was observed that the existence of a giant

component resembles a behavior -,I--.- -1. l by the random graphs theory of Erdos

and R6nyi, but by the pattern of connections the call graph obviously does not fit

into this theory. This will be discussed in more detail in Subsection 4.1.3. The










maximum clique problem and problem of finding large quasi-cliques with prespecified

density were considered in this giant component. These problems were attacked using

a greedy randomized adaptive search procedure (GRASP) [85, 86]. In short, GRASP

is an iterative method that at each iteration constructs, using a greedy function,

a randomized solution and then finds a locally optimal solution by searching the

neighborhood of the constructed solution. This is a heuristic approach which gives

no guarantee about quality of the solutions found, but proved to be practically

efficient for many combinatorial optimization problems. To make application of

optimization algorithms in the considered large component possible, the authors use

some suitable graph decomposition techniques employing external memory algorithms

(see Subsection 4.1.2).




1000



freq 1 00


10



1 --- --- -- -- -- --- ------
5 10 15 20 25 30
clique size


Figure 4 1:. .. ..... :. of clique sizes in the call g. found 1 Abello et al. [4].


Abello et al. ran 100,000 GRASP iterations taking 10 parallel processors about

one and a half dv to finish. Of the 100,000 cliques generated, 14,141 appeared to be

distinct, although many of them had vertices in common. The authors s, r-.- -1- I that

the graph contains no clique of a size greater than 32. Figure 4-1 shows the number of

detected cliques of various sizes. Finally, large quasi-cliques with density parameters









62


7 = 0.9, 0.8, 0.7, and 0.5 for the giant connected component were computed. The


sizes of the largest quasi-cliques found were 44, 57, 65, and 98, respectively.


le+07 x






le+04-

Sle+03

le+02

le+01-

le+00 .., ..-o
100 le+01 le+02 e+03 le+04 le+05
Outdegree


le+06-


le+05


8 le+04
Bx

le+03-
x

le+02


le+01
le+O1


le+07

le+06-

Sle+05

le+04

le+03

le+02
le+03O
le+02

le+01

le+00
le+00


le+01


(c)


le+ 00 .-. . . . .
le+00 le+01 le+02 le+03 le+04 le+05 le+06 le+07
Component size


Figure 4-2: Number of vertices with various out-degrees (a) and in-degrees (b); the
number of connected components of various sizes (c) in the call graph,
due to Aiello et al. [11].



Aiello et al. [11] used the same data as Abello et al. [4] to show that the considered


call graph fits to their random graph model which will be discussed in Subsection 4.1.3.


The plots in Figure 4-2 demonstrate some connectivity properties of the call graph.


Figure 4-2(a,b) shows plots of the number of vertices for every out-degree and in-


degree, respectively. Frequencies of the sizes of connected components are represented


in Figure 4-2(c).


(b)


l102 le+03 le+04 le+05
Indegree









The Internet and Web graphs

The role of the Internet in the modern world is difficult to overestimate; its

invention changed the way people interact, learn, and communicate like nothing

before. Alongside with increasing significance, the Internet itself continues to grow

at an overwhelming rate. Figure 4-3 shows the dynamics of growth of the number of

Internet hosts for the last 13 years. As of January 2002 this number was estimated

to be close to 150 million. The number of web pages indexed by large search engines

exceeds 2 billion, and the number of web sites is growing by thousands daily.


160,000,000
140,000,000
120,000,000
100,000,000
80,000,000
60,000,000
40,000,000
20,000,000
0


P oO < oC ,< CP < ,< s s,< ,


Figure 4-3: Number of Internet hosts for the period 01/1991-01/2002. Data by
Internet Software Consortium [126].


The highly dynamic and seemingly unpredictable structure of the World Wide

Web attracts more and more attention of scientists representing many diverse

disciplines, including graph theory. In a graph representation of the World Wide

Web, the vertices are documents and the edges are hyperlinks pointing from one

document to another. Similarly to the call graph, the Web is a directed multigraph,

although often it is treated as an undirected graph to simplify the analysis. Another

graph is associated with the physical network of the Internet, where the vertices are







64














-- ^ -
--s















Figure 4-4: A sample of paths of the physical network of Internet cables created by W.
C : : i and H. Burch :]. Courtesy of Lumeta Corporation. Patent(s)
Pending. Copyright @Lumeta Corporation ::: All Rights Reserved.


routers navigating packets of data or groups of routers (domains). The edges in

this graph represent wires or cables in the physical network. Figure 4-4 illustrates

the tremendous complexity of this network. This and many other maps aiming to

visualize the Internet topology are products of the ilii, i, 1 Mapping Project" [60].

They are created from data obtained by tracing routes from one terminal to a set of

other Internet domains.

Graph theory has been applied for web search [42, 59, 141], web mining [160, 161]

and other problems arising in the Internet and World Wide Web. In several recent

studies, there were attempts to understand some structural properties of the Web

graph by investigating large Web crawls. Adamic and Huberman [8, 124] used










le-I O II I l e4 7 ,
le+09
le+-fl -
a7
le+08 a

Le-06 I -.*
le4437

1e-04 + le03
+ + I+
+eI-0+ e-02
le44+02 + leI



oM"W-deg stz of compondmt

Figure 4-5: Number of vertices with various out-degrees (left) and distribution of
sizes of strongly connected components (right) in Web graph [43].


crawls which covered almost 260,000 pages in their studies. Barabdsi and Albert [26]

analyzed a subgraph of the Web graph approximately 325,000 nodes representing

nd.edu pages. In another experiment, Kumar et al. [146] examined a data set

containing about 40 million pages. In a recent study, Broder et al. [43] used two

Altavista crawls, each with about 200 million pages and 1.5 billion links, thus

significantly exceeding the scale of the preceding experiments. This work yielded

several remarkable observations about local and global properties of the Web graph.

All of the properties observed in one of the two crawls were validated for the other as

well. Below, by the Web graph we will mean one of the crawls, which has 203,549,046

nodes and 2130 million arcs.

The first observation made by Broder et al. confirms a property of the Web

graph -i--.- -1. I in earlier works [26, 146] claiming that the distribution of degrees

follows a power law. That is, the number of vertices of degree k is proportional to k-^

for some 7 > 1. Interestingly, the degree distribution of the Web graph resembles the

power-law relationship of the Internet graph topology, which was first discovered by

Faloutsos et al. [82]. Broder et al. [43] computed the in- and out-degree distributions

for both considered crawls and showed that these distributions agree with power laws.

Moreover, they observed that in the case of in-degrees the constant 7y 2.1 is the









same as the exponent of power laws discovered in earlier studies [26, 146]. In another

set of experiments conducted by Broder et al., directed and undirected connected

components were investigated. It was noticed that the distribution of sizes of these

connected components also obeys a power law. Figure 4-5 illustrates the experiments

with distributions of out-degrees and connected component sizes.

The last series of experiments discussed by Broder et al. [43] aimed to explore

the global connectivity structure of the Web. This led to the discovery of the

so-called Bow-Tie model of the Web [44]. Similarly to the call graph, the considered

Web graph appeared to have a giant connected component, containing 186,771,290

nodes, or over 9i' of the total number of nodes. Taking into account the directed

nature of the edges, this connected component can be subdivided into four pieces:

strongly connected component (SCC), In and Out components, and "T i, ,..1"..

Overall, the Web graph in the Bow-Tie model is divided into the following pieces:

The .-li.o.'l;l, connected components the part of the giant connected component
in which all nodes are reachable from one another by a directed path.

The In component consists of nodes which can reach any node in the SCC but
cannot be reached from the SCC.

The Out component contains the nodes that are reachable from the SCC, but
cannot access the SCC through directed links.

The Tendrils component accumulates the remaining nodes of the giant
connected component, i.e., the nodes which are not connected with the SCC.

The Disconnected component is the part of the Web which is not connected with
the giant connected component.

Figure 4-6 shows the connectivity structure of the Web, as well as sizes of

the considered components. As one can see from the figure, the sizes of SCC, In,

Out and Tendrils components are roughly equal, and the Disconnected component is

significantly smaller.






























16,777,756


Figure 4-6: Connectivity of the Web due to Broder et al. [43].


Broder et al. [43] have also computed the diameters of the SCC and of the whole

graph. It was shown that the diameter of the SCC is at least 28, and the diameter

of the whole graph is at least 503. The average connected distance is defined as the

pairwise distance averaged over those directed pairs (i,j) of nodes for which there

exists a path from i to j. The average connected distance of the whole graph was

estimated as 16.12 for in-links, 16.18 for out-links, and 6.83 for undirected links.

Interestingly, it was also found that for a randomly chosen directed pair of nodes, the

chance that there is a directed path between them is only about 2,!' These results

are not in agreement with the prediction of Albert et al. [12], who s, -l-:. -1. I that the

average distance between two randomly chosen documents of the Web is 18.59. Let

us mention that the property of a large network to have a small diameter has been

observed in many real-life networks and is frequently referred to as the small world

phenomenon [15, 198, 199].









4.1.2 External Memory Algorithms

In many cases, the data associated with massive graphs is too large to fit entirely

inside the fast computer's internal memory, therefore a slower external memory

(for example disks) needs to be used. The input/output communication (I/O)

between these memories can result in an algorithm's slow performance. External

memory (EM) algorithms and data structures are designed with aim to reduce the

I/O cost by exploiting the locality. Recently, external memory algorithms have

been successfully applied for solving batched problems involving graphs, including

connected components, topological sorting, and shortest paths.

The first EM graph algorithm was developed by Ullman and Yannakakis [194]

in 1991 and dealt with the problem of transitive closure. Many other researchers

contributed to the progress in this area ever since [2, 18, 19, 49, 61, 147, 197].

Chiang et al. [61] proposed several new techniques for design and analysis of

efficient EM graph algorithms and discussed applications of these techniques to

specific problems, including minimum spanning tree verification, connected and

biconnected components, graph drawing, and visibility representation. Abello et

al. [2] proposed a functional approach for EM graph algorithms and used their

methodology to develop deterministic and randomized algorithms for computing

connected components, maximal independent sets, maximal matching, and other

structures in the graph. In this approach each algorithm is defined as a sequence of

functions, and the computation continues in a series of scan operations over the data.

If the produced output data, once written, cannot be changed, then the function is

said to have no side effects. The lack of side effects enables the application of standard

checkpointing techniques, thus increasing the reliability. Abello et al. presented a

semi-external model for graph problems, which assumes that only the vertices fit in

the computer's internal memory. This is quite common in practice, and in fact this

was the case for the call graph described in Subsection 4.1.1, for which efficient EM









algorithms developed by Abello et al. [2] were used in order to compute its connected

components [4].

For more detail on external memory algorithms see the book [6] and the extensive

review by Vitter [197] of EM algorithms and data structures.

4.1.3 Modeling Massive Graphs

The size of real-life massive graphs, many of which cannot be held even by a

computer with several gigabytes of main memory, vanishes the power of classical

algorithms and makes one look for novel approaches. External memory algorithms

and data structures discussed in the previous subsection represent one of the research

directions aiming to overcome difficulties created by data sizes. But in some cases not

only is the amount of data huge, but the data itself is not completely available. For

instance, one can hardly expect to collect complete information about the Web graph;

in fact, the largest search engines are estimated to cover only 3-'-. of the Web [148].

Some approaches were developed for studying the properties of real-life massive

graphs using only the information about a small part of the graph. For instance,

Goldreich [105] proposes two randomized algorithms for testing if a given massive

graph has some predefined property. These algorithms analyze a part of the graph

and with some probability give the answer if this graph has a given property or not,

based on a certain criterion.

Another methodology of investigating real-life massive graphs is to use the

available information in order to construct proper theoretical models of these graphs.

One of the earliest attempts to model real networks theoretically goes back to the

late 1950's, when the foundations of random graph theory had been developed. In

this subsection we will present some of the results produced by this and other (more

realistic) graph models.









Uniform random graphs

The classical theory of random graphs founded by Erdis and Rinyi [78, 79] deals

with several standard models of the so-called uniform random graphs. Two of such

models are G(n, m) and G(n,p) [35]. The first model assigns the same probability to

all graphs with n vertices and m edges, while in the second model each pair of vertices

is chosen to be linked by an edge randomly and independently with probability p.

In most cases for each natural n a probability space consisting of graphs with

exactly n vertices is considered, and the properties of this space as n -- o are studied.

It is said that a typical element of the space or almost every (a.e.) graph has property

Q when the probability that a random graph on n vertices has this property tends to

1 as n -- oo. We will also -;iv that the property Q holds '- ;,,,11i. i,/ ill:l almost surely

(a.a.s.). Erdis and R6nyi discovered that in many cases either almost every graph

has property Q or almost every graph does not have this property.

T iiry properties of uniform random graphs have been well studied [34, 35, 129,

143]. Below we will summarize some known results in this field.

Probably the simplest property to be considered in any graph is its con,. ../.; .:,

It was shown that for a uniform random graph G(n,p) E G(n,p) there is a !i,, -I, .1"

value of p that determines whether a graph is almost surely connected or not. More

specifically, a graph G(n,p) is a.a.s. disconnected if p < bg. Furthermore, it turns

out that if p is in the range 1 < p < 2g, the graph G(n,p) a.a.s. has a unique

i.:,.i/ connected component [35]. The emergence of a giant connected component in a

random graph is very often referred to as the p!i .i- transition".

The next subject of our discussion is the diameter of a uniform random graph

G(n,p). Recall that the diameter of a disconnected graph is defined as the maximum

diameter of its connected components. When dealing with random graphs, one

usually speaks not about a certain diameter, but rather about the distribution of

the possible values of the diameter. Intuitively, one can -i- that this distribution









depends on the interrelationship of the parameters of the model n and p. However,

this dependency turns out to be rather complicated. It was discussed in many papers,

and the corresponding results are summarized below.

It was proved by Klee and Larman [140] that a random graph .-i!' lcally

almost surely has the diameter d, where d is a certain integer value, if the following

conditions are satisfied

d-1 pd
-- 0 and -- oo, n-- 0.
n n
Bollobds [35] proved that if np- log n -i o then the diameter of a random graph

is a.a.s. concentrated on no more than four values.

Luczak [151] considered the case np < 1, when a uniform random graph a.a.s.

is disconnected and has no giant connected component. Let diamr(G) denote the

maximum diameter of all connected components of G(n,p) which are trees. Then if

(1 np)n1/3 -- o the diameter of G(n,p) is a.a.s. equal to diamT(G).
Chluin and Lu [62] investigated another extreme case: np oc. They showed

that in this case the diameter of a random graph G(n,p) is a.a.s. equal to


log n
(1 + o(1)) .
log(np)
Moreover, they considered the case when np > c > 1 for some constant c and got a

generalization of the above result:


( 10c
log n log n (-1 )2 + logn
(1 + o(1)) < diam(G(n, p)) < + 2c- ). + 1.
log(np) log(np) c -log(2c) np

Also, they explored the distribution of the diameter of a random graph with respect

to different ranges of the ratio np/log n. They obtained the following results:









For np/ logn = c > 8 the diameter of G(n,p) is a.a.s. concentrated on at most
two values at log n/ log(np).

For 8 > np/ logn = c > 2 the diameter of G(n,p) is a.a.s. concentrated on at
most three values at log n/ log(np).

For 2 > np/logn c > 1 the diameter of G(n,p) is a.a.s. concentrated on at
most four values at log n/log(np).

For 1 > np/ logn c > co the diameter of G(n,p) is a.a.s. concentrated on a
finite number of values, and this number is at most 2 + 4. More specifically,
in this case the following formula can be proved:

log(con/11) log 4n log n 1
log(con/il) log(np) log(np) C

As pointed out above, a graph G(n,p) a.a.s. has a giant connected component
for 1 < up < logn. It is natural to assume that in this case the diameter of G(n,p)

is equal to the diameter of this giant connected component. However, it was strictly

proved by Chluin and Lu [62] that it is a.a.s. true only if np > 3.5128.

Potential drawbacks of the uniform random graph model

There were some attempts to model the real-life massive graphs by the uniform

random graphs and to compare their behavior. However, the results of these

experiments demonstrated a significant discrepancy between the properties of real

graphs and corresponding uniform random graphs.

The further discussion ain J!v. l the potential drawbacks of applying the uniform

random graph model to the real-life massive graphs.

Though the uniform random graphs demonstrate some properties similar to

the real-life massive graphs, many problems arise when one tries to describe the

real graphs using the uniform random graph model. As it was mentioned above, a

giant connected component a.a.s. emerges in a uniform random graph at a certain

threshold. It looks very similar to the properties of the real massive graphs discussed

in Subsection 4.1.3. However, after deeper insight, it can be seen that the giant









connected components in the uniform random graphs and the real-life massive graphs

have different structures. The fundamental difference between them is as follows:

it was noticed that in almost all the real massive graphs the property of so-called

clustering takes place [198, 199]. It means that the probability of the event that two

given vertices are connected by an edge is higher if these vertices have a common

neighbor (i.e., a vertex which is connected by an edge with both of these vertices).

The probability that two neighbors of a given vertex are connected by an edge is called

the clustering coefficient. It can be easily seen that in the case of the uniform random

graphs, the clustering coefficient is equal to the parameter p, since the probability

that each pair of vertices is connected by an edge is independent of all other vertices.

In real-life massive graphs, the value of the clustering coefficient turns out to be

much higher than the value of the parameter p of the uniform random graphs with

the same number of vertices and edges. Adamic [7] found that the value of the

clustering coefficient for some part of the Web graph was approximately 0.1078, while

the clustering coefficient for the corresponding uniform random graph was 0.00023.

Pastor-Satorras et al. [172] got similar results for the part of the Internet graph. The

values of the clustering coefficients for the real graph and the corresponding uniform

random graph were 0.24 and 0.0006 respectively.

Another significant problem arising in modeling massive graphs using the uniform

random graph model is the difference in degree distributions. It can be shown that

as the number of vertices in a uniform random graph increases, the distribution of

the degrees of the vertices tends to the well-known Poisson distribution with the

parameter np which represents the average degree of a vertex. However, as it was

pointed out in Subsection 4.1.3, the experiments show that in the real massive graphs

degree distributions obey a power law. These facts demonstrate that some other

models are needed to better describe the properties of real massive graphs. Next,









we discuss two of such models; namely, the random graph model with a given degree

sequence and its most important special case the power-law model.

Random graphs with a given degree sequence

Besides the uniform random graphs, there are more general v--iv of modeling

massive graphs. These models deal with random guI .l,- with a given degree sequence.

The main idea of how to construct these graphs is as follows. For all the vertices

i = 1... n the set of the degrees {ki} is specified. This set is chosen so that the

fraction of vertices that have degree k tends to the desired degree distribution pk as

n increases.

It turns out that some properties of the uniform random graphs can be

generalized for the model of a random graph with a given degree sequence.

Recall the notation of so-called p! .-i' transition" (i.e., the phenomenon when

at a certain point a giant connected component emerges in a random graph) which

happens in the uniform random graphs. It turns out that a similar thing takes place

in the case of a random graph with a given degree sequence. This result was obtained

by Molloy and Reed [162]. The essence of their findings is as follows.

Consider a sequence of non-negative real numbers po, pi, such that Y pk 1.
k
Assume that a graph G with n vertices has approximately pkn vertices of degree k.

If we define Q = Ek>1 k(k 2)pk then it can be proved that G a.a.s. has a giant

connected component if Q > 0 and there is a.a.s. no giant connected component if

Q< 0.

As a development of the an i1-i; of random graphs with a given degree sequence,

the work of Cooper and Frieze [66] should be mentioned. They considered a

sparse directed random graph with a given degree sequence and analyzed its strong

connectivity. In the study, the size of the giant strongly connected component, as

well as the conditions of its existence, were discussed.









The results obtained for the model of random graphs with a given degree sequence

are especially useful because they can be implemented for some important special

cases of this model. For instance, the classical results on the size of a connected

component in uniform random graphs follow from the aforementioned fact presented

by Molloy and Reed. Next, we present another example of applying this general result

to one of the most practically used random graph models the power-law model.

Power-law random graphs

One of the most important special cases of the model of random graphs with a

given degree sequence is the power-law random graph model. The power-law random

graphs are also sometimes referred to as (a, 3)-graphs. This model was recently

applied to describe some real-life massive graphs such as the call graph, the Internet

graph and the Web graph mentioned above. Some fundamental results for this model

were obtained by Aiello, Chuiin and Lu [10, 11].

The basic idea of the power-law random graph model P(a, 3) is as follows. If

we define y to be the number of nodes with degree x, then according to this model



y = e/x. (4.1)

Equivalently, we can write



logy = a f log x. (4.2)

This representation is more convenient in the sense that the relationship between y

and x can be plotted as a straight line on a log-log scale, so that (-0) is the slope,

and a is the intercept. This implies the following properties of a graph described by

the power law model [11]:


* The maximum degree of the graph is e"/.









The number of vertices is


S(e", > t,
e"
n = 0en 9 (4.3)
xli



where ((t) E 1 is the Riemann Zeta function.
n=1
The number of edges is


S-(/3 1-)e,/3 > 2,
E = XC~e a = 2, (4.4)
X=1
C- (2-3) O < < 2.

Since the power-law random graph model is a special case of the model of a

random graph with a given degree sequence, the results discussed above can be applied

to the power law graphs. We need to find the threshold value of 3 in which the "phase

transition" (i.e., the emergence of a giant connected component) occurs. In this case

Q = ,>l x( 2)p. is defined as

Q2 Ex(x -2)L aE -2E P [(/ 2)-2(/3 )]a for 3 > 3.
x=1 x=1 x=1
Hence, the threshold value /3 can be found from the equation



(/3 2) 2(3 1) 0,

which yields/3o 3.47875.

The results on the size of the connected component of a power-law graph were

presented by Aiello, Chuiin and Lu [11]. These results are summarized below.

If 0 < 3 < 1, then a power-law graph is a.a.s. connected (i.e., there is only one
connected component of size n).

If 1 < 3 < 2, then a power-law graph a.a.s. has a giant connected component
(the component size is 0(n)), and the second largest connected component







77

a.a.s. has a size 0(1).

If 2 < 3 < 3o = 3.47875, then a giant connected component a.a.s. exists, and
the size of the second largest component a.a.s. is 0 (log n).

3 = 2 is a special case when there is a.a.s. a giant connected component, and
the size of the second largest connected component is 0(log n/log log n).

If 3 > /o = 3.47875, then there is a.a.s. no giant connected component.

The power-law random graph model was developed for describing real-life massive

graphs. So the natural question is how well it reflects the properties of these graphs.

Though this model certainly does not reflect all the properties of real massive

graphs, it turns out that the massive graphs such as the call graph or the Internet

graph can be fairly well described by the power-law model. The following example

demonstrates it.

Aiello, Chuiig and Lu [11] investigated the same call graph that was analyzed by

Abello et al. [4]. This massive graph was already discussed in Subsection 4.1.3, so it

is interesting to compare the experimental results presented by Abello et al. [4] with

the theoretical results obtained in [11] using the power-law random graph model.

Figure 4-2 shows the number of vertices in the call graph with certain in-degrees

and out-degrees. Recall that according to the power-law model the dependency

between the number of vertices and the corresponding degrees can be plotted as

a straight line on a log-log scale, so one can approximate the real data shown in

Figure 4-2 by a straight line and evaluate the parameter a and 3 using the values

of the intercept and the slope of the line. The value of 3 for the in-degree data

was estimated to be approximately 2.1, and the value of e" was approximately

30 x 106. The total number of nodes can be estimated using formula (4.3) as

((2.1) x e" = 1.56 x e" 47 x 106 (compare with Subsection 4.1.3).

According to the results for the size of the largest connected component presented

above, a power-law graph with 1 < 3 < 3.47875 a.a.s. has a giant connected









component. Since /3 2.1 falls in this range, this result exactly coincides with

the real observations for the call graph (see Subsection 4.1.3).

Another aspect that is worth mentioning is how to generate power-law graphs.

The methodology for doing it was discussed in detail in the literature [10, 48, 65].

These papers use a similar approach, which is referred to as a random p ,'l, evolution

process. The main idea is to construct a power-law massive graph "step-by--l p : at

each time step, a node and an edge are added to a graph in accordance with certain

rules in order to obtain a graph with a specified in-degree and out-degree power-law

distribution. The in-degree and out-degree parameters of the resulting power-law

graph are functions of the input parameters of the model. A simple evolution model

was presented by Kumar et al. [145]. Aiello, Chuing and Lu [10] developed four more

advanced models for generating both directed and undirected power-law graphs with

different distributions of in-degrees and out-degrees. As an example, we will briefly

describe one of their models. It was the basic model developed in the paper, and the

other three models actually were improvements and generalizations of this model.

The main idea of the considered model is as follows. At the first time moment a

vertex is added to the graph, and it is assigned two parameters the in-weight and

the out-weight, both equal to 1. Then at each time step t + 1 a new vertex with

in-weight 1 and out-weight 1 is added to the graph with probability 1 a, and a new

directed edge is added to the graph with probability a. The origin and destination

vertices are chosen according to the current values of the in-weights and out-weights.

More specifically, a vertex u is chosen as the origin of this edge with the probability

proportional to its current out-weight which is defined as w, = 1 + 6, where 6,

is the out-degree of the vertex u at time t. Similarly, a vertex v is chosen as the

destination with the probability proportional to its current in-weight ,. = 1 + 6"

where 6, is the in-degree of v at time t. From the above description it can be seen

that at time t the total in-weight and the total out-weight are both equal to t. So for









each particular pair of vertices u and v, the probability that an edge going from u to

v is added to the graph at time t is equal to


(1 + 6T,)(1 + )
t2
In the above notations, the parameter a is the input parameter of the model. The

output of this model is a power-law random graph with the parameter of the degree

distribution being a function of the input parameter. In the case of the considered

model, it was shown that it generates a power-law graph with the distribution of

in-degrees and out-degrees having the parameter 1 + 1

The notion of the so-called scale invariance [26, 27] must also be mentioned. This

concept arises from the following considerations. The evolution of massive graphs can

be treated as the process of growing the graph at a time unit. Now, if we replace all the

nodes that were added to the graph at the same unit of time by only one node, then

we will get another graph of a smaller size. The 'i.-..-- r the time unit is, the smaller

the new graph size will be. The evolution model is called scale-free (scale-invariant)

if with high probability the new (scaled) graph has the same power-law distribution

of in-degrees and out-degrees as the original graph, for any choice of the time unit

length. It turns out that most of the random evolution models have this property.

For instance, the models of Aiello et al. [10] were proved to be scale-invariant.

4.1.4 Optimization in Random Massive Graphs

Recent random graph models of real-life massive networks, some of which

were mentioned in Subsection 4.1.3 increased interest in various properties of

random graphs and methods used to discover these properties. Indeed, numerical

characteristics of graphs, such as clique and chromatic numbers, could be used as one

of the steps in validation of the proposed models. In this regard, the expected clique

number of power-law random graphs is of special interest due to the results by Abello

et al. [4] and Aiello et al. [10] mentioned in Subsections 4.1.1 and 4.1.3. If computed,









it could be used as one of the points in verifying the validity of the model for the call

graph proposed by Aiello et al. [10].

In this subsection we present some well-known facts regarding the clique and

chromatic numbers in uniform random graphs.

Clique number

The earliest results describing the properties of cliques in uniform random graphs

are due to Matula [159], who noticed that for a fixed p almost all graphs G E G(n,p)

have about the same clique number, if n is sufficiently large. Bollobas and Erdos [37]

further developed these remarkable results by proving some more specific facts about

the clique number of a random graph. Let us discuss these results in more detail by

presenting not only the facts but also some reasoning behind them. For more detail

see books by Bollobas [34, 35] and Janson et al. [129].

Assume that 0 < p < 1 is fixed. Then instead of the sequence of spaces

{Q(n,p),n > 1} one can work with the single probability space g(N,p) containing

graphs on N with the edges chosen independently with probability p. In this way,

G(n,p) becomes an image of g(N,p), and the term !ii. (,. i' is used in its usual

measure-theory sense. For a graph G E g(N,p) we denote by G, the subgraph of G

induced by the first n vertices {1, 2,..., n}. Then the sequence w(G,) appears to be

almost completely determined for a.e. G E g(N,p).

For a natural 1, let us denote by kl(G,) the number of cliques spanning 1 vertices

of G,. Then, obviously,


uw(G,) = max{l : ki(G,) > 0}.

When I is small, the random variable ki(G,) has a large expectation and a rather

small variance. If I is increased, then for most values of n there exists some number lo

for which the expectation of ko0 (G,) is fairly large (> 1) and k0+l (G,) is much smaller

than 1. Therefore, if we find this value 10 then w(G,) = 10 with a high probability.









The expectation of ki(G,) can be calculated as


E(k1(G.))= p .

Denoting by f(l) = E(kl(Gn)) and replacing (') by its Stirling approximation we

obtain
nn+1/2
f(l) 2 --(-1)/2
f 2ir(n 1)nl-+1/2 1+1/2
Solving the equation f(l) = 1 we get the following approximation lo of the root:

lo 2 logl/ 2logpl/p+21og +2log/p(e/2) + 1 + o(1)
(4.5)
2 log1/p + O(loglogn).

Using this observation and the second moment method, Bollobds and Erdos [37]

proved that if p = p(n) satisfies n-' < p < c for every c and some c < 1, then there

exists a function cl : N N such that a.a.s.

cl(n) < w(G,) < cl(n) + l,

i.e., the clique number is .i- mptotically distributed on at most two values. The

sequence cl(n) appears to be close to lo(n) computed in (4.5). Namely, it can be

shown that for a.e. G E g(N,p) if n is large enough then


[lo(n) 2 log log n/ log n] < w(G,) < Llo(n) + 2 log log n/ log n]

and
3
w(G,) 2 logi/p n + 2 log/p logl/p 2 logi/(e/2) 1 < 2

Frieze [91] and Janson et al. [129] extended these results by showing that for C > 0

there exists a constant ce, such that for < p(n) < log-2 n a.a.s.


[2 log1/p n 2logi/ log/p n + 2 log/p(e/2) + 1 /p] < w(Gn) <











[2 log,/ n 2 log/ log/p n + 2 log/(e/2) + 1 + e/pj.

C('-lii i iic number

Grimmett and McDiarmid [106] were the first to study the problem of coloring

random graphs. Many other researchers contributed to solving this problem [13, 36].

We will mention some facts emerged from these studies.

Luczak [150] improved the results about the concentration of X(G(n,p))

previously proved by Shamir and Spencer [182], proving that for every sequence

p = p(n) such that p < n-6/7 there is a function ch(n) such that a.a.s.

ch(n) < x(G(n,p)) < ch(n) + 1.


Alon and Krivelevich [13] proved that for any positive constant 6 the chromatic

number of a uniform random graph G(n,p), where p = n- 6, is a.a.s. concentrated

in two consecutive values. Moreover, they proved that a proper choice of p(n) may

result in a one-point distribution. The function ch(n) is difficult to find, but in some

cases it can be characterized. For example, Janson et al. [129] proved that there exists

a constant Co such that for any p = p(n) satisfying < p < log-7 n a.a.s.

np V np
P< X(G(n, 2 log np 2 log log np + 1 2 log np 40 log log np

In the case when p is constant Bollobds' method utilizing martingales [35] yields the

following estimate:


l(2logb n 2log log n +O(1)'

where b 1/(1 -p).

4.1.5 Remarks

We discussed advances in several research directions dealing with massive graphs,

such as external memory algorithms and modeling of massive networks as random









graphs with power-law degree distributions. Despite the evidence that uniform

random graphs are hardly suitable for modeling the considered real-life graphs,

the classical random graphs theory still may serve as a great source of ideas in

studying properties of massive graphs and their models. We recalled some well-known

results produced by the classical random graphs theory. These include results for

concentration of clique number and chromatic number of random graphs, which would

be interesting to extend to more complicated random graph models (i.e., power-law

graphs and graphs with arbitrary degree distributions). External memory algorithms

and numerical optimization techniques could be applied to find an approximate value

of the clique number (as it was discussed in Subsection 4.1.1). On the other hand,

probabilistic methods similar to those discussed in Subsection 4.1.4 could be utilized in

order to find the ..- i, i ii i ical distribution of the clique number in the same network's

random graph model, and therefore verify this model.

4.2 The Market Graph

Although not so obviously as in the examples in Subsection 4.1.1, financial

markets can also be represented as graphs. For a stock market one natural

representation is based on the cross correlations of stock price fluctuations. A market

graph can be constructed as follows: each stock is represented by a vertex, and two

vertices are connected by an edge if the correlation coefficient of the corresponding

pair of stocks (calculated for a certain period of time) is above a prespecified threshold

0, -1 <0< 1.

4.2.1 Constructing the Market Graph

The market graph that we study in this chapter represents the set of financial

instruments traded in the U.S. stock markets. More specifically, we consider 6546

instruments and analyze daily changes of their prices over a period of 500 consecutive

trading dv4 in 2000-2002. Based on this information, we calculate the cross-

correlations between each pair of stocks using the following formula [156]:


























-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Figure 4-7: Distribution of correlation coefficients in the stock market



(RiRj)- (R) (Rj)
(R2 (R)2)(2 j) 2)

where Ri(t) = in P() defines the return of the stock i for di- t. Pi(t) denotes the

price of the stock i on di- t.

The correlation coefficients Cij can vary from -1 to 1. Figure 4-7 shows the

distribution of the correlation coefficients based on the prices data for the years 2000-

2002. It can be seen that this distribution is nearly symmetric around the mean,

which is approximately equal to 0.05.

The main idea of constructing a market graph is as follows. Let the set of

financial instruments represent the set of vertices of the graph. Also, we specify

a certain threshold value 0, -1 < 0 < 1 and add an undirected edge connecting the

vertices i and j if the corresponding correlation coefficient Ci is greater than or equal

to 0. Obviously, different values of 0 define the market graphs with the same set of

vertices, but different sets of edges.


0.07

0.06

0.05

0.04










60.00%


50.00%


40.00%


S30.00%


20.00%


10.00%


0.00% *
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
correlation threshold


Figure 4-8: Edge density of the market graph for different values of the correlation
threshold.


It is easy to see that the number of edges in the market graph decreases as the

threshold value 0 increases. In fact, our experiments show that the edge density of the

market graph decreases exponentially w.r.t. 0. The corresponding graph is presented

on Figure 4-8.

4.2.2 Connectivity of the Market Graph

In Subsection 4.1.3 we mentioned the connectivity thresholds in random graphs.

The main idea of this concept is finding a threshold value of the parameter of the

model (p in the case of uniform random graphs, and f in the case of power-law

graphs) that will define if the graph is connected or not. Moreover, if the graph is

disconnected, another threshold value can be defined to determine if the graph has a

giant connected component or all of its connected components have a small size.

For instance, in the case of the power-law model f = 1 is a threshold value that

determines the connectivity of the power-law graph, i.e., the graph is a.a.s. connected











-o 7000

i 6000
8 5000

C 4000
-o
E 3000

0 2000

i 1000
.4-
0
-1 -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

correlation threshold


Figure 4-9: Plot of the size of the largest connected component in the market graph
as a function of correlation threshold 0.


if 3 < 1, and it is a.a.s. disconnected otherwise. Similarly, /3 w 3.47875 defines the

existence of a giant connected component in the power-law graph.

Now a natural question arises: what is the connectivity threshold for the market

graph? Since the number of edges in the market graph depends on the chosen

correlation threshold 0, we should find a value 00 that determines the connectivity of

the graph. As it was mentioned above, the smaller value of 0 we choose, the more

edges the market graph will have. So, if we decrease 0, after a certain point, the graph

will become connected. We have conducted a series of computational experiments for

checking the connectivity of the market graph using the breadth-first search technique,

and we obtained a relatively accurate approximation of the connectivity threshold:

00 0.14382. Moreover, we investigated the dependency of the size of the largest

connected component in the market graph w.r.t. 0. The corresponding plot is shown

on Figure 4-9.







87

4.2.3 Degree Distributions in the Market Graph

As it was shown in the previous section, the power-law model fairly well describes

some of the real-life massive graphs, such as the Web graph and the call graph. In

this subsection, we will show that the market graph also obeys the power-law model.

It should be noted that since we consider a set of market graphs, where each

graph corresponds to a certain value of 0, the degree distributions will be different

for each 0.

The results of our experiments turned out to be rather interesting.

If we specify a small value of the correlation threshold 0, such as 0 = 0, 0 = 0.05,

0 = 0.1, 0 = 0.15, the distribution of the degrees of the vertices is very i'_ y" and

does not have any well-defined structure. Note that for these values of 0 the market

graph is connected and has a high edge density. The market graph structure seems

to be very difficult to analyze in these cases.

However, the situation changes drastically if a higher correlation threshold is

chosen. As the edge density of the graph decreases, the degree distribution more and

more resembles a power law. In fact, for 0 > 0.2 this distribution is approximately

a straight line in the log-log scale, which is exactly the power law distribution, as it

was shown in Section 4.1. Figure 4-10 demonstrates the degree distributions of the

market graphs for some values of the correlation threshold.

An interesting observation is that the slope of the lines (which is equal to the

parameter 3 of the power-law model) is rather small. It can be seen from formula (4.1)

that in this case the graph will contain many vertices with a high degree. This fact

is important for the next subject of our interest finding maximum cliques in the

market graph. Intuitively, one can expect a large clique in a graph with a small value

of the parameter 3. As we will see next, this assumption is true for the market graph.

Another combinatorial optimization problem associated with the market graph is

finding maximum independent sets in the graphs with a negative correlation threshold





















4....


1 10 100
Degree


100


S10


1


1 10 100
Degree


1000 10000


*"^K- .



1 10 100
Degree


Figure 4-10: Degree distribution of the market graph for (a) 0 = 0.2; (b) 0

0 = 0.4; (d) 0 = 0.5


0.3; (c)


0. Clearly, instruments in an independent set will be negatively correlated with each


other, and therefore form a diversified portfolio.


However, we can consider a complementary wi,'li for a market graph with a


negative value of 0. In this graph, an edge will connect instruments i and j if the


correlation between them Cij < 0. Recall that a maximum independent set in the


initial graph is a maximum clique in the complementary graph, so the maximum


independent set problem can be reduced to the maximum clique problem in the


complementary graph.


Therefore, it is also useful to investigate the degree distributions of these


complementary graphs. As it can be seen from Figure 4-7, the distribution of the


correlation coefficients is almost symmetric around 0 = 0.05, so for the values of 0


S100-



E 10


-


1000
-

100



E 10
z


10000


1 10 100
Degree


10000


1000
-

100



E 10



1


1000 10000


~,:+1.*
"rst.t *


1


~
.