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Exact and Heuristic Approaches to Solving Sensor Placement, Routing, and Tracking Problems

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Title:
Exact and Heuristic Approaches to Solving Sensor Placement, Routing, and Tracking Problems
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Vogiatzis, Chrysafis
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[Gainesville, Fla.]
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
PARDALOS,PANAGOTE M
Committee Co-Chair:
GEUNES,JOSEPH PATRICK
Committee Members:
BOGINSKIY,VLADIMIR L
HAGER,WILLIAM WARD
Graduation Date:
8/9/2014

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Subjects / Keywords:
Algorithms ( jstor )
Cost allocation ( jstor )
Distance functions ( jstor )
Heuristics ( jstor )
Intelligent vehicles ( jstor )
Mathematics ( jstor )
Optimal solutions ( jstor )
Robotics ( jstor )
Sensors ( jstor )
Transportation ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
centrality -- cluster -- optimization -- routing -- sensors -- tracking
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

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Abstract:
In this thesis, the problem of information tracking, dissemination, and spread in sensor networks is studied. At first, an extensive literature review for sensor networks in transportation and logistics networks is presented, along with the major techniques of data collection and utilization. Then, the study proceeds to the well-known multi sensor multi-target tracking problem; a new graph partitioning scheme is presented that leads to exact and heuristic approaches for tackling the underlying data association problem. The computational results depict the success of these approaches and show that they are indeed viable alternatives to obtaining optimal and other, high quality solutions in a fast and efficient manner. The study also proceeds to propose extensions to the classical multidimensional assignment problem, from both the integer and graph theoretic viewpoint. In the last part of the thesis, the problems of information dissemination and routing are tackled. First, a novel problem that aims to detect highly centralized cohesive subgraphs is investigated. The complexity of the problems is derived, along with mathematical formulations to model the problems at hand. An interesting result that was obtained has to do with the ability to model the problems without having an exponential number of constraints, making the formulations compact and easy to use. Computational results show that, even though the hunt for the largest clique has been ongoing since decades ago, in most practical graphs (Erdos-Renyi, Barabasi-Albert, online social networks) it is much smaller cliques that can spread information the fastest. An extension to these centrality-based formulations is then presented, and (k,l)-Influential Cliques are investigated. Once more, the complexity of the problems is derived, along with a mathematical formulation. Further cuts are investigated that speed up the optimization phase, and a combinatorial branch-and-bound approach is proposed. In the numerical experiments, it is easy to derive that the combinatorial branch-and-bound is always faster, especially after applying pre-processing schemes to the original network. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: PARDALOS,PANAGOTE M.
Local:
Co-adviser: GEUNES,JOSEPH PATRICK.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-08-31
Statement of Responsibility:
by Chrysafis Vogiatzis.

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Embargo Date:
8/31/2016
Resource Identifier:
969977021 ( OCLC )
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EXACTANDHEURISTICAPPROACHESTOSOLVINGSENSORPLACEMENT,ROUTING,ANDTRACKINGPROBLEMSByCHRYSAFISVOGIATZISADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014ChrysasVogiatzis

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Idedicatethisdissertationtomyparents,RitsaandTasos,andmybrother,Panagiotis.

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisorandmentor,Dr.PanosM.Pardalos,forhisimmensehelpandsupportduringmyPh.D.studiesintheUniversityofFlorida.IwouldalsoliketoextendawarmthankyoutotheprofessorsandgraduatestudentsintheDepartmentofIndustrialandSystemsEngineering.ItisbecauseofyouandyoursupportthatIhaveenjoyedeverysecondofthelast4andahalfyears.Morespecically,IwouldliketothankDr.Geunesforgivingmetheopportunitytoteach,Dr.Smithforhissupportandacumen,Dr.Richard,andalltherestoftheprofessorsinthedepartment.Ofcourse,Ihavetoacknowledgemyfriendandcollaborator,JoseL.Walteros,forourcountlessconversations,researchideas,andallthetimewehavespenttogether.Last,butdenitelynotleast,Iwouldn'thavebeenabletonishmyPh.D.ifitwasn'tfortheunconditionalandconstantsupportofmyparentsandmybrother.Mum,Dad,andPanagiotis,thisoneisforyou.Thankyouforeverything. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 1.1SensorsandOptimization ........................... 11 1.2Background ................................... 14 1.2.1Multi-sensormulti-targettrackingproblem .............. 14 1.2.2Findingconnectedsubgraphswithmaximumcentrality ....... 15 1.2.3Inuentialclique ............................. 16 2SENSORSINTRANSPORTATIONANDLOGISTICSNETWORKS ...... 18 2.1SensorsinTransportation ........................... 18 2.2DatacollectionandStatisticalAnalyses .................... 19 2.2.1Overviewofsensormethods ...................... 19 2.2.2Vehicledetectionsensors ........................ 20 2.2.3Accidentdetectionthroughimageprocessing ............. 21 2.2.4Sensornetworksfortracmonitoring ................. 23 2.3VehicleRoutingandTracAssignment .................... 25 2.3.1Sensor-basedroboticvehicleroutingcomplexity ........... 26 2.3.2Intelligentvehicleroutingthroughacentralizedhighwaysystem .. 28 2.3.3Vehicleroutingandtracmonitoringusingpersonalsensors .... 35 2.4SmartphonesandNovelApproaches ...................... 38 3GRAPHPARTITIONSFORTHEMULTIDIMENSIONALASSIGNMENTPROBLEM ...................................... 39 3.1Preliminaries .................................. 39 3.2ElementDecomposition ............................ 43 3.2.1Twodisjointsubgraphsforelementpartitioning ........... 43 3.2.2Elementaugmentation ......................... 45 3.2.3Elementpartitioning .......................... 46 3.2.4Divideandconquer ........................... 47 3.2.5Preprocessing .............................. 48 3.2.6Discussion ................................ 49 3.3DimensionDecomposition ........................... 49 3.3.1Twodisjointsubgraphsfordimensionpartitioning .......... 49 5

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3.3.2Dimensionaugmentation ........................ 51 3.3.3Dimensionpartitioning ......................... 54 3.4AnalysisandaHybridMethod ........................ 56 3.4.1Analysis ................................. 56 3.4.2Hybridmethod ............................. 57 3.5ComputationalResults ............................. 58 4CLIQUECENTRALITY ............................... 66 4.1Preliminaries .................................. 66 4.2Denitions,Notations .............................. 69 4.3MixedIntegerProgramming(MIP)Formulations .............. 72 4.3.1DegreeCentrality ............................ 73 4.3.2ClosenessCentrality ........................... 74 4.3.3BetweennessCentrality ......................... 76 4.3.3.1Standard"Probabilistic"Case ................ 76 4.3.3.2PessimisticCase ....................... 78 4.3.3.3OptimisticCase ........................ 80 4.4ComputationalExperiments .......................... 81 5THEINFLUENTIALCLIQUEPROBLEM .................... 89 5.1Preliminaries .................................. 89 5.2Complexity ................................... 90 5.2.1(k;l)-InuentialClique ......................... 90 5.2.2(k;0)-InuentialClique ......................... 92 5.2.3(0;l)-InuentialClique ......................... 93 5.2.4Inapproximability ............................ 94 5.2.5Adierentcomplexityapproach .................... 94 5.3Solutionmethod ................................ 98 5.3.1Formulation ............................... 98 5.3.2Bounds .................................. 100 5.3.3CombinatorialBranch-and-Bound ................... 103 5.4Anotherformulation .............................. 105 5.5ComputationalResults ............................. 106 5.6Futurework ................................... 107 6CONCLUSION .................................... 109 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 121 6

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LISTOFTABLES Table page 2-1NotationforthemodelofColerietal. ....................... 24 2-2NotationforthemodelofSharmaetal. ...................... 27 2-3Notationusedintheautonomousvehiclerouting. ................ 30 3-1Worst-casetimecomplexityoftheapproachespresented. ............. 57 3-2PropertiesoftheAP(m;n)anditsdecompositionschemes. ............ 58 3-3Timespent(inseconds)computingcliquecosts. ................. 60 3-4Runtimesandoptimalitygapsfortheexactandheuristicapproachesinvestigated. 62 3-5Averageoptimalitygapsreportedifeachofthemethodswasallowedaruntimeofatmost18seconds. ................................ 63 3-6Verylarge-scaleinstances. .............................. 65 4-1Maximumandminimumcliquecentralitiesinreal-lifesocialandpowergridnetworkinstances. ....................................... 84 4-2Computationaltimes(inseconds)forsolvingthemaximumandminimumcliquecentralityproblemsinreal-lifesocialandpowergridnetworkinstancesfromTable 4-1 . ....................................... 85 4-3Maximumandminimumcliquecentralitiesinbookgraphs. ............ 86 4-4MaximumandminimumcliquecentralitiesinrandomlygeneratednetworkinstancesaccordingtoErdos-RenyiG(n;p)preferentialattachmentsmodel. ........ 87 4-5MaximumandminimumcliquecentralitiesinrandomlygeneratednetworkinstancesaccordingtoBarabasi-Albertpreferentialattachmentsmodel. .......... 88 5-1Computationalresultsfordierentvaluesofkandl. ............... 107 7

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LISTOFFIGURES Figure page 1-1Sensorobservationsandtheirmatching. ...................... 14 3-1ElementPartitioning ................................. 44 3-2DimensionPartitioning ................................ 50 4-1AnillustrativeexamplethatcomputingindividualnodebetweennesscentralitiesisnotenoughtocomputeCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((S). .......................... 72 5-1Dierencebetweena(k;l)-inuentialcliqueandadominatingclique. ...... 90 5-2Thegadgetofthereductionfork=3. ....................... 91 5-3ExampleofthereductionfromGto^Gfor(k;0){INFLUENTIALCLIQUE. .. 92 5-4ExampleofthereductionfromGto^Gforagraphwith4nodes. ........ 93 5-5ThegadgetforthereductionfromaninstanceofMAXSAT. ........... 96 5-6TheconstructedgraphfromtheinstanceofMAXSATwith2literals(x1andx2)and4clausesC1=x1_x2;C2=x1_:x2;C3=:x1_x2;C4=:x1_:x2. 97 5-7AnexampleofhowthedecreaseofthesizeofN(C)canbeasbigasjCj. ... 101 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXACTANDHEURISTICAPPROACHESTOSOLVINGSENSORPLACEMENT,ROUTING,ANDTRACKINGPROBLEMSByChrysasVogiatzisAugust2014Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineeringInthisdissertation,theproblemofinformationtracking,dissemination,andspreadinsensornetworksisstudied.Atrst,anextensiveliteraturereviewforsensornetworksintransportationandlogisticsnetworksispresented,alongwiththemajortechniquesofdatacollectionandutilization.Then,thestudyproceedstothewell-knownmulti-sensormulti-targettrackingproblem;anewgraphpartitioningschemeispresentedthatleadstoexactandheuristicapproachesfortacklingtheunderlyingdataassociationproblem.Thecomputationalresultsdepictthesuccessoftheseapproachesandshowthattheyareindeedviablealternativestoobtainingoptimalandother,highqualitysolutionsinafastandecientmanner.Thestudyalsoproceedstoproposeextensionstotheclassicalmultidimensionalassignmentproblem,fromboththeintegerandgraphtheoreticviewpoint.Inthelastpartofthedissertation,theproblemsofinformationdisseminationandroutingaretackled.First,anovelproblemthataimstodetecthighlycentralizedcohesivesubgraphsisinvestigated.Thecomplexityoftheproblemsisderived,alongwithmathematicalformulationstomodeltheproblemsathand.Aninterestingresultthatwasobtainedhastodowiththeabilitytomodeltheproblemswithouthavinganexponentialnumberofconstraints,makingtheformulationscompactandeasytouse.Computationalresultsshowthat,eventhoughthehuntforthelargestcliquehasbeenongoingsincedecadesago,inmostpracticalgraphs(Erdos-Renyi,Barabasi-Albert,online 9

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socialnetworks)itismuchsmallercliquesthatcanspreadinformationthefastest.Anextensiontothesecentrality-basedformulationsisthenpresented,and(k;l)-InuentialCliquesareinvestigated.Oncemore,thecomplexityoftheproblemsisderived,alongwithamathematicalformulation.Furthercutsareinvestigatedthatspeeduptheoptimizationphase,andacombinatorialbranch-and-boundapproachisproposed.Inthenumericalexperiments,itiseasytoderivethatthecombinatorialbranch-and-boundisalwaysfaster,especiallyafterapplyingpre-processingschemestotheoriginalnetwork. 10

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CHAPTER1INTRODUCTION 1.1SensorsandOptimizationNetworksandtheirtheoryhavebeenontheforefrontofresearchandpracticefordecadesnow,astheyentailingredientsfromnumerouselds(discretemathematics,computerscience,mathematicalprograming)andcanbeusedtomodelseveralapplications.Theseapplicationscanbeasdiverseasevacuationplanningandsensorlocationtoonlinesocialnetworkmonitoringandassignmentproblems.ForanoverviewofnetworkapplicationswerefertheinterestedreadertotheseminalworkofPhillipsandGarcia-Diaz[ 103 ]and,morerecently,thecollectionofworkspresentedinGoldengorinetal.[ 61 ].Further,recenttechnologicaladvancementshavebeenthedrivingforcebehindadierentever-growingresearcheld:sensorusetocollect,analyze,monitor,anddiuseinformationandknowledge.Ingeneral,asensorisadevicethatrespondstoexternalevents.Its\response"canbedierent,basedonitstypeanduse.However,oftensensorsareemployedtomonitoreventsandcollectdata.Inthiswork,Iwillinterchangeablyrefertosensorsasmonitorswhenevertheyareusedtotrackinformation.Theseadvancementshaveenabledsensingdevicestotransmit/storeinformationandperformcalculations.Thisrapidimprovementofthetechnicalcharacteristicsofsensorshavemadethemprimecandidatesforaseriesofapplications,likethemutli-sensormulti-targettrackingproblem[ 8 , 106 ],areasurveillance,telecommunications,andtransportationandlogistics[ 120 ],amongothers.Foranoverviewofdierentsensorapplicationsandthestate-of-the-artmethodstheinterestedreaderisreferredto[ 19 ].Herein,thefocuswillbeonasetofsensorsthatareinterconnected.Werefertosuchanetworkasasensornetwork.Sensornetworksarecategorizedasstaticordynamic.Instaticsensornetworks,locationandplacementareoftheessence,sincetheyneedtobechoseninadvance.Hence,arobustandecientsensorplacementisvitalinordertoensureconnectivityduringthelifespanofthesensors. 11

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Propernetworkandgraphtheoreticoperationsarecrucialtoguaranteethatthequalityandlifetimeofasensornetworkbemaximized.Intheliterature,twoparamountnotionsarecoverageandconnectivity(robustness).Ingeneral,anecientsensornetworkdesignshouldbebothwell-connectedandmaximizeareasurveillance,evenundersystemfailureswhichcanbeduetoanadversary,sensorfaults,orqualityofcommunicationservice.Inthisdissertation,threedistinctproblemsarestudied:multi-sensormulti-targettrackingproblems(dataextractionandtransformationtoknowledge);communicationprotocolsbasedoncentralitymetrics(networkrobustness/design);andinuentialcardinality-constrainedsubgraphdetection(informationdispersion,ecientsensorplacement).Themulti-sensormulit-targettrackingproblemisawell-knownoptimizationproblemwhereasetofsensorsSisdeployedtotrackasetoftargetsT,resultinginaseriesofmeasurementsjSjjTj,thatarethentobeassignedtoeachotherinordertominimizedatadissimilarity.Inthissetting,themulti-sensormulti-targettrackingproblemisadataassociationproblem,andisoftenmodeledasaMultidimensionalAssignmentProblem(MAP)[ 104 ].Secondly,communicationprotocolsanddataroutinghasbeenawell-studiedprobleminsensorliteratureforyears[ 4 , 48 ].Routinginwirelesssensornetworksusuallyinvolvestwooperations,namelydatacollectionanddissemination.Theseprotocolsoftenusetree-basedrouting[ 48 ].Insuchprotocols,theuseoftopologicalfeaturestodesigntransmissionpatternsacrosssensors(nodes)hasbeenproposedin[ 94 ].Theinsightbehindtheseprotocolsisthathigherstructuralvalueisoftenassociatedwithcentralsensors,ratherthansensorsthatarefoundclosetotheboundary.Further,theimportance/relevanceofasensorgrowsproportionallywithitsparticipationinpaths[ 34 ].Thedenitionofcentralityfornodesisextendedtoconnectedclustersofrestricteddiameter.Thereasonfordoingsoistwo-fold:rst,therestricteddiameterensuresthat 12

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communicationofdataandpotentialre-routingcanoccurfaster,andsecond,usingmultiplenodesinsteadofsingletonsincreasesthecentralitymetricsignicantly.Last,datadiusionisanindispensableprotocolcomponent,whichisoftenmotivatedbyrobustness,scalingandenergyeciency[ 47 ].Whenallnodesneedtoreceiveinformation/queriesfromauthorizedexternalusers(i.e.,reportanymovement),severalnodesarefedthequery,andthentheinformationisdisseminated,usingthenetworkstructure.Mostemployedprotocolsfordatadiusionincludeooding[ 69 , 70 ],randomwalkapproaches[ 23 , 115 ]andinformationgradientdiusion[ 32 ].Thelattertakesadvantageofdomain-specicknowledge.Ananalysisoftheperformanceofgradient-basedroutingprotocolscanbefoundin[ 47 ].Intheapproachpresentedherein,theaimistodetectmaximum\inuence"connectedclustersinsensornetworks.Suchclustersareimportantbecausetheycanbeusedtofeedinformationintothesystemandexpectitsdispersiontobefast.Westartbyinvestigatinginuentialcliques,theircomplexityalongwithsomespecialcaseresults,andtheirmathematicalformulations.Wethenproceedtoproposeacombinatorialbranchandboundandageneralrandomizedadaptivesearchprocedure(GRASP)tosolvetheproblemexactlyandheuristically.Thisdissertationisoutlinedasfollows.Chapter 1 introducestheproblemsathandandbrieyoersabackgroundstudy.Then,inChapter 2 ,anoverviewofsensorsintransportationandlogisticsnetworksispresented.Chapter 3 introducesnoveldecompositionschemesforthemultidimensionalassignmentproblem.InChapter 4 ,Iproposecentrality-basedroutingschemesforinformationdispersionthroughasensornetwork.Inthatpart,Ialsogeneralizecentralitymetricstoconnectedclusters.Furthermore,Chapter 5 generalizesthenotionofdegreecentralityandintroducesthenovelproblemofdetecting(k;l)-inuentialcliquesinagraph.Last,Chapter 6 concludesthisdissertationandprovidesinsightinpossiblefuturework. 13

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Figure1-1. Sensorobservationsandtheirmatching. 1.2Background 1.2.1Multi-sensormulti-targettrackingproblemDataassociationisafundamentaloptimizationproblemthataimstomatchobservationswiththeleastdissimilarity.Themulti-sensormulti-targettrackingproblemisaspecialcase,wheresensorobservationsarematchedaccordingtotheirorigin[ 31 ].Figure 1-1 oersanexampleofsuchsensorobservationsandtheirpossiblematching.Inthatgure,thenodesinthegrapharegivenbyapair(target,sensor),andhence,target1wassensedtobeinthepositionofnode(1,1)bysensor1,inthepositionofnode(1,2)bysensor2,andsoon.Thematchingpresentediscalledacliquecostmatchingandiswidelyusedinthemulti-sensormulti-targettrackingproblemframework.Themulti-sensormulti-targettrackingproblemisusuallymodeledasamultidimensionalassignmentproblem(MAP)[ 106 ].TheMAPcanbewrittenasaninteger(0-1)program(showninFormulation 1{1 throughout 1{5 ). 14

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minmXi1=1mXi2=1mXin=1ci1i2:::inxi1i2:::in (1{1)s:t:mXi2=1mXi3=1mXin=1xi1i2:::in=1;8i1=1:::m (1{2)mXi1=1mXij)]TJ /F8 5.978 Tf 5.75 0 Td[(1=1mXij+1=1mXin=1xi1i2:::in=1;8ij=1:::m;8j=2;:::;n)]TJ /F1 11.955 Tf 11.95 0 Td[(1 (1{3)mXi1=1mXi2=1mXin)]TJ /F8 5.978 Tf 5.76 0 Td[(1=1xi1i2:::in=1;8in=1;:::;m (1{4)xi1i2:::in2f0;1g;8i1=1;:::;m;:::;in=1;:::;m: (1{5)Overtheyears,aseriesofmethodshavebeenproposedtosolvetheMAP.GiventheinherentNP-hardnessoftheproblem,heuristicmethodshaveattractedmostpracticalinterest.MostnotableamongthemaretheLagrangianrelaxationproposedbyPooreetal.[ 108 ],whichyieldedhighqualityresults,theapproximationalgorithmsofSpieksma[ 117 ],GRASPwithpathrelinking[ 3 ],andrandomizedapproachessuchastheonepresentedbyOliveiraandPardalos[ 93 ].Herein,wewillconcentrateonthemultidimensionalassignmentproblemwithdecomposablecosts[ 7 ](clique,andstarcostsnamely),wherethecostofassignmentcanbecalculatedasafunctionofthepairwiseelementassignments,i.e.,ci1i2:::in=f(ci1i2;:::;cinin)]TJ /F8 5.978 Tf 5.75 0 Td[(1). 1.2.2FindingconnectedsubgraphswithmaximumcentralityCentralityisaveryimportantpropertyofanodeinanygraph,sinceitgivesusanunderstandingofhowcentralthatnodeis.Ithasbeenextensivelystudiedintheliteraturefrombothatheoreticalandexperimentalstandpoint,startingfromtheseminalpapersofBavelas[ 15 ],Leavitt[ 85 ],andSabidussi[ 114 ]sincethe1950s.However,centralityhasingeneralbeenappliedtonodes,ratherthansetsofnodes.Inthiswork,weareextendingthedenitionstoconnectedclusters.Themostused 15

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centralitynotionsinliteraturearethefollowing:degree(ndanodewithmaximumdegree);closeness(ndanodewhichisclosesttoeveryothernode);betweenness(ndanodethatisusedbyamaximumnumberofshortestpaths);Katz(extensionofdegreecentralitytoincludepathconnectednodes);eigenvector(extensionofdegreecentralitytoincludehigh-scoringandlow-scoringnoderelations).Morerecently,centralityindiceshavebeenstudiedinasurveyby[ 77 ].Inthesamebook,Jacobetal.presentthestate-of-the-artforderivingcentrality[ 71 ].Asismentionedtherein,computationaltimeisoftheessencewhencomputingcentralityofanode.However,applyingtheirdenitionstraighttothenodemightprovetobeanaiveandinecientapproach.In[ 24 ],theauthorsshowthatthebetweennesscentralityofanodecanbecomputedinO(nm+n2logn),basedoncomputingashortestpathstreeforeverynodei2V,whichcanbeprecomputed.Theyalsoshowthatasimilarcalculationleadstothesametimeboundforclosenesscentralityinthesamepaper.Inthiswork,Iproceedtoextendthedenitionsofcentralitytocliques,derivethecomplexityoftheproblems,andproposemathematicalformulationstotacklethem. 1.2.3InuentialcliqueInthispart,weinvestigatetheproblemofdetectinginuentialcliques,i.e.cliquesthatsatisfyinuence(connectivity)constraints.Theproblemarisesinthecontextofsensorplacement,whereasetofsensorsneedstobedeployedinanetworktoobserveatleastknodes.If,inaddition,thesensorsarerequiredtocommunicatewitheachother(i.e.,theyneedtobeplacedcloseenough),thenthesubsetsoughtonthegraphisaclique.Observethattherearetwoproblemsinvolvedwiththisframework.Therstoneseekstomaximizethesizeofthecliquesubjecttoaninuenceconstraint,whilethelatterdoestheconverse:maximizesthenumberofinuenced(adjacenttotheclique)nodes,subjecttoaconstraintonthesizeoftheclique.Bothproblemswillbestudiedherein. 16

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Thisworkisanaturalextensionoftherecentresearchontheeldsofconnectedmaximumcoverage,andconnecteddomination.Itiscloselyrelatedtothedominatingcliqueproblem[ 35 ]onagraphG(V;E),whenweseekasetofnodesDV,whichformadominatingsetandacliqueatthesametime.Inthiswork,thecliqueneedsnotbedominating,solongasitisadjacenttoatleastknodesoutsidetheclique.Boththeminimum,andthemaximumversionsofthedominatingcliqueproblemareknowntobeNP-hardoptimizationproblems[ 79 ],thatcanbefoundecientlyinchordalgraphs[ 78 ].Anexactalgorithmfortheminimumdominatingcliquecanbefoundin[ 80 ].Otherrelatedproblemsstemfromtheeldofconnecteddominatingsets.Arecentpublicationontheeldrelaxesthecliquerequirementtoak-club,aconnectedstructureofadiameterthatislessthanorequaltok[ 25 ].Withintheeldofsensorplacement,researchhasbeenfocusedondetectingrobustconnecteddominatingandmaximumcoveragesets.TheinterestedreaderisreferredtotheworksofZhengetal.[ 124 ],Lietal.[ 86 ],andZhouetal.[ 125 ]. 17

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CHAPTER2SENSORSINTRANSPORTATIONANDLOGISTICSNETWORKS 2.1SensorsinTransportationInthelastdecades,theeconomicsoftransportationhavebecomeofincreasingimportanceforcorporations,municipalitiesandcommuters.Duetothelargecongestionlevelseverydayusershavetoface,beingabletondevenanapproximateshortestpathisvital.However,theanalysesthatprovideduswithveryusefulobservationsandinsightwouldneverbepossibleifitwerenotforanautomatedsystemofdatacollectionusedaroundthetransportationnetworks.Sincethe1950s,theneedfordatacollectiontoobservethelevelsofcongestionortheconditionoflinksinthenetworkhasappeared,asitcanbenotedfromtheBureauofTransportationStatistics( http://www.rita.dot.gov/bts/sites/rita.dot.gov.bts/files/publications/national_transportation_statistics/index.html ).Researchersinstalledsimpleorsophisticatedsensorsinordertomeasurethevolumeofthetracowonspeciclinks,theaveragevehiclevelocityandthetimerequiredtotraverseastreetamongother.Nowadays,themajorityofthevehiclesusingthenetworkareequippedwithroutingguidance(usuallyGPSdevices).Inadditiontothat,theincreasedwirelesscapacitiesofcellphonesandPDAdevicesinthelastyearshavemadetrackingandwirelesssensingacommoneverydayphenomenon.Theattemptstocombinetheseincreasedcapabilitieswithpracticaltransportationproblemsaregoingtobedescribedindetailinthefollowingsections.Moreover,itisimportanttonotethatespeciallyintheUnitedStates,theappearanceofhighcongestionlevelsisnoticedmoreoftenthanever.Thiscomesasaresultofthelargenumberofcommutersthatusethetracnetworkonaneverydaybasis[ 95 ].FromthereportsoftheBureauofTransportationStatistics( http://www.rita.dot.gov/bts/sites/rita.dot.gov.bts/files/publications/national_transportation_ 18

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statistics/index.html ),itcanalsobenotedthatpublictransportationusershaveshrunktoinsignicantlevels,whileautomobilecommutershaveneverceasedtoincreasesince1950.ItcanbeseenintheworkofO'Toole[ 95 ]thatthelastfewyears,averagevehiclespeedisrapidlydecreasing,whileenergyconsumptionandCO2levelsareincreasing.Itbecomesevidentthatitisnowmorethaneverthatoptimalroutingandtracassignmenttechniquesneedtobeapplied.Thechapterisoutlinedasfollows.Inthenextsection,apresentationofsensor-basedmethodsforcollectingusefuldataforanalysisintheeldsoftransportationwillbegiven.Section 2.3 willthenpresentthemostrecentadvancementsintheinfamousproblemsofvehicleroutingandtracassignment.Inthatsection,specialattentionwillbegiventomodernGPSsystemsandintelligentvehiclesensors.Last,insection 2.4 ,theconclusionsoftheauthoralongwithfutureresearchpossibilitieswillbepresented. 2.2DatacollectionandStatisticalAnalysesAmongthemostimportantandvitalongoingresearchinanytransportationsystemisthedevelopmentandimplementationofIntelligentTransportationSystems.Systemsofthiskindcanbeusedtoreducethehumanfactorasfarastracassignmentandvehicleroutingareconcerned.Inadditiontothat,optimizationoftracowinintersectionswillalsoresultinlesscongestionandaccidents[ 72 ]. 2.2.1OverviewofsensormethodsFirstofall,letusfocusonthetrackingmethodsthathavebeenputtopracticeinthelastdecade.Thesealgorithmshaveextensivelyinvestigatedcomputervisionasameansofvehicletracking.AseminalcontributionintheeldoftracmonitoringhasbeenpresentedbyPeterfreund[ 102 ]in2002.Theauthorfocusesonthesnakesactivecontourmodels,introducedbyKass,WitkinandTerzopoulos[ 74 ]andproposesastochasticvelocitysnakemodelinordertotrackvehiclesinintersectionsandtraclights.EquallyusedisthemethodofGardnerandLawton[ 55 ]whichiscommonintracimages. 19

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However,alltheabovemethodsarediculttoputintopracticewhen,becauseofcongestion,thereisalargenumberofvehiclesatagivenintersectionatanytime.ThatistheinsightbehindtheKamijoetal.approach[ 72 ]whereeachvehicleistrackedindividuallyevenifinthesecongestionlevels,certainvehiclesmightbenotcompletelywithinthesensor/camerarange.Thisphenomenonisreferredtoasocclusionandhasbeenamajorobstacleintracmonitoring. 2.2.2VehicledetectionsensorsFirstofall,letusstartbyprovidingthereaderwithadetailedpresentationofthemajorsensorsthatarecurrentlyavailableforvehicledetection,astheywerenotedbyLuzandMimbella[ 90 ]. InductiveLoop:Oneoftheintrusivesensors,mostusuallyinstalledonthepavementsurface.Theycanalsobeinstalledunderneaththemonitoredroadbytunnelingunderthesurfaceofthestreet.Eventhoughtheyrepresentatechnologythatisgreatlyunderstoodandveryaccurate,theirinstallationandmaintenanceishardsinceitrequiresthedisruptionofowforabigtime. Magnetometer:Similartotheinductiveloop,amagnetometerisaexiblesensorthatcanbeusedforavarietyofpurposes.Unfortunately,itispronetoerroneousmeasurementswhenthetracisheavyand,hence,ithasbecomeunattractive.Inadditiontothat,anotherdrawbackthatamagnetometerinheritsfromaninductiveloopisthatitinvolvesahugecostofinstallationandmaintenance. ActiveandPassiveInfrared:Activeandpassiveinfraredsensorscanbeusedincombinationsinordertocoverallmeasurementsofthevehiclesonagivenroadsegment.Anactiveinfraredsensorcanmeasureaccuratelythevehiclepositionandclass,whileitcanalsoprovideuswithanestimateonthespeed.Apassive,ontheotherhand,canbeusedforspeedmeasurements.Bothcategoriesofinfraredsensorsshareacommondisadvantage:theyaresusceptibletofalsemeasurementswhenvisibilityislimited(i.e.,lessthan20feet)becauseofweatherconditions. Ultrasonic:Ultrasonicpermitmultiplelanemonitoringatthesametime.Thisisthemainadvantagethatmakesthemattractivetouse.However,theypresentproblematicbehaviorwhenfunctioningundersuddentemperaturechangesandairturbulences. Acoustic: 20

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Similarlytoultrasonicsensors,forapparentreasons,acousticsensorscanalsomonitormorethanonelanesatatimewithaccuracy.Theirmajorproblemshavetodowithcoldtemperaturesandspecictracpatterns.Forexampletheyarenotrecommendedforuseinlargecitycrossroadswherethephenomenonofstopandgotracisusual. Videoimageprocessing:Avideoimageprocessingsensoris,inmostcases,theidealcandidate.Providesmonitoringforasmanylanesasthevideoimagecapturercantakecareofanditiseasytoinstallandmaintainwithoutdisruptingthenormalvehicularow.Amajordrawbackistheincreasingcomplexityatdecodingthesensormeasurementsintoactualdatathatcanbeused.Also,asfarasrequirementsfornormalinstallationareconcerned,itneedstobeensuredthattheyareinstalledatahighpoint,usually50-60feetabovethemonitoredroad. 2.2.3AccidentdetectionthroughimageprocessingThefundamentalideaofKamijoetal.[ 72 ]istheapplicationofastochasticrelaxationalgorithminthedetectionprocess.Thisisvitalforthepractice,sincevehicleshaveaseriesofverydistinctcharacteristics,includingbutnotlimitedtotheirappearance,theirdirectionandtheircolor.Giventhatinanintersection,congestionlevelsarehigherandocclusionoccursoften,itisimpracticaltoadoptasimplecontourmethodfortracking.Forthemodelingpurposesoftheiralgorithm,thetrackingofasinglevehicleistransformedintoalabelingmethod,whereeachpixelisassigned(i.e.,labelled)toaspecicvehicleatanytime.Afterallpixelshavebeenlabelled,itispartforthedeductivemethodtobeappliedinordertoobtainaninitial,butaccurate,objectmapping.Therearevemajorcomponentstothedeductiveprocess: Initialization:Thebackgroundimageissetbyusinga20minuteimagesequence.Also,theentrancepointstotheintersectionaresetbydeningslitswherenewintensitiesoftheimage(i.e.incomingtrac)appear. Generationofnewvehicles:Wheneveralongtheslitsdened,anewintensityisobserved,anewIDisassignedtotheincomingvehicle.AllthepixelssharingthatintensityareassignedthesameID. Vehiclevectorestimation: 21

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Ateachblockoftime,thesimilarityofthevehiclemotionisestimatedby:(x(t+1);y(t+1))=(x(t)+u(t);y(t)+v(t)) (2{1)andthemotionvectorisapproximatedasthemostfrequentmotionvectorofallpixelsassignedwiththesamelabel,D=X0di8;0dj8jI(i+di+ui;j+dj+vj;t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(I(i+di;j+dj;t)j: (2{2)Inthiscase,I(x;y;t)istheintensityofpixel(x;y)attimet. Vehicleregionupdate:Allthevehicleblocksareupdatedattimet+1comparedtotimetasperthemotionvectorobtainedinthepreviousstep.Ifthenewintensitydierenceissmallerthanathresholdthenthevehicleisconsideredtobeoutoftheintersection.Inthecasewherethedierenceisbigger,thenitisassumedthataneighboringvehicleishidingapartofthevehicletracked. Vehicleblocksdivision:Attheentrancepointofthecamera(theslitsdenedduringinitialization),simultaneousarrivalsmayresultincharacterizingmultiplevehiclesasone.Thisisthereasonwhyitischeckedalwaysifthemotionvectorsobtainedbyalabelledvehiclematch.Ifthatisnotthecasethenthepixelscorrespondingtothesamelabelaredividedinordertodistinguishbetweendierentvehicles.Thelastpartofthismethodologyconsistsofthestochasticrelaxationinordertoassigncertainpixelstomorethanonevehicles,becauseofocclusion.TheauthorsthenextendtheMRFmodeltoincludeanddealwithnotonlyimagesasintheworkbyGemanandGeman[ 57 ]andbyAndreyandTarroux[ 5 ],butalsotime-axisdistribution.Theirspatio-temporalMRFmodelprovidesanestimationofthecurrentobjectmapbasedontheinformationprovidedinthepreviousobjectmapalongwiththecurrentandpreviousimages.Afterapplyingthestochasticrelaxationproceduretotheirmethodology,thetracmonitoringsystemfordetectingaccidentsisreadytobeexaminedandtestedwiththeresultsbeingsuccessfulinaspecicintersection,butcanbegeneralizedtoeachtopologyandgeometry[ 72 ]. 22

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2.2.4SensornetworksfortracmonitoringBeforetherecentadvancementsinimageprocessinganddataextraction,theuseofinductiveloopdetectorswasandstillremainsthemostcommonwaytocollectinformationontracconditionsmainlybecauseoftheirhighlevelsofaccuracyandreliability.However,theirinstallationandtheirmaintenancerequiresasignicantdown-timeofthearcbeingupdatedandhence,theybecomeexceedinglyexpensive.Thatisthemainreasonwhyinthelastyears,moresophisticatedsensorsarebeingused,suchassurveillancecameras,microwaveradars,ultrasoundandinfraredsensors.However,thesesensorsarelessreliable,eventhoughthemethodshavebecomemoreinvolvedovertheyears,andtheyalsoarecostly.Therefore,numerousscienticeortshavebeenmadeinordertoincorporatecheaper,wirelesssensorsintheexistinginfrastructureforstatisticalandmonitoringpurposes.TheapproachesthatwillbefocuseduponinthissubsectionaretheonesbyColerietal.[ 33 , 43 ].IntheworkofColerietal.[ 33 ],theTrac-Dotsensormodelconsistsofthefollowingmajorcomponents: processor, radio, magnetometer, battery.Themagnetometer(magneticsensor)isusedforvehicledetection.Thepowerconsumptionofsuchasensorbasedmodelisverysmall,makingitanidealcandidatefortracmeasurementsanddatacollectionforstatisticalanalysis.Itsaccuracyisalsoveryhigh,reachingthe97%accuracythatisachievedbyinductiveloopdetectors.Inordertoincreaseeciencyandbatteryutilizationtheauthorshaveproposedthefollowinglinearprogrammingmodel[ 43 ],thatisdescribedinFormulation 2{3 throughout 2{9 ,whilethenotationisprovidedinTable 2-1 . 23

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Table2-1. NotationforthemodelofColerietal. NotationDenition G=(V;E)Thegraphrepresentingthesensornetwork,withV=f1g[Vs[Vr.Node1istheaccesspoint,Vs=[2;N]arethesensornodesandVr=[N+1;M]therelaynodes.Ifnodesiandjarewithintransmissionrangethen(i;j)2E.giRateofpacketsperunittimethatnodei2[1;N]cantransmit.psEnergyspentinsensorwhenobtainingpacketsinonepacket.ptx;ijEnergyspentforthetransmissionofapacketfromnodeitonodejpertimeunitfijAveragetimerequiredtoreceivepacketsatnodejfromnodei.eiThebatteryenergyofeachofthepartsofthenodesi2[1;M]. minMXi=1ei (2{3)s:t:Xjfij)]TJ /F7 11.955 Tf 11.96 11.36 Td[(Xjfji=gi;8i2[2;N] (2{4)td(Xjptx;ijfij+Xjprxfji+psgi)ei;8i2[2;N] (2{5)Xjfij)]TJ /F7 11.955 Tf 11.96 11.36 Td[(Xjfji=0;8i2[N+1;M] (2{6)td(Xjptx;ijfij+Xjprxfji)ei;8i2[N+1;M] (2{7)fij0;8i;j2[1;M] (2{8)ei0;8i2[1;M]: (2{9)Theabovelinearprogramhasthelimitationthatitconstrainstherelaysensorstobeinaxedpositioninthenetwork.Inreality,itisdesiredtobeabletodeterminetheoptimalplacementoftherelaynodes,thusalteringdynamicallythetopologyofthenetwork.TheresultingformulationispresentedinFormulation 2{10 throughout 2{18 . 24

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minMXi=1ei (2{10)s:t:Xjfij)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xjfji=gi;8i2[2;N] (2{11)td(Xjptx;ijfij+Xjprxfji+psgi)ei;8i2[2;N] (2{12)Xjfij)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xjfji=0;8i2[N+1;M] (2{13)td(Xjptx;ijfij+Xjprxfji)ei;8i2[N+1;M] (2{14)ptx;ij=ptx(d(i;j));8i;j2[1;M] (2{15)d(i;j)2=jli)]TJ /F3 11.955 Tf 11.95 0 Td[(ljj2;8i;j2[1;M] (2{16)fij0;8i;j2[1;M] (2{17)ei0;8i2[1;M]: (2{18)However,theresultingformulationascanbeseenbythereaderisanonlinear,nonconvexproblemandhence,theauthorsproposeanapproximatealgorithmforitssolution.Theirproposedalgorithmisthentestedthroughsimulationwithremarkableresults.Theirnovelapproachinenergymanagementforsensorsinawirelessnetworkcanresultinamoreecientmethodofdatacollectionintracengineering,ascanbeseenbytheirworkattheTrac-Dot[ 33 ]. 2.3VehicleRoutingandTracAssignmentVehicleroutingandtracassignmentproblemsaretwoofthemostimportantproblemsintransportationengineeringandplanning.Manyattemptstosolvetheminadynamicallychangingenvironmentsuchasthemodernurbangridhaveappearedinliterature,howeveritisinthelastfewyearswiththeboomofsmartphonesandsensorsthatthereexistthetoolstotacklethemsuccessfully. 25

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Onlinealgorithms[ 73 ]userecentinformationasfeedbacktoaltertheirsolutionsinordertoremainoptimalateveryinstance.Forthesealgorithmicapproachestofunctioneectivelyreliableinformationneedstobeprovidedinafastandecientway.Withtherecentincreaseofguidancedevices,suchasGPS,andcheapwirelesssensors,thepossibilitytoobtainthesedataappeared. 2.3.1Sensor-basedroboticvehicleroutingcomplexityBeforegeneralizingthevehicleroutingproblemtorealistic,practicalapplicationsinvolvingdecisionmakersintheurbanenvironment,itisimportanttoreviewtheroboticvehiclesensorbasedrouting.Thisisacommonapplicationinautomatedcontrolsystemsinindustry[ 26 , 39 , 111 ]and,thus,therehavebeenattemptstomodelandoptimizetheirbehavior.Theresultsofthisscienticresearchhasprovidedinsighttoresearchersasfarasonlineguidancethroughintelligentautomobilesystemsequippedwithsensorsareconcerned.Usually,inroboticapplications,insteadofanoptimalshortestpathforallthevehiclesinvolved,researchersareinterestedinlimitingtheselectiontoasetofavailablepathsandselectingthebestamongthose.ThisapproachhasbeenstudiedbyInalhanetal.[ 68 ]wherexedroutesweregiventothevehiclesandbyGerkeyetal.[ 59 ],whereaxedroadmapwitharcsandnodeswasemployed.Theaboveapproaches,eventhoughtheyarepracticalandpresentgoodresults,arenottheoreticallyguaranteeingoptimality.So,thatledSharmaetal.[ 116 ]toresearchthetimecomplexityofthissensor-basedvehicleroutingproblemwithnolimitationsontheselectionofroutestotheagentsutilizingthenetwork.Fortheirwork,previousresearchoncommunicationbetweenroboticsystems[ 75 ].Thenotationforthesetupoftheproblemdiscussedin[ 116 ]ispresentedinTable 2-2 .TheexclusionzoneCisadiskofanonconstantradiuscenteredatthepositionoftheagent,wheretherecanbenoothervehicle.Ifthereisanothervehicleatthesametime,thenaconictissaidtoappearandnewroutinghastooccur.Thediskisdenedin 26

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Table2-2. NotationforthemodelofSharmaetal. NotationDenition QThesquareenvironmentwhereagentsareallowedtomoveofareaA.(O;D)iTheorigin-destinationpairofagenti.t0;iThetimewhenanagentisdispatchedinthenetwork,t0;i0.TiThetimeanagentrequiresinthenetworkuntilitreachesitsdestination.iThetime-dependentpaththatagentiisfollowing,:[0;Ti]!Q.vi(t)Thevelocityofagenti,vi(t)vmax.Ci(t)TheexclusionzoneCofanagentattimet. Equation 2{19 .Asitcanbeseen,theradiusisdependentonthevelocityoftheagentiatthattimet.Morespecically,aconictoccursitthereexistsatimetcsuchthat: agentsiandjareactiveattimetcand Ci(tc)\Cj(tC)6=;.Ci(t)=fz2R2:jjz)]TJ /F3 11.955 Tf 11.96 0 Td[(xi(t)jjr0+kjjvi(t)jjg (2{19)Theobjectiveofthesensorbasedvehicleroutingproblemwithroboticagentsistondanoptimalroutingpolicy.Assuchwedeneamapping:(O;D)!(t0;T;)whichissafe,thatisnoconictoccursatanytime.TheauthorsthendeneTtobethetimewhenallagentshavesafelyarrivedattheirdestinationsaccordingtopolicy.Then,thetimecomplexityisdenedasT(O;D)=infsafeT(O;D):Thisformulationleadstointerestingtheoreticalresultswhenitcomestotheupperandlowerboundsinthetimecomplexityoftheproblem.Therearetwocasesthatwereexamined: Bestcasescenario. 27

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Averagecasescenario.Inthebestcasescenario,the(O;D)ipairsareselectedsoastominimizethetotaltimerequiredforallagentstoremainintheenvironment.Theauthorsgoaheadandprovethefollowinglemmainthatcase: Lemma1. Foranysetofn(O;D)pairs,suchthattheaveragedistancebetweenoriginanddestinationpointsisL,thetimecomplexityoftheproblemis(p nL).Theyalsoproceedtoprovethefollowinglemmaforanupperboundonthetimecomplexityofthebestcase: Lemma2. Foranyn2N,9n(O;D)pairssuchthatthetimecomplexityisO(p nL),whereLrepresentstheaveragedistanceofanytwoorigin-destinationpoints.Combiningthesetwolemmas,theauthorsobtainaveryimportanttheoremonthetimecomplexityofthesensorbasedvehicleroutingproblemasformulatedabove.However,themostimportantaspectoftheirworkisyettofollowwiththeaveragecasescenariostudy,wheretheorigin-destinationpairsarenolongerarbitrarilyselected,butarerandom.Inthatcase,thefollowinglemmaisprovedbytheauthorsonalowerboundofthetimecomplexity. Lemma3. Thetimecomplexityoftheproblemwhenthesetofnorigin-destinationpairsisrandomlyselectedfromtheuniformdistributionis,withhighprobability,(p n).Next,theauthorspresenttheiralgorithmicframeworkthatterminatesinO(p n)time,henceconcludingthatwithhighprobabilitythetimecomplexityoftheproblemisO(p n).Sensorbasedfullyautomatedvehiclesareusedfortransportationpurposesinlargefacilities[ 89 ]ordepotcenters[ 119 ]and,hence,theresultofthetimecomplexityoftheiroptimalroutingproblemisofsignicantimportanceintheeldsoflogistics. 2.3.2IntelligentvehicleroutingthroughacentralizedhighwaysystemThepreviousresultsmayapplyonlytosupplychainnetworksandtotheoptimizationoftheautomatedproceduresregardingstoring,handlingandshipping,howeverinspiredanumberofresearchersingeneralizingthenotionsinthelarge-scale,real-lifetracsystem. 28

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Theideaofreceivingfeedbackthatprovidesthetripmakerwithinformationonthestateoftheroadsthattheyareplanningtouseisnotnewatall.Televisionandradiochannelsspendtimeoninformingtheaudienceswhichstreetsshouldbeavoided,wheretheusersareexperiencingnormaltracowsoriftherehasbeenanaccident.Inthelastyears,guidancedevices,suchastheGPSroutingsystem,providethepossibilitytoobtaintracdatainreal-time[ 91 ].Aswasmentionedintheprevioussubsection,manyattemptstomodelrealisticallytheproblemforrealvehicularowsemanatefromtheroboticworldandafullyautomatedandcontrolledsystemofvehicles.Thatwas,also,theinsightofBaskar,DeSchutterandHellendoorn,whorstproposedanhierarchicaltraccontrolsystemforintelligentvehicles[ 10 , 12 ]andthenadapteditsformulationtoobtainamathematicalprogrammingprobleminordertocomeupwiththeoptimalroutingofthevehcles[ 11 ].Theframework,describedinBaskaretal.[ 10 ]consistsofthefollowingelements: 1. thevehiclecontrollers,whichcontrolthespeedandsteeringofthevehiclesbyreceivingordersfromtheplatooncontrollers; 2. theplatooncontrollers,whichtakecareofthemergesandsplitsofplatoonsandthevehicletovehicledistancesbyreceivingcontrolcommandsfromtheroadsidecontrollers; 3. theroadsidecontrollers,whichareinchargeofasegmentofthewholenetworkintheinfrastructure; 4. andthehigher-levelcontrollers,whichcoordinatethewholenetworkandsuperviseallothercontrollers.Usingthisinfrastructure,theauthorsfocusonoptimalroutingtoeachplatoonthatiscurrentlyinthenetwork.ThenotationthatisusedbyBaskaretal.[ 11 ]isgivenintheTable 2-3 . 29

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Table2-3. Notationusedintheautonomousvehiclerouting. NotationDenition OOriginnodesDDestinationnodesIInternalnodesVThesetofallnodes,V=O[I[DLThesetoflinksinthenetwork(o;d)Oneorigin-destinationpair,(o;d)2ODLo;dThesetoflinksthatbelongtoarouteconnectingotodDo;dThedemandofthepair(o;d)ClThecapacityofthelinkl2LvlThespeedonlinkl2LlThetraveltimeonlinkl2LLinvThesetofalllinksincomingtonodevLoutvThesetofalllinksleavingnodevxl;o;dDecisionvariablesdenotingtheowsforeverypair(o;d)2ODTThesimulationperiod ThesimplelinearmodelthatcorrespondstotheframeworkdescribedbeforeisgiveninFormulation 2{20 throughout 2{23 .minJlinks=X(o;d)2ODXl2Lo;dxl;o;dlT (2{20)s:t:Xl2Louto\Lo;dxl;o;d=Do;d;8o2O;8d2D (2{21)Xl2Linv\Lo;dxl;o;d=Xl2Loutv\Lo;dxl;o;d;8v2;8(o;d)2OD (2{22)X(o;d)2Iod;lxl;o;dCl;8l2L: (2{23)Themodelissimpletounderstandsinceitinvolvesonlytheowbalanceinthenetworkandthecapacityconstraintsforeachofthelinks.TheobjectivefunctioninEquation 2{20 isameasureofthetimethatthevehicleshavetospendwhiletravelinginthenetwork.Inordertomorerealisticallymodeltheproblem,theauthorsthenproceedtoincludequeuesthatcanbeformedattheentriesoftheinfrastructure.Sonowthemodelcanbe 30

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rewrittenas:minJlinks+Jqueue=X(o;d)2ODXl2Lo;dxl;o;dlT+ (2{24)X(o;d)2OD1 2(Do;d)]TJ /F3 11.955 Tf 11.96 0 Td[(Fouto;d)T2 (2{25)s:t:Xl2Linv\Lo;dxl;o;d=Xl2Loutv\Lo;dxl;o;d;8v2;8(o;d)2OD (2{26)X(o;d)2Iod;lxl;o;dCl;8l2L (2{27)Xl2Louto\Lo;dxl;o;dDo;d;8o2O;8d2D (2{28)Fouto;d=Xl2Louto\Lo;dxl;o;d: (2{29)Jqueueisameasureofthetimespentbythevehiclesinthequeuesformedintheoriginnodes.Inordertocomeupwithanestimateforthesemeasures,theauthorsnotethatthequeuesizeincreaseswithtimewitharateofDo;d)]TJ /F3 11.955 Tf 12.52 0 Td[(Fouto;d.Hence,attheendofthesimulationthetotallengthis(Do;d)]TJ /F3 11.955 Tf 12.72 0 Td[(Fouto;d)Tandtheaveragecanbecalculatedas1 2(Do;d)]TJ /F3 11.955 Tf 12.69 0 Td[(Fouto;d)T.ThatishowthetermofJqueueiscomputedintheobjectivefunctionabove.Itisimportanttonotethatoncemorethemathematicalprogramobtainedislinear.Eveninthislastmodel,theapproachishighlyunrealistic.Itisnotavalidassumptionthatthedemandsarestatic,buthavetobeconsidereddynamicinordertoaccommodatemostpracticalapplications.Inordertodoso,adiscretizationofthetimespentoneachlinkisintroduced,withtheelementarymeasurementofTs.Thiscanbewrittenmoreclearlyasl=lTs;wherel2Z+: (2{30)Lettingqo;d(k)bethepartialqueuelengthofvehiclestravelingfromotodattimek,i.e.t=kTs,andbyassumingthatthenetworkisinitiallyempty,i.e.qo;d(k)=0and 31

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xl;o;d(k)=0fork0wecannowhaveforeachoftheoriginnodeso:Xl2Louto\Lo;dxl;o;d(k)Do;d(k)+qo;d(k) Ts8d2D; (2{31)andbydenitionDo;d(k)=0forkK.Now,byconsideringthefactthateveryvehicleonlinklwillreachtheendofthelinkafterltimesegments,weobtainthatXl2Linv\Lo;dxl;o;d(k)]TJ /F3 11.955 Tf 11.96 0 Td[(l)=Xl2Loutv\Lo;dxl;o;d(k);8v2I8(o;d)2OD: (2{32)Alsoforeverylink,wehavethecapacityconstraints,howevertakingintoconsiderationthetimewearein.So,Constraint 2{27 isnowtransformedinthedynamiccaseintoX(o;d)2Iod;lxl;o;d(k)Cl;8l2L: (2{33)Theimportantpartofthismodelingapproachisthedescriptionofthequeuesformed.TheowisgivenbyasimilarconstrainttotheonepresentedinConstraint 2{29 ,whichhoweveristransformedinordertoaccommodatethetimefactorintoFouto;d(k)=Xl2Louto\Lo;dxl;o;d(k): (2{34)Therefore,thequeuelengthisincreasinglinearlywiththerateofDo;d(k))]TJ /F3 11.955 Tf 12.03 0 Td[(Fouto;d(k)forthetimeinterval[kTs;(k+1)Ts)andwegetthefollowingequationforthequeuelength:qo;d(k+1)=max)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(0;qo;d(k)+(Do;d(k))]TJ /F3 11.955 Tf 11.96 0 Td[(Fouto;d(k))Ts: (2{35)ThereexisttwocaseswhichcanbedistinguishedforthedeterminationofthetimeJqueue;o;d(k)thatavehiclehastospendinthequeueformedatanorigino: a.Thequeuelengthbecomeszerowhiletheinterval[kTs;(k+1)Ts). b.Thequeuelengthremainspositiveinthesameinterval. 32

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Letusconsiderthesecondcase.DeningthetimeafterkTsatwhichthequeuelengthbecomeszeroasTo;d(k)=qo;d(k) Fouto;d(k))]TJ /F3 11.955 Tf 11.96 0 Td[(Do;d(k); (2{36)JqueuecannowbeestimatedasJqueue;o;d(k)=8><>:1 2(qo;d(k)+Qo;d(k+1))Tsfortherstcase1 2qo;d(k)To;d(k)forthesecondcase. (2{37)Ingeneralnow,wehaveJqueue=Kend)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0X(o;d)2ODXl2Lo;dJqueue;o;d(k) (2{38)andJlinks=Kend)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0X(o;d)2ODXl2Lo;dxl;o;d(k)lT2s: (2{39)So,nallythemathematicalformulationbecomesmin(Jlinks+Jqueue) (2{40)s:t:Constraints 2{31 throughout 2{35 (2{41)Thismodelisforthesecondcaseanonlinear,nonconvexandnonsmoothproblem.Assuch,thisproblemishardtosolveandhence,theauthorspresentanapproximatesolutionalgorithm.Eitherway,bytransformingtheaboveproblemintoamixedintegerlinearprogram,thereexistseveralsolversthatcansolveiteciently[ 113 ].Forthistransformationtotakeplace,thepropertiesofBemporadandMorari[ 16 ]canbeused:[f0]()[=1]istruei 8><>:fM(1)]TJ /F3 11.955 Tf 11.96 0 Td[()f+(m)]TJ /F3 11.955 Tf 11.96 0 Td[();whereisasmallpositivenumber.y=fisequivalentto 33

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8>>>>>>><>>>>>>>:yMymyf)]TJ /F3 11.955 Tf 11.95 0 Td[(m(1)]TJ /F3 11.955 Tf 11.96 0 Td[()yf)]TJ /F3 11.955 Tf 11.95 0 Td[(M:Then,thetermofFouto;d(k)fromEquation 2{35 canbeeliminated,thusqo;d(k+1)=max)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(0;qo;d(k)+(Do;d(k))]TJ /F7 11.955 Tf 26.88 11.36 Td[(Xl2Louto\Lo;dxl;o;d(k))Ts; (2{42)whichstillisnonlinear.However,bylettingDmax;o;dbethemaximumdemandfor(o;d)2OD,Fmax;o;dbethemaximumfeasibleow(i.e.Fmax;o;d=Pl2Louto\Lo;dCl)andqmax;o;dbethemaximumqueuelengthformedfromoriginotodestinationdandequaltoDmax;o;dTsKend,thentwonewparameterscanbedenedasmlowo;d=)]TJ /F3 11.955 Tf 9.3 0 Td[(Fmax;o;dTs (2{43)muppo;d=qmax;o;d+Dmax;o;dTs; (2{44)hencethefollowingalwaysstands:mlowo;dqo;d(k)+(Do;d(k))]TJ /F7 11.955 Tf 26.88 11.36 Td[(Xl2Louto\Lo;dxl;o;d(k))Tsmuppo;d: (2{45)Now,byintroducingthebinaryvariableso;d(k)aso;d(k)=8>><>>:1iqo;d(k)+(Do;d(k))]TJ /F7 11.955 Tf 29.2 8.96 Td[(Pl2Louto\Lo;dxl;o;d(k))Ts00otherwise. (2{46)So,applyingproperty1,Equation 2{42 takestheformofqo;d(k+1)=o;d(k)(qo;d(k)+Do;d(k))]TJ /F7 11.955 Tf 26.87 11.35 Td[(Xl2Louto\Lo;dxl;o;d(k))Ts): (2{47) 34

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2.3.3VehicleroutingandtracmonitoringusingpersonalsensorsAspersonalsensorswedenotethesedevicesthatcanprovideontheyfeedbackandinformation,suchassmartphones,PersonalDigitalAssistants(PDA)andGPSsystems.Themaindrivingideaisthatthenumberofthetracnetworkusersthatalsoownsoneoftheaforementioneddeviceshassignicantlyincreasedinthelastyears,makingthepropagationofinformationeasier.ThishasbeentheinsightofThiagarajanetal.[ 118 ]increatingaprototypeapplicationthatgatherstheinformationrequiredthroughmobilephonesandprovidesonlineroutingbasedonrecenttractrends.Intheirwork,theypresentreal-timetracmonitoringsystemtrackingthevehicletrajectoriesusingahiddenMarkovchain.Ingeneral,threearethemajorpillarsofahighqualityonlineroutingalgorithm: 1. Accuracy:thetimeestimationsandthecongestionlevelsthatareusedtorerouteandguidevehiclesinthenetworkhavetobecloseenoughtorepresenttherealsituation. 2. Energyeciency:itisanimportantnotethatasmartphonebatteryisconsumedmuchfasterwhenanalgorithmthatrequiresaccesstoonlinedataisexecutedcontinuously.Therefore,atrade-obetweenthesamplingtimeofthedataandahighqualitysolutionneedstobeagreed. 3. Timeeciency:thealgorithmappliedneedstobefastandecient.Analgorithmthatistoocomputationallyexpensivemaypresentoptimalroutes,howeveritisunrealistictoassumethatitcanprovideontheyrouteswhenneeded.However,theaboveobjectivesofanonlineroutingalgorithmpresentanumberofchallenges,themostdicultandcommonofwhicharementionedbelow: Mapmatchingofthetracetotheroadsegmentitcorrespondsto[ 67 , 83 ]. Timeestimationofspecicsegmentsinaroute. Accuracyisenergyconsumptive.SamplingGPShasbeenshowntobe[ 53 ]farmoreexpensiveintermsofenergythatWiFisampling,whichunfortunatelyislessaccurate. 35

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Forthetrackingalgorithmproposed,theauthorsuseahiddenMarkovchain.Bythat,theyimplythatthepositionsateachsamplingperiodoftimeareknown,howevertheroadsegments(i.e.,thetransitionsbetweenpositions)areunknown.Givenasetofknownpositionsoverthetimeofthevehiclemovement,thegoalistodetectthemaximumlikelihoodroadsegmentsthatwereused.Thealgorithmthattheyproposecanbesummarizedasfollows: Computetransmissionprobabilities. Computeemissionprobabilities. EmploytheViterbidecodingalgorithm. Badzonedetectionandremoval.Inordertocomputethetransitionprobabilities,thefollowingnotionsneedtobeconsidered.Firstofall,thereexistsaprobabilitythatthecarwillstillbeinthesameroadsegmentforthenextsamplingperiod.Also,acarcanonlychangeroadsegmentsifthereexistsanintersectionbetweenthesegmentitwasonandthesegmentitisobservedtobeon.Last,therearelimitsonthevehiclespeedsthatprohibitthecartogoextremelyfastinanygivenroadsegment.Mathematically,thenotionsabovearesummarizedtoConstraints 2{48 to 2{50 representingtheprobabilitypforavehiclewhosepositionatsamplingtimet)]TJ /F1 11.955 Tf 11.95 0 Td[(1isiwhileatsamplingtimetisj:Ifi=j;p=: (2{48)Ifjandisharenointersection;p=0: (2{49)Ifiandjshareanintersection;p=orp=0: (2{50)Constraint 2{48 denestheprobabilitythatacarisstillfoundinthesameroadsegmentandisdenedas1 dmax+1: 36

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Thethirdconstrainteectivelyprohibitsthevehiclefrommovingextremelyfastatanyroadsegment.Ifavehicleisdetectedatpositioniattimet)]TJ /F1 11.955 Tf 12.77 0 Td[(1andthenattimetisfoundatj,thenthealgorithmcomputesthetimeitwouldnormallytakethevehicletotraversethisroute.IfthatimpliesthatthecaristravelingataspeedthatisgreaterthanthethresholdspeedSoutlierdenedat200mph,thentheprobabilityissettobeequalto0;otherwiseitisequalto.Thenextstepofthetrackingalgorithminvolvestheemissionprobabilitiesofthemodel.Theemissionprobabilitynotionisemployedtocoverthefactthatitispossibleforapointtobeobservedfromaroadsegmentthatisclosebutisnotnecessarilytheclosestone.So,usingaGaussianfunctionwithzeromeanN,theemissionprobabilityoftheroadsegmentiatpositionlisdenedasN(dist(i;l))wheredist(i;l)istheEuclideandistance.ThevarianceofNisdependentonthesensorthatproducedthepositionand,hence,dierentvariancesareusedforWiFiandGPSpositionsampling.ThemostimportantcomponentofthetechniqueappliedistheViterbidecodingalgorithm[ 67 ]whichndsthemostlikelysequenceofhiddenstates,i.e.roadsegments,thatthevehicleisrequiredtopassthrough.Lastinthesequence,afterhavingobtainedavalidroutebyapplyingtheViterbialgorithm,the"badzones"aredetectedandremovedfromtheroute.Thatway,theauthorscanensurethattherouteisasrealisticaspossibleandtheycanuseittoobtainusefulinformationonthetracstatusandthetimerequiredtotraversethesearcsinrealtime.Overall,thealgorithmpresentedwasappliedtorealdata,withimportantresults,includingthefactsthat: UsingWiFilocalization,90%oftheroutespredictedwerewithina10-15%oftheoptimalroute.GPSlocalizationpresentedoptimalresultswithhighaccuracyoverasamplingperiodof30seconds. 37

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Ahybridalgorithmemployingthe30secondsGPSsamplewithWiFilocalizationinbetweenhasanimprovedperformanceoverthetwomethodsmentionedabove,howeverthegainsaremuchlessthantheenergyconsumptions. UsingGPSlocalizationoverasamplingperiodof20secondsoutperformsthehybridapproach. 2.4SmartphonesandNovelApproachesTherecentboomthatsmartphoneshaveseeninthelastfewyearshasmadecheapsensorsavailabletoanumberofusers.Especiallywhenitcomestotransportationsystems,thereisnowthepossibilityofcollectinginformationfastthroughthewirelessnetworksthatsupportthesephones.Thisinsightdroveanumberofresearcherslikein[ 118 ]toinvestigatethemethodsthatfeedbackcanbederivedfromthesedevicesandprovidedtosophisticatedalgorithms.Currentresearchisfocusingonsensor-basedvehicleroutingproblems,wherevehiclesarebothprovidingandprovidedfeedbackonthetracstatusoftheirrouteandthetimethatisrequiredtotraverselinksinthenetwork.Animportantpartofthealgorithmthatwouldbenetthetripmakers'decisionswouldbetheincorporationofhistoricaldataofthedayortheperiodinthepredictionmodel.Inordertodoso,sophisticatedtime-seriesapproaches[ 122 ]and/orkernelregressionmachinelearningareappliedtothemodel.Dataminingtechniques[ 66 ]canalsoprovideuswithusefulremarksonthetracbehaviorthroughoutatimeperiod.Anothercomponentofthesealgorithmsthatneedstobeimprovedsignicantlyisenergyconsumption.Nowadays,itisknownthattrackingandroutingdevicesareexpensiveanduseupasignicantamountofbattery.Therefore,itisnotonlyimportanttoprovidetravelerswithreliable,on-the-yalgorithmsforrouting,butalsoalgorithmsthatuseupaslittleenergyaspossible.Thesearethemajordirectionsforfutureresearchthatwilloptimizetheproceduresofusingsensorsinroutingandtracassigning.Ifweweretoimprovetheseconditions,thenthealgorithmsdiscussedinthischapterwouldcertainlybemuchmoreaccessibletoanumberofusers. 38

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CHAPTER3GRAPHPARTITIONSFORTHEMULTIDIMENSIONALASSIGNMENTPROBLEM 3.1PreliminariesThemulti-sensormulti-targettrackingproblemisaparticularcaseofthedataassociationproblem.Theproblemconsistsofselectingthemostprobableofassociationsamongseveralsensormeasurements.Assumingtherearensensorsandmmeasurements,thepossibleassociationsare(m!)ninnumber,whichisintractableinlarge-scalescenarios.Thedataassociationproblemspecicsmayvarydependingontheapplicationandassumptions,howeveritisoftenformulatedasamultidimensionalassignmentproblem.Beforeweproceedtogivetheformulationforthemultidimensionalassignmentproblem,itisofinteresttoshowthesimpleassignmentproblemversion.TheAssignmentProblem(2AP)isawell-knownprobleminoptimization,wheretwodisjointsetsIandJ,eachcontainingnelements,areassignedtoeachotherelementwise.Ifelementsiandjfromthetwosetsareassignedtogether,theyyieldacostofcij.Asolutionisvalid,ifandonlyifeachelementisassignedexactlyonce.Letvariablexijbedenedasfollowsxij=8><>:1;ifelementiisassignedtoelementj0;otherwise.The2APmaynowbeformulatedasinFormulation 3{1 throughout 3{4 .minnXi=1nXj=1cijxij (3{1)s:t:nXj=1xij=1;8i=1;2;:::;n (3{2)nXi=1xij=1;8j=1;2;:::;n (3{3)xij0;8i;j=1;2;:::;n; (3{4) 39

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wheretheconstraintsguaranteethateachelementisassignedexactlyonce.Denotethebijectivefunction:I!J,where(i)istheelementfromthesecondsetassignedtoelementi.Wecanensurethateveryelementisonlyassignedoncebyrequiringtobeapermutationvector.Thisleadstothefollowingpermutationformulation,showninEquations 3{5 throughout 3{7 .minnXi=1ci(i) (3{5)s:t:(i)6=(j)8j6=i (3{6)(i)2f1;2;:::;ng;8i=1;2;:::;n: (3{7)The0-1integerprogrammingandpermutationformulationsareequivalent,buttheyoerdierentapproachestondinganoptimalassignment.Whenthenumberofdimensionsinanassignmentproblemisgreaterthantwo,theproblemisreferredtoasamultidimensionalassignmentproblem(MAP),introducedbyPierskala[ 104 ].Asanexample,considertheproblemofassigningjobs,workers,andmachines.TheMAPisawell{studiedprobleminliteraturewithnumerousapplications,includingmulti{sensormulti{targettracking[ 17 , 107 ]anddataassociationproblems[ 6 , 9 ].TheMAPistypicallyformulatedasinFormulation 3{8 throughout 3{12 . 40

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minmXi1=1mXi2=1mXin=1ci1i2:::inxi1i2:::in (3{8)s:t:mXi2=1mXi3=1mXin=1xi1i2:::in=1;8i1=1:::m (3{9)mXi1=1mXij)]TJ /F8 5.978 Tf 5.75 0 Td[(1=1mXij+1=1mXin=1xi1i2:::in=1;8ij=1:::m;8j=2;:::;n)]TJ /F1 11.955 Tf 11.96 0 Td[(1 (3{10)mXi1=1mXi2=1mXin)]TJ /F8 5.978 Tf 5.76 0 Td[(1=1xi1i2:::in=1;8in=1;:::;m (3{11)xi1i2:::in2f0;1g;8i1=1;:::;m;:::;in=1;:::;m: (3{12)SurveysontheMAPanditsapplicationscanbefoundinGilbertandHofstra[ 60 ],BurkardandCela[ 28 ],andmorerecently,Spieksma[ 117 ].Inaddition,anexcellentcollectionofarticlesonextensionsoftheclassicalassignmentproblemisgivenbyPardalosandPitsoulis[ 97 ].AnothershortintroductiontoassignmentproblemsandtheirapplicationsispresentedbyCela[ 30 ].BurkardandCela[ 27 ]alsogiveanannotatedbibliography.Inthischapter,weconsiderthegraphtheoreticaldescriptionoftheMAPwithdecomposablecosts.Thisimpliesthatthecostofeveryassignmentisafunctionofthepairwiseassignmentcostsoftheelements,orci1i2:::im=f(ci1i2;ci1;i3;:::;cim)]TJ /F8 5.978 Tf 5.76 0 Td[(1im):InBandeltetal.[ 7 ],themosttypicallyuseddecomposablecostfunctionsarepresented.Oneofthemisthefamilyofcliquecosts,whichisoftenusedforrepresentingMAPassignmentcostsandisalsofollowedbytheauthorsherein.Thecliquecostofanassignmentisconsideredtobethesummationofalltheedgecoststhatbelongtotheclique/assignment.Otherfamiliesconsideredbytheauthorsin[ 7 ]areminimumstar,minimumtree,andminimumtourcosts.Oneofthegoalsofthischapteristoshowthat 41

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whenconsideringadecomposablecostMAP,asignicantamountofthecomputationaltimeisspentcalculatingassignment/cliquecosts,basedonthegivenedgecostsasaninputtotheproblem.Computationalresultswillindeedverifythatthisisaseriousbottleneck,whichincreasestheruntime.Anumberofmethodshasbeenproposedovertheyearstosolve,eitherexactlyorheuristically,theMAP.However,duetotheinherentNP{hardnessoftheproblem,heuristicapproacheshaveattractedthemostpracticalinterest.Fromthose,werstnotethehigh{qualitysolutionsforlarge{scaleinstancesprovidedbytheLagrangianrelaxationfortheMAPproposedbyPooreandRobertson[ 108 ].Branch{and{boundtechniqueshavealsobeenaviableoptionforsolvingMAPs,asshownin[ 84 , 98 , 100 ].Furthermore,greedyrandomizedadaptivesearchprocedures(GRASP[ 105 ]andGRASPwithpath{relinking[ 3 ])andotherrandomizedtechniques[ 93 ]haveproventobeecient.Lastly,PasiliaopresentsaseriesoflocalneighborhoodsfortheMAPin[ 99 ].WeconsiderthevariantoftheMAPwithdecomposablecosts,whereeachassignmentisrepresentedbyaclique.Weproposetwonoveldecompositionschemesthatpartitiontheproblemintodisjointsubproblemsbasedonboththenumberofdimensions(m)andthenumberofelementsperdimension(n).Thispartitioningwilldividethefeasibledomainintosmaller,moremanageablesizes,whichcanbesolvedexactlyorheuristically.Then,thesolutionsmayberecombinedinordertoobtainupperandlowerboundstotheoriginalproblem.Throughoutthechapter,werefertothesetofdimensionsasM,andthesetofelementsperdimensionasN.TheirrespectivesizesareconsideredtobejMj=m,andjNj=n,accordingly.Hence,wheneverwerefertoafullm{dimensionalassignmentproblemwithnelementsperpartition,wewilldenoteitaseitherAP(m;n),orAP(M;N)interchangeably.Sincethegraphathandism{partite,werefertoeachpartitionasPi;i=1;:::;m.Clearly,wehavethatthesetofallelementsisequaltoP1[P2[[Pm.Theremainderofthechapterisoutlinedasfollows:Section 3.2 introducesanelementpartitioningscheme;itshowsthatthesolutionsobtainedarefeasibleandcanserveas 42

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upperbounds.Usingthis,weproceedtoproposeanexact(elementaugmentation)andaheuristicmethodology(elementpartitioning)tosolvetheoriginalMAP.InSection 3.3 ,weinvestigateadimensionpartitioningschemeandprovethatthesolutionobtainedbythesubproblemsisalowerbound.Wealsopresentanexactmethodology(dimensionaugmentation)andtwoheuristicapproaches(2{AP,anddimensionaugmentation).Section 3.4 summarizestheresultsdiscussedinSections 3.2 and 3.3 ;ahybridmethodisalsopresentedtotakeadvantageofbothdecompositionschemes.ComputationalresultsaregiveninSection 3.5 ,whichalsoconcludesthisstudyandoersinsightonfuturework. 3.2ElementDecompositionInthissection,wediscusstherstdecompositionscheme,whichpartitionstheMAPalongitselements.Werstpresenttheideaofthedecompositionandshowthatitprovidesuswithafeasiblesolution(upperbound).Then,weproceedtopresenttwoalgorithms,elementaugmentationandelementpartitioning,whichprovideuswithanexactandheuristicsolution,respectively,andanalyzetheirworst{casetimecomplexity. 3.2.1TwodisjointsubgraphsforelementpartitioningFirstofall,letuselaborateontheideaofdecomposingtheproblembypresentinganexamplewheretheoriginalm{partitegraphispartitionedintotwodisjointsubgraphs.Asimplewayofdecomposingtheproblemintotwosubproblemswouldbetohalvethenumberofelementsineachofthem.ThiswouldresultintwodisjointKmn=2subgraphswithcorrespondingoptimizationproblemsdenotedbyAP(m;n=2).ThedecompositionisshowninFigure 3-1 .Observethatintherelaxedproblem,alledgesconnectingelementsfromdisjointsubsetsarecut,andhencearenotconsideredinanyfeasiblesolutionoftheproblem.Alsonotethatthesolutionobtainedfromthetwosubproblemsisfeasibletotheoriginalproblem.WeproceedtostatethisrelationshipbetweenthesolutionsobtainedfromthesubproblemsandtheoriginalprobleminProposition 3.1 . 43

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Figure3-1. ElementPartitioning ATwocompleteAP(m;n=2) BFeasiblesolutionoftwoAP(m;n=2) Proposition3.1. ForanytwononemptyelementpartitionsN1andN2suchthatN1\N2=;andN1[N2=N,z[AP(M;N1)]+z[AP(M;N2)]z[AP(M;N1[N2)]whereMisthesetofdimensions,Nisthesetofelements,andzistheoptimalvalueofagivenassignmentproblem. Proof. ConsiderG0(V;E),whereV0=N1[N2=N,andE0=(N1N1)[(N2N2).ObservethatwehaveEE0,henceforanoptimalsolutionovertheoriginaledgeset,weobtainz(AP(M;N))z(G0(V0;E0)):Ontopofthat,itiseasytoseethatz(G0(V0;E0))=z(AP(M;N1))+z(AP(M;N2)):Combining,wehavetheresult. Proposition 3.1 remainstrueregardlessofthenumberofthepartitionedsubproblems,orthenumberofelementsperpartition.Theonlyrequirementforthepropositiontohold 44

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isthat,forksubproblems,wehave,N1\N2\\Nk=;:Bydecomposingtheproblem,weseethateachsubproblemcanbesolvedfaster.Formally,theoriginalproblemAP(m;n),hasaworst{casetimecomplexityofO((n!)m)]TJ /F6 7.97 Tf 6.59 0 Td[(1).Ontheotherhand,thetwosubproblemsarestillhardtosolve,butwithaworst{casetimecomplexityofO((n=2!)m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)each.Hence,wecancomeupfasterwithafeasiblesolution(cf.Proposition1),andobtainanupperbound.Thisupperboundwillbeusedinourexperimentsasaboundforanimplicitenumerationalgorithm. 3.2.2ElementaugmentationFurtherexpandingontheideapresentedintheprevioussubsection,weproposeaniterativealgorithm.Ineachiteration,weconsideranextraelementperdimension,untiltheoriginalproblemissolved.LetS=fP1;P2;:::;Pmg,wherePireferstothesetofelementsofdimensioni(partitioni).Now,wecanpresentElementAugmentationinAlgorithm 1 . Algorithm1ElementAugmentation n 2 fori=1!mdo N fu;vg2Pi S S=fu;vg endfor whileS6=;do SolveAP(m;n)overN,using^z(AP(m;n)]TJ /F1 11.955 Tf 11.95 0 Td[(1))asUB fori=1!mdo N fug2Pi S S=fug endfor n n+1 endwhile return^z ThealgorithmbeginswithaverysimpletosolveAP(m;2)withonlytwoelementsperdimension.Theoptimalsolutionoftheproblemwouldpresentuswithasetof2 45

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disjointcliques.Wethenproceedtoaddanotherelement(atrandom)fromthesetofelementsperpartitionthathavenotbeenconsidered.Thenewproblemcanagainbesolvedtooptimality,usingthesetofcliquesobtainedinthepreviousiterationasastartingsolution.Observethat,intheory,theworst{casetimecomplexityoftheapproachpresentedinAlgorithm1(ElementAugmentation)isworsethantheonederivedfortheoriginalproblem.ThiscomplexityisshowninEquation 3{13 . T(m;n)=2(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+6(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)++[(n)]TJ /F1 11.955 Tf 11.95 0 Td[(1)!](m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+[n!](m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(3{13)Theincreasedcomplexitycanbeattributedtothefactthatinsteadofsolvingonefullm{dimensionalassignmentproblem,wesolveaseriesofsubproblemsofincreasingsize,AP(m;2),AP(m;3),:::,AP(m;n).Asitwillbeshowninthenumericalexperiments,though,theperformanceofthealgorithmissignicantlybetter,dueinparttothetighterupperboundsandthehighqualityinitialsolutionateachsubproblem. 3.2.3ElementpartitioningAnotherapproachwouldbetoimmediatelycreateaseriesofkpartitionsoftheoriginalproblem,thatwouldthenbecombinedtoprovideuswithanupperbound.Sucharandomizedapproachwouldbefastertoprovideuswithafeasiblesolution.However,thequalityofthesolutionwouldbedependentontheinitialselectionofthepartitions.Themethodology(ElementPartitioning)isdescribedinAlgorithm 2 .Atrst,werandomlyselectksimilarlysizedpartitions.Notethatthefunctiondenotedasrand(n;N)isessentiallyselectingnelementsfromeachsetofdimensions.Incasen=kisnotinteger,theresultistruncatedandthelastsubproblemisallowedtohavemorethann=kelementsperpartition.Last,observethatitisimportantforthesubproblemstobeofsimilarsize,inordertoensurethatnosubproblemismuchhardertosolve. 46

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Algorithm2ElementPartitioning k k0 fori=1!kdo Pi rand(n=k;N) N N=Pi endfor fori=1!kdo ^zi AP(M;n=k)overPi ^z [^z^zi] endfor return^z Assumingthatallsubproblemshavethesamesize(n=k),theoverallworst{casetimecomplexityoftheapproachinAlgorithm 2 isgiveninEquation 3{14 . T(m;n)=k((n=k)!)(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1):(3{14)Wewillshowinthecomputationalresults,thatthisapproachissignicantlyfaster,howeverthequalityofresultsisworse,mainlybecauseoftherandomselectionofpartitions. 3.2.4DivideandconquerFinally,theelementpartitionapproachcanbeimplementedasarecursivedivideandconqueralgorithmtoeitherndboundstothem-dimensionalassignmentproblemorasamethodforsolvingtheproblemexactly.Althoughthedivideandconquerapproachmayactuallysolvetheworst-caseassignmentprobleminlongertime,thisapproachprovidesatighterupperboundforthepartialsolutions. RecursiveRelation.Thedivideandconquerapproachisdescribedbytherecursiverelationgivenbelow: T(m;n)=2T(m;n=2)+(n!)m)]TJ /F6 7.97 Tf 6.59 0 Td[(1(3{15)TheMAPisiterativelybrokenintotwohalvesuntilallthesubproblemsareoftypeAP(m;2).Fromthere,webegintoconstructthecompletesolutionbysolvingasetofAP(m;4)problemsbyusingthesolutionsoftwoAP(m;2)problemswiththesame 47

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elementsasupperbounds.WecontinuecombiningsolutionsuntilwecansolvetheoriginalAP(m;n)byusingthesolutionsoftwoAP(m;n=2)asupperbounds. 3.2.5PreprocessingTheperformanceofeachoftheapproachesisdependentonhowwellwecanbreakaparttheelementsinthedimensions.Wecansimplyandquicklypartitionthedimensionsbyarbitrarilyorrandomlychoosingwhichdimensionsgoineachpartition.However,wecansignicantlyimprovetheperformanceofthealgorithmsbypartitioningwisely.Ideally,wewouldliketopartitiontheelementsinsuchawaythattheedgesofanoptimalsolutionarenotcut.Inotherwords,wewantalltheelementsofacliqueinanoptimalsolutiontobeinthesamesubproblem.Anapproachwouldbetoquicklyndagoodfeasiblesolutionthroughafastheuristicandthen,usetheheuristicsolutionasaguideforpartitioningtheelements.Noedgesthatareinthefeasiblesolutionaretobecut.Theelementpartitionapproachshouldresultinafasterconvergencerateastheinitialheuristicsolutionimproves.Inthecasewheretheheuristichappenstondanoptimalsolution,thenalloftheelementpartitionprocedureswouldconvergeextremelyfastsincenoreassignmentsarenecessary.Theprocedureswouldthenprovideapolynomialtimealgorithmforcheckingtheoptimalityofthesolution.FortheElementAugmentationapproach,wecanpreprocessthedatasetsuchthattheelementswiththelowercliquecostsareplacedatthetoppositions.Therearrangementoftheelementsineachdimensionshouldsatisfytheconditionthatthecliquedenedbythetopelementineachdimensionislowerthanthecliquedenedbythesecondineachdimension.Thecliquefromthesecondelementsisoflowercostthanfromthethirdelements,andsoon.Preprocessingthedatasettosatisfyingthisconditionwilldecreasethelikelihoodofathatacliquewillberemovedfromthecurrentsolutionwhennewelementsareaddedtotheproblem.Thisapproachisactuallythepurelygreedyapproachinwhichthelowestcostfeasiblecliquesareplacedinthesolution.Thegoal 48

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istomaximizetheintersectionbetweenthecliquesetsolutionsofAP(m;n)]TJ /F1 11.955 Tf 12.56 0 Td[(1)andofAP(m;n)totightentheboundsandincreasethealgorithm'sconvergencerate. 3.2.6DiscussionThissectionpresenteddierentapproachestopartitioningtheelementsofanm-partitegraphtosolvesmallerproblems,whichleadtoupperbounds.Section 3.2.1 describesthemethodologyforpartitioningtheoriginalAP(m;n)intotwosmallerinstancesofAP(m;n=2).Section 3.2.2 showshowwecanstartfromthebasicAP(m;2)andsequentiallyaugmentingthenumberofelementsuntilwehavethecompletesolutionforAP(m;n).Section 3.2.4 illustrateshowwecancombinecharacteristicsoftheElementPartitionandAugmentationapproachestoimplementadivideandconquerprocedureforsolvingAP(m;n).ThedivideandconquerapproachcontinuallypartitionstheproblemuntilwehaveasetofsubproblemsofsizeAP(m;2).Wethensequentiallyaugmentthenumberofelementsbycombiningtwosimilarlysizedsubproblemsatatime. 3.3DimensionDecompositionInthissection,wepartitionthedimensionsetoftheMAP,inordertoobtainsubproblemsofsmallerdimensionality.Thiswilldecreasethecomputationaltimerequiredtosolveeachsubproblem.However,anyproblemobtainedbyavaliddimensiondecompositionisinfeasibletotheoriginalproblem.Hence,itcanserveasalowerbound.Weinvestigateanexactmethod(dimensionaugmentation),whichissimilartotheelementaugmentationpresentedbefore,initssolvingiterativelysubproblemsoflargerdimensionalityuntiltheoriginalMAPissolved.Then,weproposetwoheuristicmethods.Therstoneconsiderstwodimensionsatatime,solvingaseriesof(polynomiallysolvable)AP(2;n),whiletheotherpartitionsthesetintodisjointsubproblemsandthenmatchesthecliquesobtainedtoobtainafeasiblesolution. 3.3.1TwodisjointsubgraphsfordimensionpartitioningLetusbegin,similarlytobefore,byconsideringtwodisjointsubproblems.EachofthemcanberepresentedasAP(m=2;n).Observethatinthetwosubproblems,the 49

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numberoffeasiblesolutions(cliques)issignicantlyreducedto2(n!)m=2)]TJ /F6 7.97 Tf 6.58 0 Td[(1fromtheoriginal(n!)m)]TJ /F6 7.97 Tf 6.58 0 Td[(1.SuchadecompositionisshowninFigure 3-2 . Figure3-2. DimensionPartitioning ATwocompleteAP(m=2;n) BFeasiblesolutionoftwoAP(m=2;n) IntheexampleofFigure 3-2 ,anoptimalsolutiontothetwosubproblemsyieldsatotaloftwelvedisjointcliquesofsmallersize.IntheoriginalMAP,though,werequireafeasiblesolutiontoconsistofsixdisjointcliques,spanningthroughalldimensions.Hence,anysolutionobtainedfromthetwosubproblems,isinfeasibletotheoriginalproblem.ThisisestablishedinProposition 3.2 . Proposition3.2. ForanytwononemptydimensionpartitionsM1andM2suchthatM1\M2=;,z[AP(M1;N)]+z[AP(M2;N)]z[AP(M1[M2;N)]whereMisthesetofdimensions,Nisthesetofelements,andzistheoptimalvalueofagivenassignmentproblem. Proof. TheoptimalsolutionforapartitionM1;z[AP(M1;N)]containsjNjdisjointcliques,eachofasizeofjM1j.Similarly,z[AP(M2;N)]wouldcontainjNjdisjointcliquesofsizejM2j.Notethatinordertotransformthissolutionintoafeasiblesolutionfortheoriginalproblem,wewouldneedtoreconcilethecliquesandcreatejNjdisjointcliques 50

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ofsizejM1[M2j.Thisimpliesthatatleastsomeedgesconnectingthe2jNjdisjointcliquesneedtobeadded.Sincethecostsbetweenanytwonodesarenonnegative,thatalsoimpliesthatz[AP(M1;N)]+z[AP(M2;N)]^z[AP(M1[M2;N)];where^z[AP(M2;N)]isafeasiblesolutiontotheAP(M1[M2;N).Last,sincethisisafeasiblesolution,wecaneasilyobtainthat^z[AP(M1[M2;N)]z[AP(M1[M2;N)];andtheresultisathand. 3.3.2DimensionaugmentationTheDimensionAugmentationalgorithmbegins,asintheElementAugmentation,byselectingtwodimensionsoftheproblem.Thesubproblemisa2-dimensionalassignmentproblemwhichiseasilysolvedinpolynomialtime.Inthenextiteration,weproceedtoaddanewdimensiontotheproblem.Thenewsubproblemisstilla2{dimensionalassignmentproblem,whichmatchesthecliquesobtainedinthepreviousiteration,withtheelementsoftheincomingdimension,byconsideringthecostofaddingtheelementstotheexistingcliques.Weiterateonthat,untilalldimensionshavebeensuccessfullyaddedtotheproblem.ObservethatateveryiterationonlyanAP(2;n)issolved.Becauseofthat,afeasiblesolutioncanbereadilyfound,withouttheneedtofullyenumerateallcliques/assignmentsintheoriginalproblem.ThisapproachispresentedinAlgorithm 3 .Contrarytotheelementaugmentationapproach,Algorithm 3 hasaworst{casetimecomplexity(showninEquation 3{16 )thatismuchsmallerthantheoneoftheoriginalAP(m;n). T(m;n)=(m)]TJ /F1 11.955 Tf 11.96 0 Td[(1)O(n3)(3{16) 51

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Algorithm3DimensionAugmentation(2-APsolver) i rand(1;:::;m) j rand(1;:::;m) N Pi[Pj S S=fPi;Pjg whileS6=;do SolveAP(2;n) N ^z(AP(2;n)) k rand(i:Pi2S) N N[Pk S S=Pk endwhile return^z Ontheotherhand,insteadofcontinuouslysolving2{dimensionalassignmentproblemsbetweentheprevioussetofcliquesandtheincomingdimension'selements,wecouldproceedasfollows.AfterobtainingthesolutionoftherelaxedsubproblemAP(2;n),wecangoaheadandusethelowerboundobtainedtosolveexactlytheAP(3;n)withtheaddednewdimension.Thiswouldresult,asitcanbeclearlyseen,inamuchworsetimecomplexity(seeEquation 3{17 ).Thisistobeexpected,sincewesolveexactlyeachoftheAP(2;n);AP(3;n);:::;AP(m;n).ThepseudocodeispresentedinAlgorithm 4 . T(m;n)=2(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+6(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)++[(n)]TJ /F1 11.955 Tf 11.95 0 Td[(1)!](m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+[n!](m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(3{17)Theperformanceofeachofthedimensionpartitionapproachesisdependentonhowwellwecanbreakaparttheproblemintodimensionsubsets.Wecanarbitrarilyandrandomlyselecthowthedimensionswillbepartitioned.However,wecanimprovetheperformanceofthealgorithmsbypartitioningwisely.SimilarlytotheElementPartitionpreprocessing,wecanperformafastheuristictondagoodsolutionbeforepartitioningthedimensions.Wewouldthenlooktopartitionthedimensionssuchthatthecutedgesareofhighesttotalcosts.A"brute"forceapproachistosimplysumalltheedgecostsfromonedimensiontoanother.Ifthetotal 52

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Algorithm4DimensionAugmentation(exact) m0 2 i rand(M) M M=fig j rand(M) M M=fjg N Pi[Pj S S=fPi;Pjg whileS6=;do SolveAP(m0;N),using^z(AP(m0)]TJ /F1 11.955 Tf 11.96 0 Td[(1;N))asLB i rand(M) M M=fig N N[Pi S S=fPig m0 m0+1 endwhile return^z ishigh,thenweplacethedimensionsindierentpartitions.Inotherwords,weplacethedimensionswithalowtotaledgecostsbetweentheminthesamepartition.Inamulti-targetmulti-sensortrackingapplication,wewouldliketogrouptogetherthedimensionsorsensorsthatarephysicallyclosetoeachother.Thisisbecauseifoneofthesensorspickedupatarget,thereisagoodchancethatothersensorsclosebywouldhavealsopickedupthesametarget.Forthemulti-targetmulti-scantrackingapplication,wewanttogroupthedimensionsorscansthataretemporallyclosetoeachother.Again,thisissothatatargetthatispickedupinonescanhasahighprobabilityofalsobeingpickedupbythescansimmediatelybeforeandafterthecurrentscan.Fromapurelymathematicalperspective,groupthedimensionsthataremostcompatiblewitheachotherinasomesense.Thegoalistopartitionthedimensionsinsuchawaysothattheoptimalcliquesolutionsfromtwodierentpartitionsstayintactwhenthepartialproblemsarecombined.Oneapproachistousethe2APRelaxationdescribedbefore.Thepairwisematchingprovidesameasureof"anity"betweentwodimensions.Alowoverallsolutioncostindicatesastrong"anity"betweentwodimensions.Fromthere,weessentiallyperformaclusteringofthethedimensionsintopartitions.Itmaywellbethata 53

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non-balancedpartitioning,wherenotallpartitionshavethesamenumberofdimensions,ismoreappropriate.Thisapproachwouldimprovethechancesthatthecliquesfoundinthepartitionswillnotbebrokenupwhenthepartialsolutionsarecombined.Ideally,wewouldliketopartitionthedimensionsinsuchawaythattheoptimalcliquesolutionsofthesubproblemsaresubsetsoftheoveralloptimalsolution.Inotherwords,wewantalltheelementsofacliqueinanoptimalsubproblemsolutiontobeintheoptimalcompletesolution.Anapproachwouldbetoquicklyndagoodfeasiblesolutionthroughafastheuristicandtousetheheuristicsolutionasaguideforpartitioningtheelements.Thedimensionpartitionapproachshouldresultinafasterconvergencerateastheinitialheuristicsolutionimproves.Inthecasewherethesubproblemcliquesareallinthesamesubsettheheuristichappenstondanoptimalsolutionandthenalloftheelementpartitionprocedureswouldconvergeextremelyfastsincenoreassignmentsarenecessary.Theprocedureswouldthenprovideapolynomialtimealgorithmforcheckingtheoptimalityofthesolution. 3.3.3DimensionpartitioningWenowpresentadimensionpartitioningscheme,basedonadivide{and{conqueralgorithm.ObservethathalvingtheproblemintwoAP(m=2;n)subproblems,signicantlydecreasestheruntime.However,asshowninProposition2,theobtainedsolutionisinfeasible.Inthissubsectionwediscussawaytoreconcilethecliquesofeachsubproblem,henceobtainingafeasiblesolutiontotheinitialAP(m;n).Thealgorithmdividestheoriginalproblemintokdisjointsubproblems.Thenthesolutionsobtainedarereconciled,usingtheReconcileCliquessubroutineshowninAlgorithm6.ThepseudocodeofthisapproachispresentedinAlgorithm 5 .Inthecaseofjusttwodisjointsubproblem,eachrepresentedbyAP(m=2;n),theoverallworst{casetimecomplexityisderivedinEquation 3{18 . 54

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Algorithm5DimensionPartitioning k k0 fori=1!kdo Mi rand(m=k;M) M M=M1 endfor fori=1!kdo ^zi AP(Mi;N) endfor ^z Reconcile(z1;z2;:::;zk) return^z T(m;n)=2T(m=2;n)+O(n3)(3{18)InAlgorithm 5 ,werefertoasubroutinecalledReconcile,whichispresentedinAlgorithm 6 .Aftersolvingtwosubproblems,AP(M1;N)andAP(M2;N),theirsolutionsetscanbereconciledinordertoprovideuswithonesetoflargercliques.ThatsolutionwillbefeasibletotheproblemAP(M1[M2;N). Algorithm6ReconcileCliques M1 Q1M M2 Q2M:Q16=Q2 D1 AP(M1;n) D2 AP(M2;n) ^z AP(2;D1[D2) return^z ObservethatthetimecomplexityofcomputingallpossibleassignmentsbetweenthecliquesinthehypersetD1andthoseinD2isO(m2n2).Theworst{casetimecomplexityofthealgorithmforpartitioningthedimensionsintotwosubsets,solvingeachsubproblemseparately,andreconcilingthecliquesfromthesubproblemsisgivenbythefollowingexpression:T(m;n)=2(n!)(m=2)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+(m=2)2n2+O(n3) 55

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3.4AnalysisandaHybridMethod 3.4.1AnalysisIntheprevioussectionsofthechapter,weintroducedtwonoveldecompositionschemesfortheclassicalMAP.Foreachofthem,bothexactandheuristicapproacheswerediscussed.Insummary,weproposedtheElementAugmentationandDimensionAugmentation(exact)algorithmstooptimallysolvetheproblem,andtheDimensionAugmentation(2-AP),ElementandDimensionPartitioningalgorithmstoobtainaheuristicsolution.ThisdistinctionisreectedinTable 3-1 ,wheretheworst-casetimecomplexitiesoftheoriginalMAPandourapproachesareconcerned.ObservethatboththeElementandDimension(exact)Augmentationalgorithmshaveaworseoveralltimecomplexity.However,asitwillbeshowninthecomputationalresults,themethodsstilloutperformingeneraltheclassicalMAPformulation.Thisis,inpart,duetoboundsobtainedfromthepreviousiterations(thesmallersubproblems),whichleadtofasterconvergence.Ontheotherhand,therestoftheapproaches(Element/DimensionPartitioning,DimensionAugmentation(2-AP))haveabetterworst{casetimecomplexity,whichisexpected,sincetheyprovideaheuristicsolution.Thequalityofthesesolutions,alongwiththeruntimeswillbediscussedinthefollowingsection.Table 3-2 presentssomeusefulproperties,thathavetodowiththestructureofthegraphconsideredforeachofthepartitioningschemes.Itisinterestingtonotethedecreaseinthenumberofcliquesthatneedtobeconsidered,whentheoriginalproblemisdecomposedintodisjointsubgraphs.Basedontheseremarks,wecannowproceedtointroduceahybridalgorithmthatemploysbothpartitioningschemes,inanattempttofasterconvergetoahighqualitysolutiontotheoriginalMAP. 56

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Table3-1. Worst-casetimecomplexityoftheapproachespresented. ExplicitEnumerationT(m;n)=(n!)(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)ElementAugmentationT(m;n)=2(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+6(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)++[(n)]TJ /F1 11.955 Tf 11.95 0 Td[(1)!](m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+[n!](m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)ElementPartitioning(overkpartitions)T(m;n)=k(n=k!)(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)DimensionAugmentation(2{APsolver)T(m;n)=(m)]TJ /F1 11.955 Tf 11.96 0 Td[(1)O(n3)DimensionAugmentation(exactsolver)T(m;n)=O(n3)+[O(n3)+(n!)2]++O(n3)+(n!)(m)]TJ /F6 7.97 Tf 6.59 0 Td[(2)+O(n3)+(n!)(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)DimensionPartitioning(2disjointsubgraphs)T(m;n)=(m)]TJ /F1 11.955 Tf 11.96 0 Td[(1)O(n3)] 3.4.2HybridmethodThehybridmethodemploysbothdimensionandelementdecompositionschemestofastersolvelarge{scale(i.e.,highdimensionalityand/orbiggercardinalityofpartitions).ThealgorithmispresentedinAlgorithm 7 .Thekeyadvantageofthemethodisthatitprovidesuswithasystemicwayofdecomposinglarge{scaleMAPs,byemployingthelowerandupperboundsobtainedateveryiteration.Basedonthesebounds,weobtainanoptimalitygap(andafeasiblesolution)thatissignicantlysmaller(ofhigherquality)comparedtotheelementand 57

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Table3-2. PropertiesoftheAP(m;n)anditsdecompositionschemes. Edges1 2m(m)]TJ /F1 11.955 Tf 11.96 0 Td[(1)n2OneAP(m;n)21 2m(m)]TJ /F1 11.955 Tf 11.95 0 Td[(1))]TJ /F4 7.97 Tf 6.67 -4.97 Td[(n 22TwoAP(m;n=2)21 2)]TJ /F4 7.97 Tf 6.67 -4.98 Td[(m 2(m 2)]TJ /F1 11.955 Tf 11.96 0 Td[(1)n2TwoAP(m=2;n)CliquesnmOneAP(m;n)2)]TJ /F4 7.97 Tf 6.68 -4.98 Td[(n 2mTwoAP(m;n=2)2n(m 2)TwoAP(m=2;n)FeasibleSolutions[n!](m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)OneAP(m;n))]TJ /F4 7.97 Tf 13.65 -4.98 Td[(n 2![2(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TwoAP(m;n=2)[n!][n!](m 2)]TJ /F6 7.97 Tf 6.58 0 Td[(1)TwoAP(m=2;n) Algorithm7HybridAlgorithm ElementPartition(AP(m;n))intokpartitions fori=1!kdo DimensionPartition(AP(m;ni)) endfor ^z=ReconcileCliques(^z1(AP(2;n1));^z2(AP(2;n2));:::;^zk(AP(2;nk))) return^z dimensionpartitioning,whenemployedbythemselves.Ontopofthat,asitwillbeshowninthecomputationalresults,thismethodcansolvelargerproblemsthantheoriginalIPformulationoftheMAP. 3.5ComputationalResultsInthissection,wewilldepictthesuccessofourdecompositionschemesbysolvingaseriesof4-dimensional,6-dimensional,and8-dimensionalassignmentproblems.All 58

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experimentswereperformedonanIntelCore2Duoat2.4GHzwith4GBofRAM.AllcodeswerewritteninC++andtheresultsobtainedwerecontrastedagainsttheresultsofGurobi5.0.1andCPLEX12.4.Theimplementedalgorithms,aswellasthecommercialsolver,wereallowedtoutilizebothcores,thusspeedinguptheprocess.AGPUimplementationwasalsocreated,howeveritdidnotprovetobecomputationallyecientsincethesolverscouldnotusetheextraresourcestondoptimalsolutionstothedecomposed/originalproblems.Thegoalofourexperimentswastoshowhowwellourdecompositionalgorithms(bothperdimensionandperelement)perform.Anotherobjectivewastoshowthatcalculatingclique(assignment)costsisaseriousbottleneckoftheproblem.Especiallyinlargerinstances,asignicantfractionoftheoperationstimewasdevotedtocomputingthesecosts.Inourapproaches,wediscardalargenumberofcliquespersubproblemand,hence,thecomputationsaremuchfaster.ThiswillbebettershowninTable 3-3 .Beforeweproceedtothecomputationalresultsthough,letusdescribethewaythatthetestbedwasset.Foreachoftheexperiments,50randominstancesofeachsizewerecreated.These50instanceswerethensolvedbyacommercialsolver(Gurobi5.0.1andCPLEX12.4)andthefastest(ortheonewiththesmallestoptimalitygap,ifnotsolvedoptimally)solutiontimewasreported.Allinstanceswererandomlygenerated,followingauniformdistribution,basedontheinstancegeneratorpresentedin[ 62 ].Inthepaper,theauthorsshowthattheseinstancesarehardtosolve,andhowtheycanserveasbenchmarkinstances.First,allinstanceswerepartitionedintosmallersubproblemsperelement.Inthisimplementation,n=5)]TJ /F3 11.955 Tf 12.42 0 Td[(n=10elementsperpartitionwereconsideredatrandom.Clearly,employingasmarterwaytodenethepartitionswouldyieldbetterresults,howeveritwasourintentiontotesthowwelllarge-scaleMAPscanbesolvedevenwhenusingarandompartitioningscheme.Afterobtainingthesmallersubproblems,wetackleeachinstanceusingthealgorithmsofElementAugmentationandElementPartitioning. 59

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Then,wepartitionedtheoriginalproblemsonthesetofdimensions.FortheDimensionAugmentationapproach,boththe2-APsolver,andtheexactmethodologywereappliedandcompared.Inadditiontothat,wesolvedtheproblems,utilizingtheDimensionPartitioningmethod.Forthelatter,allpossible(m2) 2dimensionpairingswereconsidered.Last,wecontrastallresultstotheonesobtainedbythehybridmethod,andtheexactsolver(CPLEXandGurobi).TheinstancesaredenotedbymDn,wheremisthenumberofdimensionsoftheMAP,andnthenumberofelementsperpartition. Table3-3. Timespent(inseconds)computingcliquecosts. ProblemElementDimensionFullCliqueInstancePartitioningPartitioningComputation 4D50:0030:020:124D100:0040:030:524D150:0040:042:554D200:0040:457:824D250:0050:8850:534D300:0051:07493:854D350:011:95711:214D400:015:63988:434D450:0111:331076:044D500:0114:101386:926D50:0040:100:246D100:0050:380:986D150:0060:793:956D200:011:1311:076D250:014:0780:126D300:027:81713:506D350:0213:191109:026D400:0219:751533:248D50:0060:140:558D100:010:673:768D200:027:94981:47 Sincetheinputoftheproblemswasthesetofalledgecostsamongthenodesbelongingtodierentpartitionsofthem{partitegraph,sometimeneedstobedevotedtobuildtheassignmentcosts,beforeproceedingtosolvingtheproblem.Table 3-3 presents 60

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ourndingsonthecomputationaltimerequiredtocomputeallclique/assignmentcosts,whentheyareprovidedindecomposedform(i.e.,peredge),intheoriginalproblem,andinthesubproblemsafterpartitioningthegraphathandinboththeelementsanddimensions.ItiseasytoseethatinthefullMAP,computingallpossiblecliquesofsizem(feasibleassignments)inthegraphiscomputationallyexpensive,especiallyasthenumberofelementsperpartitionincreases.Ontheotherhand,decomposingtheprobleminboththeelementsandthedimensionsdiscardsasignicantnumberofedges,hencedecreasingtheoverallcomputationaltimerequiredtocalculateallassignmentcosts.InTable 3-4 ,wepresentourndingsfortheeciency,andthequalityofsolutionsobtainedforthetwopartitioningschemes,andeachofthemethodsdevisedforthem.Runtimesareinseconds,whiletheoptimalitygapsaregiveninparentheses(whenpresent).First,wepartitionedtheoriginalprobleminitselements,andsolvedtheproblemexactly(ElementAugmentation)andheuristically(ElementPartitioning).Then,partitioningtheprobleminitsdimensionsgivesustheopportunitytotestthethreemethods:2-APDimensionAugmentation(heuristic),DimensionAugmentation(exact),andDimensionPartitioning(heuristic).Last,wepresentthecomputationalruntimesobtainedbythehybridmethodofSection 3.4 ,andthecommercialsolver(CPLEXandGurobi). 61

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Table3-4. Runtimesandoptimalitygapsfortheexactandheuristicapproachesinvestigated. ProblemElementElementDimensionDimensionDimensionHybridCommercialInstanceAugmentationPartitioningAugmentationAugmentation(2{AP)PartitioningMethodSolver 4D50.210.00(19.02%)0.070.01(18.72%)0.01(7.06%)0.01(5.12%)0.134D100.310.01(21.05%)0.220.03(18.36%)0.04(10.11%)0.04(7.81%)0.274D152.970.02(19.95%)2.250.04(19.04%)0.06(9.52%)0.09(9.99%)3.354D2060.110.50(19.26%)51.060.10(19.50%)0.09(14.97%)0.32(10.30%)77.024D25234.431.01(29.32%)201.330.16(19.51%)0.16(16.24%)0.98(11.88%)271.654D30990.721.71(30.18%)1,103.280.26(23.25%)0.25(17.57%)1.12(12.37%)1,106.844D351,284.352.15(36.35%)1,743.900.54(24.48%)0.55(29.87%)2.90(15.10%)1,663.704D401,597.443.96(38.14%)2,411.202.81(29.11%)2.76(18.20%)4.11(17.63%)2,306.194D457,668.477.10(27.71%)14,512.113.17(31.94%)3.05(27.71%)5.67(18.12%)12,649.904D5015,331.95(0.03%)13.23(34.12%)21,600.00(8.60%)6.09(33.05%)5.29(23.97%)9.03(20.90%)18,835.00(1.02%)6D50.910.01(27.23%)0.460.05(19.98%)0.02(4.54%)0.01(2.24%)0.596D106.330.03(11.90%)3.240.41(21.35%)0.34(12.09%)0.09(7.83%)4.096D15407.110.60(17.18%)366.251.17(21.50%)0.87(17.45%)0.78(12.14%)405.706D201,821.901.89(21.44%)1,577.681.36(22.47%)1.08(21.78%)0.50(15.04%)1,900.126D252,601.872.70(21.94%)2,701.204.02(23.03%)1.69(22.91%)1.11(16.30%)2,713.116D3015,976.108.56(27.08%)21,600.00(1.25%)7.74(24.77%)3.17(24.43%)4.89(19.11%)18,912.80(0.13%)6D3519,304.42(0.08%)10.13(27.34%)21,600.00(2.07%)11.08(31.20%)4.96(27.78%)6.01(19.97%)21,600.00(0.91%)6D4021,600.00(0.83%)14.97(33.89%)21,600.00(6.16%)16.11(36.81%)6.89(31.05%)8.30(21.00%)21,600.00(1.37%)8D52.130.01(23.14%)1.090.08(20.09%)0.04(9.82%)0.03(5.08%)1.678D101,320.91.04(18.36%)1,012.382.15(16.40%)1.39(23.91%)2.07(19.44%)1,324.58D2021,600(1.39%)17.12(34.91%)21,600(1.89%)22.85(32.74%)15.38(31.09%)17.70(28.81%)21,600(2.03%) 62

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Table3-5. Averageoptimalitygapsreportedifeachofthemethodswasallowedaruntimeofatmost18seconds. ProblemElementElementDimensionDimensionDimensionHybridCommercialInstanceAugmentationPartitioningAugmentationAugmentation(2{AP)PartitioningMethodSolver 4D50%19.02%0%18.72%7.06%5.12%0%4D100%21.05%0%18.36%10.11%7.81%0%4D150%19.95%0%19.04%9.52%9.99%0%4D203.34%19.26%2.19%19.50%14.97%10.30%4.25%4D258.89%29.32%8.02%19.51%16.24%11.88%9.03%4D3018.33%30.18%21.07%23.25%17.57%12.37%20.27%4D3522.47%36.35%29.76%24.48%29.87%15.10%32.98%4D4036.59%38.14%43.17%29.11%18.20%17.63%41.70%4D4540.55%37.71%59.61%31.94%27.71%18.12%62.20%4D5049.21%34.12%83.11%33.05%23.97%20.90%79.30%6D50%27.23%0%19.98%4.54%2.24%0%6D100%11.90%0%21.35%12.09%7.83%0%6D155.61%17.18%5.44%21.50%17.45%12.14%7.33%6D2019.40%21.44%15.11%22.47%21.78%15.04%19.56%6D2527.12%21.94%27.91%23.03%22.91%16.30%31.08%6D3048.75%27.08%62.09%24.77%24.43%19.11%59.87%6D3555.23%27.34%76.44%31.20%27.78%19.97%74.21%s6D4080.91%33.89%94.11%36.81%31.05%21.00%79.52%8D50%23.14%0%20.09%9.82%5.08%0%8D109.82%18.36%9.97%16.40%23.91%19.44%11.06%8D2041.56%34.91%46.12%32.74%31.09%28.81%39.15% 63

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Inthebiggestofinstances,thecommercialsolverswereunabletosolveallofthemwithinthelimitof6hours(21600seconds).Fortheseinstances,thebestsolutionobtainedbyeitherGurobiorCPLEXwasreportedand,hence,thereisalsoanoptimalitygapassociatedwiththem.Theaverageoptimalitygapforeachmethodispresented(whenitexists)inparentheses.FromTable 3-4 ,wenotethefollowing.Themethodologiespresentedfortacklingtheproblemexactlyoutperformthecommercialsolverinmostinstances.ItisinterestingtoobservethatDimensionAugmentationisfasterthanElementAugmentationintheinstanceswithasmallernumberofelementsperpartition.However,asthenumberofelementsincreases,ElementAugmentationismoreecient,asexpectedfromthecomplexityanalysis.Ontheotherhand,theheuristicapproachesdevised,provideuswithhighqualitysolutions,especiallywhenweighingintheirextremelyfastruntimes.Fromtheheuristicapproaches,theHybridMethodhasconsistentlyoutperformedtherestasfarastheaverageoptimalitygapsreportedareconcerned.Anotherinterestingfactorthatweweighedinourexperimentshastodowiththequalityofthesolutionsobtainedbythecommercialsolverifallowedtorunforexactlythesametimeasourapproaches.TheseresultsaregiveninTable 3-5 .Observethatinmostinstances,theoptimalitygapofthecommercialsolverislargerthantheonesobtainedbytherestofthemethods.Hence,whenahighqualityfeasiblesolutionisrequiredfast,theheuristicmethodspresentedhereposeabetteralternative.Insummary,weseethat,especiallyasthesizeoftheinstancesincreases,thecommercialsolversbecomelessandlesspractical,asexpected.Ontheotherhand,thereisonlyasmallcomputationaloverheadintheproposedmethodologies.Ofcourse,thesolutionsobtainedaresuboptimal,howevertheecacyofourapproachescanbeseeninthecomputationaltimesachieved.Inordertofurthershowhowtheproblemscaleswithsize,andhowthequalityofthesolutionsobtainedbycommercialsolverssignicantlydrops,wepresentthreeexemplarlarge-scaleinstances(namely10D100,20D50,and 64

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Table3-6. Verylarge-scaleinstances. ProblemElementDimensionDimensionHybridInstancePartitioningAugmentation(2{AP)PartitioningMethod 10D100121.93(39.17%)147.03(38.57%)201.33(27.13%)205.76(25.92%)20D50204.77(41.05%)254.31(38.36%)390.42(30.11%)401.93(26.81%)30D30267.31(39.95%)282.91(39.04%)448.83(34.52%)572.88(29.32%) 30D30),wherethecommercialsolverwasunabletoprovideuswithasolution,duetomemoryrestrictions.Allfourofourheuristicapproachesprovideduswithafeasiblesolutionwithin{atmost{minutes.Itisinterestingtonotethatthecommercialsolverscouldnottackletheseproblems,astheyranoutofmemory.Inparentheses,wepresenttheaverageoptimalitygapassociatedwitheachapproach.TheseresultsaregiveninTable 3-6 .Tosumup,twodecompositionschemesfortheMAParepresented.AsshowninSection 3.5 ,thenumericalexperimentsprovethatouralgorithmsperformwell,especiallyinlarge-scaleinstances.Inadditiontothat,bothdecompositionschemesadmitparallelimplementations,whichimpliesthatwiththeappropriatecomputationalpowertheresultscanbefurtherenhanced.SincetheMAPis,ingeneral,anon{separableoptimizationproblem,techniquesthatcandecomposetheproblemintosmalleronesareinherentlyofhighquality,andcanreachsuboptimalsolutionsinanecientandtimelymanner.Overall,ourtechniqueshaveprovedtobeofinterestinsolvingmulti-targetmulti-sensorand/ordataassociationproblemswithnearoptimalsolutions.Inadditiontothat,weproposedahybridmethodthatemploysbothtechniquespresentedherein.Thehybridmethodwasshowntoprovideuswithhighqualitysolutionsquickly,makingitareliablealternativewheneveragoodsolutionisreadilyrequired.Webelievethattheaforementionedmethodscanbefurtherinvestigatedtoincludesmartpre{processingandpost-processing.Especiallyinthecaseoftheelementpartitioning,thealgorithmimplementedcreatesrandompartitionsofthesamesize.Thisrandomizationcausesthealgorithmtobefast,howeverasmarterselectionofthepartitionscanbelookedintotoreachsolutionsofevenhigherquality. 65

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CHAPTER4CLIQUECENTRALITY 4.1PreliminariesCentralityisoneofthemostfundamentalconceptsinnetworkanalysis;itisameasureof\importance"or\inuence"ofanactorwithinanetworkandiscommonlyusedtoidentifyitsmost\central"actors.IthasbeenextensivelystudiedintheliteraturefromboththeoreticalandexperimentalperspectivesstartingfromtheseminalpapersofBavelas[ 14 , 15 ],Leavitt[ 85 ],andSabidussi[ 114 ]inthe1950s,andisprimarilyconsideredinsocialnetworkanalysis.Sincethen,adiversesetofindiceshasbeenproposed(see[ 20 , 22 , 77 ]forextendedreviews)andappliedinvariousareasincludingtracmonitoring[ 40 ],internet[ 110 ],electrical[ 96 ],biological[ 121 ]andphysical[ 13 ]networkedsystems.Themajorityofcentralitymeasurescanbecategorizedintothreeprimaryclassesbasedonthefollowingconcepts[ 52 ]:degree,closeness,andbetweenness.Ineachclass,thebasecentralitymetricofanodeAinanetworkisdenedasfollows: Denition1. BasicCentralityMeasures Degreecentrality:ThenumberofnodesadjacenttoA. Closenesscentrality:Theshortest(maximumshortest,averageshortest)distancetonodeAfromanyothernodeinthenetwork. Betweennesscentrality:ThefractionofshortestpathsconnectinganytwoothernodesB,CthatpassthroughA.Thesethreemainclassesofcentralitymetricscapturedierentaspectsof\inuence"ofgraphnodes,anddependingontheapplication,eachoneofthesemetricsmaybepreferredoveranother.Specically,iftheedgesinasocialnetworkrepresentdirectcontactsbetweentwopersons,thenapersonwithmoredirectcontacts(highdegree)canbeviewedasamajorcommunicationchannelofinformationorbeinvolvedinthemainstreamofinformationowinthenetwork.Conversely,apersonwithlessdirectcontacts(lowdegree)isbarelyinvolvedintheinformationowwithothersandmaylackanactive 66

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participationintheongoingcommunicationprocess.Similarly,apersonlocatedonthecommunicationpathsthatlinksalargenumberofpeopletogether(highbetweennesscentrality),mayexhibitapotentialforinuenceorcontrolovertheircommunication.Theconceptofclosenesscentralitywasoriginallysuggestedasameasureofpotentialcontrolofotherswhenapersonpassesitsinformationthroughthenetwork[ 15 ];thefewerthenumberofintermediariesinthecommunicationpathsbetweenapersonandothers,themorecentralthatpersonisconsideredtobe.Formorediscussionontheintuitionandpotentialapplicationsofthesecentralitymetrics,wereferthereaderto[ 52 ].Morerecently,EverettandBorgatti[ 45 , 46 ]extendedthesethreestandardnetworkcentralitymeasurestogroupsandclasses.Theyarguethatgroupcentralitymeasureswillenableresearcherstoanalyzetherelativeinuenceofgroupsorteamswithcertainattributes(age,ethnicity,clubmembership,oroccupation)withinanorganization.Moreover,incommunicationortransportationnetworks,agroupofnodeswiththehighestbetweennesscentralitymaycorrespondtothebestcandidatelocationsforadistributedmonitoringsystem,whichcanbeusedfornetworkmonitoringtaskssuchastracmeasurements[ 40 ],assumingthatnodepairsusetheshortestpathsforcommunication.However,asopposedtodegreeandclosenesscentrality,computingbetweennesscentralityofanygivengroupisacomputationallydemandingtask.Theproblemofcomputinggroupbetweennesscentrality,aswellastheidenticationofthemostprominentgroupwiththemaximumbetweennesscentrality,areconsideredin[ 40 , 41 , 49 , 76 , 109 ].Alternatively,groupcentralitymeasurescanbeappliedtosetsofinformalgroupsofindividualsformingcohesivesubgroups(e.g.,cliques),whichwouldallowtoanalyzewhichonesarethemostandtheleastcentralinthenetwork.Thecliqueconcept,whichwasoriginallyintroducedbyLuceandPerry[ 87 ]in1949tomodelanotionofcohesivesubgroupsinsocialnetworksanalysis,isasubsetofverticesthatarepairwiseadjacent.Itensuresperfectcommunicationbetweengroupentitiesastheyaredirectlylinkedtoeachother.Cliqueshaveavarietyofpracticalapplicationsinscienceandengineering(see[ 2 , 67

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18 , 21 , 29 , 54 ]andreferencestherein).Notonlycliqueswithmaximumcentralities,asthemostinuentialclusters,butalsocliqueswithminimumcentralitiescanbeofparticularinterest.Forinstance,inthecontextofsocialnetworks,cliqueswithverylawcentralitiesmayrepresenthighly-connectedcommunities,whichhaveverypoorcommunicationabilitywiththerestofthenetwork.Thus,themembersofsuchcommunitiesmightbemorelikelytoadheretotheopinionsoftheirgroupmembersandlesslikelytobeinuencedbyinformationwhichtheyreceivefromothernetworkmembers,whichmakesitdiculttoreachasocialconsensusinawholenetwork.Theeectofsuch\remote"communitiesonthenetworkconsensuscharacteristicsismentionedinarecentwork[ 44 ].Inthischapter,weconsidertheproblemofdetectingcliqueswithmaximumandminimumcentralities,usingtheaforementionedstandardcentralitymeasures.Specically,weaddressthefollowingquestion:whichgroupofkmembersinducingacliqueisthemostandtheleastcentralinagivennetwork?AsmostoftheproblemsrelatedtondingcliquesareNP-hard,theproblemsthatweconsiderarealsoNP-hard.Inthiswork,wedevelopmixedintegerprogrammingformulationsforallconsideredcentralitymetrics.Wedemonstratetheperformanceoftheproposedformulationsonsyntheticandreal-lifenetworkinstances.Interestingly,ourndingsindicatethatthemostcentralcliquesinreal-lifenetworksareusuallysmallerthanthesizeofthelargestcliqueinthesamegraph.Thismaybepartiallyexplainedbythefactthatlargercliqueshavelessavailablenodestoinuence.Moreover,thenumberoflargestcliquesisrelativelysmallincomparisontothenumberofcliquesofwithfewernumberofnodes.Thechapterproceedsasfollows.Section2introducesnotationsanddenitionsrelatedtothreemaingroupcentralitymeasures,formallydenesthecliquecentralityproblemandbrieydiscussesitscomplexity.InSection3,wedeveloplinearintegerprogrammingformulationsforthecorrespondingoptimizationproblems.Section4illustratestheperformanceoftheproposedformulationsonvariousreal-lifeandsyntheticnetworkinstances.Moreover,wediscusssomeinterestinginsightsconcerningcliqueswiththe 68

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maximumandminimumcentralities.Finally,wegiveourconcludingremarksandsuggestthedirectionsforfutureworkfollowingthecomputationalresultsinSection4. 4.2Denitions,NotationsConsiderasimpleundirectedgraphG=(V;E)withasetofnnodes(vertices)V=f1;:::;ngandasetofedgesEVV.TwoverticesiandjareconnectedinGifthereisapathPijbetweenthem,i.e,thereexistsasetofverticesfv0;:::;v`gVsuchthatv0=i,v`=j,and(vk;vk+1)2E;8k=0;:::;`)]TJ /F1 11.955 Tf 12.41 0 Td[(1.AgraphGisconnectedifeverypairofnodesi;j2Visconnected.Forsimplicityofexposition,weonlyconsiderconnectedgraphsG,althoughthisworkcanbegeneralizedtodisconnectedgraphs.ApathPijbetweeniandjinGistheshortestone(notethatitisnotnecessarilyunique),andiscalledthegraphgeodesicifitcontainstheleastnumberofedgesamongallpathsconnectingiandjinG.Letg(i;j)bethenumberofgeodesics(distinctshortestpaths)betweennodesiandj,andgt(i;j),ge(i;j)bethenumberofgeodesicsthatcontainnodet2Voredgee2E(e6=(i;j),i6=t,j6=t),respectively.Thelength(i.e.,numberofedges)oftheshortestpathbetweeniandjinGisreferredtoasthedistancebetweeniandjinGandisdenotedbyd(i;j);thelargestdistanceinagraphisreferredtoasthediameter,diam(G)=maxi;j2Vd(i;j).Foranynodei,letN(i)=fjj(i;j)2Egbethesetofneighborsofnodei.ForanysubsetSV,thesubgraphinducedbySinGisdenedbyG[S]=(S;(SS)\E).AnodebelongingtoSisreferredtoasagroupnode,whilenodesinVnSareconsideredtobethenon-groupnodes.LetgS(i;j)bethenumberofgeodesicsbetweennodesi;j2VnSpassingthroughS;N(S)=fj2VnSj(i;j)2E;i2SgbethesetofneighborsofS;andforanynodei2VnS,letd(i;S)=minj2Sd(i;j)bethedistancebetweeniandS.Tobeconsistentwiththeexistingstudiesonsocialnetworks,inthedenitionsbelow,wecallasubgraphG[S]asagroupS.Followingtheabovenotations,thegroupcentralitiesaredenedasfollows[ 45 , 46 ]: Denition2. GroupDegreeCentrality 69

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AgroupSdegreecentralityCd(S)isthenumberofnon-groupnodesthatareconnectedtoS: Cd(S)=jN(S)j(4{1)Notethatmultiplelinkstothesamenodearecountedonlyonce. Denition3. GroupClosenessCentralityAgroupSclosenesscentralityisbasedonthedistancesfromStoallnodesVnSoutsidethegroup.Weconsidertwovariantsofclosenesscentrality,themaximumdistanceCc1(S)andthetotaldistanceCc2(S)tooutsidenodes:Cc1(S)=maxi2VnSd(S;i)Cc2(S)=Xi2VnSd(S;i) (4{2)Observethatsinceweconsidergroupswithaxedsize,optimizingthetotaldistanceisequivalenttooptimizingtheaveragedistancefromagroupStooutsidenodesVnS,whichisequaltoCc2(S)=(n)]TJ /F3 11.955 Tf 12.26 0 Td[(k).Intheexistingliterature,themaximumdistancefromagivennodetoeveryothernodeisknownasnodeeccentricity[ 64 ].Inthissense,Cc1(S)canbeviewedasagroupeccentricitymetric. Denition4. GroupBetweennessCentralityAgroupSbetweennesscentralitymeasureistheproportionofgeodesicsconnectingpairsofnon-groupmembersthatpassthroughS: Cb(S)=Xi;j2VnSgS(i;j) g(i;j)(4{3)NotethatthetermgS(i;j) g(i;j)inthedenitionofbetweennesscentralityinEquation 4{3 isalsoknownintheexistingliterature[ 51 ]astheprobabilitythatapairofnodesi;j2VnScommunicatesusingapathgoingthroughSiftheytheyselectacommunicationpathrandomlyamongallpossibleshortestpaths.Inthiscase,Cb(S)istheaveragenumberofcommunicationsinthegraphGwhichgothroughS.Inaddition,onewouldnaturally 70

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considertheoptimisticandthepessimisticcasesofbetweennesscentralitydenedas Cb)]TJ /F1 11.955 Tf 6.76 -7.75 Td[((S)=Xi;j2VnSgS(i;j) g(i;j);Cb+(S)=Xi;j2VnSgS(i;j) g(i;j)(4{4)Intheoptimisticcase,eachpairofnodesi;j2VnSalwayspreferstocommunicateusingapathgoingthroughSifthereisageodesicgoingthroughS.Inthemostpessimisticcase,eachpairofnodesi;j2VnSalwaystriestoavoidtocommunicatethroughapathgoingthroughS;thus,theywillcommunicatethroughnodesinSifallgeodesicsgothroughS.Observethat,asopposedtostandardbetweennesscentralityCb(),computingCb+(S)isstraightforwardandcanbedoneusingcentralitiesCb+()ofindividualnodesinSsince Cb+(S)=Xi;j2VnSmaxt2Sgt(i;j) g(i;j)=Xi;j2VnSmaxt2SCb+(t)(4{5)However,Cb)]TJ /F1 11.955 Tf 6.76 -7.15 Td[((S)cannotbecomputedjustusingCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[(()forindividualnodes.Forexample,ifS=fa;bgitiseasytoconstructanexampleoftwographsinwhichthereexistsapairofnodesi;jsuchthatCb)]TJ /F1 11.955 Tf 6.76 -7.15 Td[((a)=Cb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((b)=0inbothgraphs,butCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((S)=1inonegraph,andCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((S)=0intheother(Fig. 4-1 ).AcliqueSisasubsetofVsuchthatthesubgraphG[S]inducedbySinGiscomplete[ 21 ],i.e.,allitsnodesarepairwiseadjacent.ThemaximumcliqueproblemseeksforacliqueSofmaximumcardinalityjSjinG.ThedecisionversionofthisproblemisNP-complete[ 56 ].Inthisstudy,wefocusonndingcliquesofagivensizewithmaximumandminimumcentralities.Theformaldenitionoftheproblemforndingthemostandtheleastcentralcliquesanditsdecisionversionarethefollowing: Problem1(cliquewithmaximum/minimumcentrality). GivenagraphG=(V;E),acentralitymetricC()2fCd();Cc1();Cc2();Cb();Cb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[(();Cb+()gandapositiveintegerk,ndacliqueSofsizek(k=jSj)withmaximum/minimumcentralityvalueC(S). 71

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Figure4-1. AnillustrativeexamplethatcomputingindividualnodebetweennesscentralitiesisnotenoughtocomputeCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((S). i a b j i a b j j Denition5. CLIQUECENTRALITY:GivenagraphG=(V;E),acentralitymetricC()2fCd();Cc1();Cc2();Cb();Cb)]TJ /F1 11.955 Tf 6.76 -7.15 Td[(();Cb+()g,anon-negativenumbersandanintegerk,isthereacliqueSVofsizeksuchthatC(S)s(orC(S)s)?AsanalremarkwenotethattheCLIQUECENTRALITYproblemisNP-completesinceitcontainstheCLIQUEproblemasaspecialcasewhens=0(ors=jVj2).Therefore,Problem 1 isNP-hard. 4.3MixedIntegerProgramming(MIP)FormulationsInthissection,wedeveloplinearmixed0{1programmingformulationsforndingcliqueswithmaximum/minimumcentralitiesforallconsideredcentralitymetrics:degreeCd(),closenessCc1(),Cc2(),andthreeversionsofbetweennessCb(),Cb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((),Cb+().Letxi(i2V)bea0-1variablesuchthatxi=1ii2VisinacliqueS.Tomodelcliqueconstraintsweusethestandardtechniquethatifthereisnoedgebetweennodesiandj,i.e.,(i;j)=2E,thenacliqueScannotsimultaneouslycontainbothnodesiandj: xi+xj1;8(i;j)=2E:(4{6)BelowarethemodelingtechniquesandIPformulationsforeachcentralitymetric. 72

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4.3.1DegreeCentralityTheIPformulationfordegreecentralityisstraightforward.Letyi=1ii2N(S).Then,Cd(S)=Pi2Vyiandtheformulationcanbewrittenasfollows:(MAX-Cd):maxXi2Vyi (4{7)s:t:yiXj:(i;j)2Exj;8i2V (4{8)xi+yi1;8i2V (4{9)xi+xj1;8(i;j)=2E (4{10)Xi2Vxi=k; (4{11)xi;yi2f0;1g;8i2V (4{12)whereConstraints 4{8 ensurethatyi=0ifiisnotaneighborofacliqueS(i=2N(S)),Constraints 4{9 ensurethatyi=0ifiisinacliqueS.Notethatsincethisisamaximizationproblem,wedonotneedanextrasetofconstraintstoenforceyi=1forallothercases.However,theseconstraintswillberequiredfortheminimizationversionoftheproblem.TheremainingsetofConstraints 4{10 and 4{11 enforceStobeacliqueofsizek. 73

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Similarly,weobtaintheformulationforndingacliqueofsizekwithminimumCd(S):(MIN-Cd):minjVjXi=1yi (4{13)s:t: 4{8 throughout 4{11 ;xi+yi1 iXj:(i;j)2Exj;8i2V (4{14)xi;yi2f0;1g;8i2V (4{15)whereConstraints 4{14 ensurethatyi=1ifiisnotinacliqueSandisaneighborofatleastonecliquenode,iissucientlylargeconstant,whichforthetightnessofLPrelaxationpurposescanbesettoi=min(jN(i)j;k).NotethatConstraints 4{8 and 4{9 canbeomittedduetothestructureoftheobjectivefunction. 4.3.2ClosenessCentralityTocalculatetheclosenesscentralityCc1(S)orCc2(S)ofacliqueS,weneedtocomputethedistancefromeverynodei2VnStothecliqueS.TomodelsuchdistancesinthecorrespondingIPformulations,weintroduceasetofbinaryvariablesx(`)i(i2V;`=1;:::;diam(G))whichisdenedasfollows.First,thevariablesx(0)i(i2V)determinethecliqueS,i.e,x(0)i=xi=1ii2S.Then,thevariablesx(1)i(i2V)denethesetofnodesthatiseitherinSoradjacenttoS,i.e.,x(1)i=1ii2N(S)[S.Forothervalues`,x(`)i=1id(i;S)`.Observethatthenumberofnodes,whosedistanceis`fromacliqueSisPi2Vx(`)i)]TJ /F7 11.955 Tf 13.75 8.96 Td[(Pi2Vx(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)i.Sinced(i;S)diam(G)foranycliqueSandi2VnS,thenwejustneedtoconsider`=1;:::;diam(G).TheappropriatevaluesofthesevariablesinthecorrespondingIPformulationsaremodeledrecursivelyusingthefollowingobservation:x(`)i=1ix(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)i=1,oriisadjacenttonodejsuchthatx(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j=1.Thisobservationcanbewrittenintermsoflinear 74

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constraintsasfollows:x(`)iXj:(i;j)2Ex(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j+x(`)]TJ /F6 7.97 Tf 6.58 0 Td[(1)i;8i2V;`=1;:::;diam(G) (4{16)x(`)i1 i0@x(`)]TJ /F6 7.97 Tf 6.58 0 Td[(1)i+Xj:(i;j)2Ex(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j1A;8i2V;`=1;:::;diam(G) (4{17)whereiissucientlylargeconstant,whichforthetightnessofLPrelaxationpurposescanbesettoi=1+jN(i)j.RecallthatweconsidertwoversionsofclosenesscentralitymetricsCc1(S)andCc2(S)whichconsiderthemaximumandthetotal(oraverage)distancetoacliqueS,respectively.Fortherstmetric,Cc1(S),weintroduceanextrasetofbinaryvariablesv`(`=1;:::;diam(G)),wherev`isequalto1ithereisatleastonenodewithdistance`toS.Inotherwords,v`=1iPi2Vx(`)i)]TJ /F7 11.955 Tf 13.58 8.97 Td[(Pi2Vx(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)i1.Inthiscase,Cc1(S)canbeexpressedasCc1(S)=diam(G)P`=1v`;whichleadstothefollowingformulationforndingacliqueSwiththeminimumdistancetotheoutsidenodes:(MIN-Cc1):mindiam(G)X`=1v` (4{18)s:t: 4{10 , 4{11 , 4{16 , 4{17 v`Xj2V(x(`)j)]TJ /F3 11.955 Tf 11.95 0 Td[(x(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j);1`diam(G) (4{19)v`1 jVjXj2V(x(`)j)]TJ /F3 11.955 Tf 11.95 0 Td[(x(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j);1ldiam(G) (4{20)x(`)i;v`2f0;1g;8i2V;0`diam(G)Constraints 4{19 and 4{20 ensurethatvariablev`=1ifandonlyifthereexistsatleastonenodewithdistance`fromthecliqueS.ForthecentralitymetricCc2()whichcountsthetotaldistance,theformulationisstraightforward: 75

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(MIN-Cc2):mindiam(G)X`=1` Xi2Vx(`)i)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xi2Vx(`)]TJ /F6 7.97 Tf 6.59 0 Td[(1)i! (4{21)s:t: 4{10 , 4{11 , 4{16 , 4{17 x(`)i2f0;1g8i2V;0`diam(G)ObservethatbothformulationsMIN-Cc1andMIN-Cc2identifythemostcentralcliquessincetheyndcliqueswithminimumdistanceortotaldistancetoothernodes.Theformulationscontain(ndiam(G))variablesandconstraints.Inordertondtheleastcentralcliques,theobjectivesinthecorrespondingformulationshavetobemaximized.Notethatthistechniquecanbealsousedfordevelopingformulationswithotherversionsofclosenesscentralitymetrics,suchasCc(S)=Pi2VnS1 d(i;S)orCc(S)=Pi2VnS1 2d(i;S)[ 36 ]. 4.3.3BetweennessCentrality 4.3.3.1Standard"Probabilistic"CaseAswementionedbefore,computingstandardbetweennesscentralitydenedbyEquation 4{3 foragroupisnotaneasytask.However,whenthegroupformsaclique,itsbetweennesscentralitycanbecomputedusingcentralitiesofindividualnodesandedges,aswedemonstrateinthefollowingproposition. Proposition4.1. LetSVbeacliqueinagraphG=(V;E)withthesetofedgesEs=(SS)\E.Then,foranypairofnodesi;j2VnS: gS(i;j)=Xt2Sgt(i;j))]TJ /F7 11.955 Tf 13.21 11.36 Td[(Xe2Esge(i;j):(4{22) Proof. ObservethatanygeodesicpathPijbetweennodesiandjwhichgoesthroughthecliqueSmaycontainoneortwocliquenodesonly.Otherwise,itiseasytoverifythatitcannotbegeodesic,i.e.,theshortestone. 76

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Thenumberofgeodesicpathsgoingthroughonlyonecliquenodetforallt2SisappropriatelycountedintheexpressionPt2Sgt(i;j).However,thenumberofgeodesicpathsgoingthroughtwocliquenodess,tintheexpressionPt2Sgt(i;j)iscountedtwiceforanypossibles;t2S(onceforsandoncefort).ObservethatifashortestpathPijcontainsnodess;t2Sthenitgoesthroughanedge(s;t)2EsasthecliqueScontainsallpossibleedges.Hence,togettherightnumberoftotalgeodesicpathsbetweennodesiandjgoingthroughacliqueS,weneedtosubstractPe2Esge(i;j)fromPt2Sgt(i;j),whichendstheproofoftheproposition. FortheIPformulation,weneedtocomputegt(i;j)andge(i;j)foralli;j;t2Vande2E.Forsimplicity,letgij=g(i;j),gt(i;j)=gijtandge(i;j)=gije,whereanindextreferstonodesandereferstoedges.Also,letue(e2E)beabinaryvariablesuchthatue=1ie2Es.Tocalculategijt,weusetheclassicalall-pairsshortestpathalgorithmofFloyd-Warshall[ 50 ],wherewecomputeallshortestpathsbetweenanytwonodesofthegraph.Thealgorithmwasmodiedtoaccountformultipleshortestpathsconnectingtwonodes.Foreveryedgee2E,ge(i;j)isabyproductofthepreviouscalculations.Theformulationforndingacliquewiththemaximumstandardbetweennesscentralitycanbewrittenasfollows:(MAX-Cb):maxXi;j2V0B@Pt2Vgijtxt)]TJ /F7 11.955 Tf 13.76 8.96 Td[(Pe2Egijeue gij1CA (4{23)s:t: 4{10 , 4{11 ;uexi;uexj;uexi+xj)]TJ /F1 11.955 Tf 11.95 0 Td[(1;8e=(i;j)2E (4{24)xi;ue2f0;1g;8i2V;e2E (4{25)whereConstraint 4{24 ensurethatue=1ifandonlyife2Ec. 77

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ObservethattheobjectivefunctioninEquation 4{23 alsoincludesthegeodesicsbetweencliqueandnon-cliquemembers.ThenextpropositionshowsthatitstillreturnsthecorrectvalueofbetweennesscentralitydenedbyEquation 4{3 . Proposition4.2. TheobjectivefunctionofEquation 4{23 returnsthecorrectvalueforbetweennesscentrality,i.e., Cb(S)=Xi;j2VnSgS(i;j) g(i;j)=Xi;j2V0B@Pt2Vgijtxt)]TJ /F7 11.955 Tf 13.76 8.97 Td[(Pe2Egijeue gij1CA(4{26) Proof. ByProposition 4.1 anddenitionofvariablesxtandue,Cb(S)=Xi;j2VnSgS(i;j) g(i;j)=Xi;j2VnS0B@Pt2Vgijtxt)]TJ /F7 11.955 Tf 13.76 8.97 Td[(Pe2Egijeue gij1CAHence,toprovetheproposition,weneedtoshowthat Xt2Sgijt)]TJ /F7 11.955 Tf 13.2 11.36 Td[(Xe2Esgije=0;8i2S;j2V:(4{27)Considertwopossiblecasesfornodejseparately: j2S:sinceSisaclique,thenthereisonlyoneshortestpathbetweennodesiandj(i;j2S)anditgoesthoughanedge(i;j)2Es.Hence,gije=0foralle2Esandgijt=0forallt2S,andEquation 4{27 istruefori;j2S. j2VnS:ifageodesicPijfromacliquenodeitothenon-cliquenodejdoesnotcontaincliquenodes,thengije=0foralle2Esandgijt=0forallt2S,andEquation 4{27 isalsotrue.However,ifPijgoesthroughacliqueS,itmusthavegothroughonlyoneothernodet2Sandanedge(i;t)2Es.Otherwise,theshortestpathdenitionwillbeviolated.Inthiscase,thenumberofsuchpathswillbecountedinPt2Sgt(i;j)andinPe2Esge(i;j)onlyonceandEquation 4{27 stillholds. 4.3.3.2PessimisticCaseWeintroducenew0-1variablestij(i;j2V)suchthattij=1iallgeodesicsbetweennodesi;jpassthroughacliqueS.Then,theproblemformulationwhichmaximizesCb)]TJ /F1 11.955 Tf 6.76 -7.16 Td[(()becomesthefollowing: 78

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(MAX-Cb)]TJ /F1 11.955 Tf 6.75 -7.75 Td[():maxXi;j2Vtij (4{28)s:t: 4{10 , 4{11 , 4{24 ;gijtijXt2Vgijtxt)]TJ /F7 11.955 Tf 11.95 11.36 Td[(Xe2Egijeue;8i;j2V (4{29)tij+xi1;tij+xj1;8i;j2V (4{30)xi;tij;ue2f0;1g;8i;j2V;e2E (4{31)whereConstraints 4{29 ensurethattij=0iatleastonegeodesicbetweennodesi;jdoesnotpassthroughacliqueS,andConstraints 4{30 makesurethatonlygeodesicsbetweennon-cliquenodesareconsidered.TheformulationforminimizingbetweennessCb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[(()isthefollowing:(MIN-Cb)]TJ /F1 11.955 Tf 6.75 -7.75 Td[():minXi;j2Vtij (4{32)s:t: 4{10 , 4{11 , 4{24 ;tij1+Xt2Vgijtxt)]TJ /F7 11.955 Tf 11.96 11.35 Td[(Xe2Egijeue)]TJ /F3 11.955 Tf 11.95 0 Td[(gij)]TJ /F3 11.955 Tf 11.95 0 Td[(xi)]TJ /F3 11.955 Tf 11.96 0 Td[(xj;8i;j2V (4{33)xi;tij;ue2f0;1g;8i;j2V;e2E (4{34)whereConstraints 4{33 ensurethattij=1iallgeodesicsbetweennodesi;j2VnSpassthroughacliqueS.NotethatConstraints 4{29 and 4{30 arenotrequiredduetothestructureoftheobjectivefunction. 79

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4.3.3.3OptimisticCaseFortheoptimisticcase,recallthatthebetweennesscentralitycanbecomputedusingindividualnodecentralitiesas Cb+(S)=Xi;j2VnSmaxt2Sgt(i;j) g(i;j)(4{35)Letzij(i;j2V)beabinaryvariablesuchthatzij=1ithereexistsashortestpathbetweennodesiandjwhichpassesthroughatleastonenodeinthecliqueS.Letalso, fijt=gt(i;j) g(i;j);8i;j;t2V(4{36)Then,theproblemformulationforndingacliquewiththemaximumCb+()isthefollowing:(MAX-Cb+):maxXi;j2Vzij (4{37)s:t: 4{10 , 4{11 ;zij1 jVjXt2Vfijtxt)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)]TJ /F3 11.955 Tf 11.95 0 Td[(xj;8i;j2V (4{38)zijXt2Vfijtxt8i;j2V (4{39)xi;zij2f0;1g;8i;j2V (4{40)IntheobjectivefunctioninEquation 4{37 ,weaimtomaximizethenumberofshortestpathsthatusethenodesoftheclique.Constraints 4{38 and 4{39 modeltheindicatorvariableszij,byenforcingthemtobeequalto1wheneverthereexistsashortestpathbetweennodesiandjthatpassesthroughanodeofthecliqueS,and0otherwise.Toobtaintheformulationforndingtheleastcentralcliqueswiththeminimumvalueofbetweennesscentrality,theobjectiveneedstobeminimizedwiththe 80

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samesetofconstraints.NotealsothatConstraints 4{38 and 4{39 canbeomittedforthemaximization/minimizationversionoftheproblem. 4.4ComputationalExperimentsInthissection,wepresenttheresultsofourcomputationalexperimentstodemonstratetheperformanceofthedevelopedMIPformulationsandnumericallyillustratethedierencebetweenmaximumandminimumcliquecentralitieswithrespecttothecliquesizejSj=kinreal-lifeandsyntheticnetworksusingvariouscentralitymetrics.ThecomputationalexperimentswereperformedonaserverwithtwoAMDOpteron6128Eight-CoreCPUsand12GBofRAM,runningLinuxx8664,CentOS5.9.AllformulationswereimplementedinC++andsolvedusingGurobi5.50,whilegraphoperationswereperformedusingNetworkX[ 63 ].Thereal-lifenetworkinstancesareobtainedfrompubliclyavailabledatafromtheUniversityofFloridaSparseMatrixCollectiondatabase[ 37 ](whichalsoincludesDIMACS10ChallengeCollection[ 38 ])andCOLOR02/03/04[ 1 ].Specically,theconsiderednetworksinclude:karate(socialnetworkofakarateclubwith34membersataUSuniversityinthe1970s[ 123 ]),krebs(terroristnetworkofthe9/11hijackersandtheirassociates,compiledbyKrebs[ 81 ]),dolphins(socialnetworkoffrequentassociationsbetween62dolphinsinacommunitylivingoDoubtfulSound,NewZealand[ 88 ]),ieeebus(theIEEE118BusTestCaseisaportionoftheAmericanElectricPowerSystemintheMidwesternUSasofDecember,1962[ 37 ]),andbookgraphsanna,david,huck,jeancorrespondingtofourclassicworks:Tolstoy'sAnnaKarenina,Dicken'sDavidCoppereld,Twain'sHuckleberryFinn,andHugo'sLesMiserables[ 1 ].Wealsoincludeexperimentsontwoclassesofrandomlygeneratedgraphinstances:Erdos-RenyiandBarabasi-AlbertgraphsgeneratedaccordingtotheclassicalG(n;p)[ 42 ],andpreferentialattachmentsmodels[ 112 ],respectively.TheresultsofthecomputationalexperimentsaresummarizedinTables 4-1 4-4 .Foranygraphinstance,weconsiderallpossiblevalueskforwhichacliqueofsizek 81

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existsinthisgraphandidentiedcliqueswithmaximumandminimumcentralitiesCd();Cc1();Cc2(),Cb(),Cb)]TJ /F1 11.955 Tf 6.75 -7.15 Td[((),Cb+().Forabetterrepresentation,thevaluesofdegreeandbetweennesscentralitiesarereportedasapercentageofmaximumpossiblevalues,i.e.,wescaledthembyn)]TJ /F1 11.955 Tf 11.38 0 Td[(1and(n)]TJ /F3 11.955 Tf 11.37 0 Td[(k)(n)]TJ /F3 11.955 Tf 11.38 0 Td[(k)]TJ /F1 11.955 Tf 11.38 0 Td[(1)=2,respectively.WealsoscaleCc2()byn)]TJ /F3 11.955 Tf 11.38 0 Td[(ksothatthecorrespondingvaluerepresenttheaveragedistancefromacliquetooutsidenodes.Oneoftheimmediateobservationsisthatthevariationbetweenmaximumandminimumcentralitiesofthesamesizecliquesisquitelargeevenforthecliqueswhosesizesareclosetothesizeofthemaximumclique.Moreover,forseveralreal-lifenetworkinstances(krebs,dolphins,ieeebusandsomebookgraphs),thecliqueswiththemaximumcentralitiesamongallpossiblecliquesizesarenotnecessarilythelargestones.However,nosuchpatternisobservedinsyntheticnetworks,whichmaybeattributedtotheirspecictopologicalstructure.Theseobservationssuggestthatlargecliquesarenotnecessarilythemostcentralinreal-lifenetworks.Inaddition,thecentralityvaluesofmostcentralcliquesseemtobereasonablylargetoo.Forexample,thereisacliqueofsize3inkrebsnetworksuchthatitsneighborsetcontainsmorethan50%oftheremainingnodes.Also,thereisacliqueofsize3indolphinsnetworksuchthatmorethan80%pairsofnodeshaveatleastoneshortestpathgoingthroughthisclique.Notethattheindividualnodecentralitiesarenotthatlargeinthecorrespondingexperiments.Table 4-2 providesthecomputationaltimesforsolvingproblemspresentedinTable 4-1 .Notethatthecomputationalruntimedoesnotalwaysincreasewiththesizeofthecliquesthatarebeingconsidered.Instead,thereisaconsistentspikein\average-sized"cliques.Further,wecanobservethatcomputingbetweennesscentralityusingtheoptimisticmodelusuallytakesconsiderablymoretimethanusingthestandard(probabilistic)versionofthemetric,oritspessimisticcounterpart.Inthischapter,weconsideredacentrality-basedmodeltomeasuretherelativeinuenceofcliqueswithinanetworkbyutilizingthreeclassicalgroupcentralitymeasures:degree,closeness,andbetweennessaswellastheirintuitivealternations.Foreach 82

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centralitymetricwedevelopedalinear0-1programmingformulationforidentifyingcliquesofanygivensizekwithmaximumandminimumcentralityvalues.Thenumericalexperimentsdemonstratethatlargecliquesmayexhibitasignicantvariationintheirconnectiontotherestofthenetwork.Moreover,themostcentralcliquesarenotusuallythelargestones,whichsuggeststhattheperfectcommunicationamongcliquemembersdoesnotnecessarilyimplytheirgoodintegrationwithinawholenetwork.Thedirectionsoffutureresearchmayincludethedevelopmentsofmoreadvancedexactorheuristicmethodstohandlelarge-scalenetworkinstances.Inaddition,ascliquesaresometimesviewedasoverlyrestrictivegroupstructures,thisapproachcanbegeneralizedforstudyingcentralitiesoflessrestrictivebutstillhighly-connectedsubgraphsknownasclique-relaxations[ 101 ],i.e.,quasi-cliques,k-plexes,k-clubs,etc. 83

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Table4-1. Maximumandminimumcliquecentralitiesinreal-lifesocialandpowergridnetworkinstances. Degree(%)ClosenessBetweenness(%)jSjCd()Cc1()Cc2()Cb()Cb)]TJ /F6 7.97 Tf 6.39 -5.33 Td[(()Cb+()MaxMinMinMaxMinMaxMaxMinMaxMinMaxMin karate:jVj=34;jEj=78,maxcliquesize=5151.513.12351.763.5239.393.1316.970.0054.195.88262.509.38251.382.6651.694.3621.160.0066.1311.59364.529.68241.352.6859.717.9623.370.0068.9817.11463.3340.00241.371.8067.2010.6224.791.3870.9439.48562.0758.62231.381.4567.7415.2424.952.5979.7669.16krebs:jVj=62;jEj=153,maxcliquesize=6136.071.64351.704.0049.341.6730.360.0072.713.23251.673.33251.553.2561.132.0035.280.0077.636.40355.936.78251.513.0868.383.9537.890.0081.179.52448.2712.07341.592.4872.616.2038.910.2181.8614.67547.3715.79341.602.3773.0811.3438.961.0381.7623.51642.8630.36341.642.1873.0815.5638.683.3280.1232.05dolphins:jVj=62;jEj=159,maxcliquesize=5119.671.64582.395.6116.711.679.080.0037.543.23230.003.33482.134.6833.002.0613.790.0049.346.40333.905.08472.023.7636.704.5515.500.0055.6310.34431.0313.79572.173.3833.009.1113.700.8851.0816.23528.0721.05672.723.2535.2716.5811.955.9537.1227.74ieeebus:jVj=118;jEj=179,maxcliquesize=417.690.857144.258.8527.150.8614.160.0040.561.69210.340.867143.978.8444.941.1921.040.0054.213.38310.432.618123.987.0337.172.2618.080.0049.795.8546.146.14995.365.368.625.054.121.0310.5510.55 84

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Table4-2. Computationaltimes(inseconds)forsolvingthemaximumandminimumcliquecentralityproblemsinreal-lifesocialandpowergridnetworkinstancesfromTable 4-1 . Degree(%)ClosenessBetweenness(%)jSjCd()Cc1()Cc2()Cb()Cb)]TJ /F6 7.97 Tf 6.39 -5.32 Td[(()Cb+()MaxMinMinMaxMinMaxMaxMinMaxMinMaxMin karate:jVj=34;jEj=78,maxcliquesize=510.020.030.110.040.060.050.100.120.010.010.270.0320.030.040.250.100.150.090.610.780.040.021.130.5830.650.110.440.430.160.100.580.040.030.021.241.0140.640.100.330.270.130.100.110.100.020.010.980.8650.600.090.300.260.130.100.060.090.020.010.450.37krebs:jVj=62;jEj=153,maxcliquesize=610.050.030.160.170.120.100.270.050.030.010.450.0820.160.160.170.200.250.301.445.280.360.020.671.3030.150.160.240.250.410.521.210.370.280.021.552.1740.140.090.130.130.130.161.730.600.100.012.201.7050.050.050.080.080.080.080.500.270.100.010.680.4760.040.040.040.040.070.060.250.120.080.010.830.31dolphins:jVj=62;jEj=159,maxcliquesize=510.050.030.230.220.270.170.420.030.030.011.090.0620.140.071.870.400.600.214.072.470.460.021.921.1030.310.120.210.230.770.268.796.880.470.024.741.8840.330.260.410.260.940.347.600.790.480.0119.422.2450.060.050.120.110.130.133.420.170.070.0117.530.66ieeebus:jVj=118;jEj=179,maxcliquesize=410.240.270.310.3210.9111.285.100.350.190.015.910.5620.921.0322.7221.9526.9517.1728.6431.041.740.03354.70343.0930.580.5922.8314.4713.047.5330.022.980.560.02272.58234.2840.380.5110.865.214.784.3514.531.880.420.01171.83100.29 85

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Table4-3. Maximumandminimumcliquecentralitiesinbookgraphs. Degree(%)ClosenessBetweenness(%)jSjCd()Cc1()Cc2()Cb()Cb)]TJ /F6 7.97 Tf 6.39 -5.32 Td[(()Cb+()MaxMinMinMaxMinMaxMaxMinMaxMinMaxMin huck:jVj=74;jEj=303,maxcliquesize=11172.601.37361.384.8956.941.3933.460.0077.672.70279.171.38361.314.9069.791.4438.630.0083.895.37383.101.41361.283.9678.642.9841.580.0086.308.00485.717.14351.262.2781.493.4743.070.0088.2610.59588.417.79351.232.2784.934.8344.150.0089.5613.14688.244.42351.242.2987.346.4944.940.0090.4115.66788.064.48351.242.3187.349.7244.940.0090.4118.14880.304.55351.322.3378.5312.1341.700.0087.2320.58980.0015.38351.322.0978.5615.2041.710.0587.2624.861079.6917.19451.332.0778.5620.1441.711.8887.2628.381173.0273.02441.401.4070.1244.9738.7413.3783.1983.19jean:jVj=80;jEj=257,maxcliquesize=10145.571.27361.614.2251.301.2826.360.0071.652.5261.551.28351.423.2660.511.3530.830.0080.634.97367.535.19351.363.0668.002.6834.330.0086.307.41469.743.95351.343.0973.163.1436.030.0088.969.81566.672.67351.373.1273.544.2335.310.0087.3412.18664.862.70351.393.1573.544.7534.810.0084.2114.53764.384.11351.402.7873.996.5534.900.0084.3016.84834.729.72441.862.2640.447.7219.150.0249.7219.98933.8011.27441.872.2540.579.2819.190.0449.8125.241032.8621.43441.892.1340.6612.2219.220.5849.8736.57david:jVj=87;jEj=406,maxcliquesize=11195.351.16231.052.5357.581.1838.130.0088.242.3021001.181312.5564.501.2042.630.0092.194.5731004.761312.5569.072.4544.830.0092.976.8241003.611312.5773.563.6046.650.0093.699.0451006.101311.9479.344.5648.350.0094.3311.2361004.931211.9582.016.9549.160.0094.7613.39710015.001211.8583.408.7249.600.0095.1115.75810026.581211.7383.6710.4449.870.0095.3518.47910032.051211.6884.1812.1750.060.0095.5421.721010064.941211.3584.8719.0850.230.8995.7038.6011100100111184.9029.2350.2515.5195.7595.72anna:jVj=138;jEj=493,maxcliquesize=11151.820.73351.534.2428.010.7413.660.0055.331.45272.790.74351.283.2641.720.7519.770.0073.912.89381.482.96241.192.6953.721.5624.910.0083.294.32487.312.99241.132.6968.622.3329.310.0089.875.73591.733.76341.092.4775.373.1331.660.0094.207.15693.945.30341.072.3382.124.0433.180.0096.428.71794.6628.24341.061.7486.917.9833.900.8897.2316.76888.4642.31341.121.6077.7610.4331.741.2692.8221.56988.3751.16341.121.5078.4215.7731.823.0492.8929.901076.5659.38341.251.4268.3423.3928.377.8484.1745.651176.3876.38331.251.2568.3438.1528.3721.8184.1784.17 86

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Table4-4. MaximumandminimumcliquecentralitiesinrandomlygeneratednetworkinstancesaccordingtoErdos-RenyiG(n;p)preferentialattachmentsmodel. Degree(%)ClosenessBetweenness(%)jSjCd()Cc1()Cc2()Cb()Cb)]TJ /F6 7.97 Tf 6.4 -5.32 Td[(()Cb+()MaxMinMinMaxMinMaxMaxMinMaxMinMaxMin Erdos-Renyi(n=50;p=0:20):jEj=249,maxcliquesize=4132.6510.20231.672.222.892.202.670.0017.964.47252.0814.58231.482.065.673.964.570.0028.578.05368.0931.91231.321.728.427.136.270.0536.1612.37476.0943.48221.241.5710.7810.587.330.2941.2318.57Erdos-Renyi(n=50;p=0:30):jEj=375,maxcliquesize=4151.0216.33231.491.902.222.211.420.0022.374.04275.0727.08231.251.734.564.181.660.0032.828.16387.2344.68221.131.559.569.212.250.0039.9112.12493.4860.87221.071.3912.7012.026.520.0545.8816.45Erdos-Renyi(n=75;p=0:20):jEj=528,maxcliquesize=4129.739.46231.732.051.701.491.390.0011.782.79250.6817.81231.491.853.422.812.690.0020.435.53361.1133.33231.391.685.114.683.730.0425.668.76464.7946.48221.351.546.956.804.540.1228.1811.69Erdos-Renyi(n=75;p=0:30):jEj=835,maxcliquesize=6140.5417.57231.581.851.491.410.890.0013.552.67267.1232.88221.331.672.872.831.720.0024.075.30380.5647.22231.191.534.424.382.490.0031.647.98487.3263.38221.131.376.175.933.180.0037.3310.65590.1372.86221.101.278.007.563.790.0040.5413.53691.3085.51221.091.1410.359.214.340.1043.7517.05 87

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Table4-5. MaximumandminimumcliquecentralitiesinrandomlygeneratednetworkinstancesaccordingtoBarabasi-Albertpreferentialattachmentsmodel. Degree(%)ClosenessBetweenness(%)jSjCd()Cc1()Cc2()Cb()Cb)]TJ /F6 7.97 Tf 6.39 -5.32 Td[(()Cb+()MaxMinMinMaxMinMaxMaxMinMaxMinMaxMin Barabasi-Albert(n=50;=2):jEj=96,maxcliquesize=3128.574.08351.843.3718.840.6712.240.0047.594.04241.676.25351.653.0125.714.0918.830.0059.108.29346.8112.77341.552.5529.397.9425.060.5664.7313.59Barabasi-Albert(n=50;=3):jEj=141,maxcliquesize=4141.866.12341.632.9211.142.118.330.0043.354.07258.376.25241.422.5816.753.3014.940.0059.178.14365.9614.89231.342.2320.035.8519.740.0963.9212.41463.0450.12231.371.5224.3524.0221.607.6367.3541.59Barabasi-Albert(n=75;=2):jEj=146,maxcliquesize=3127.032.70351.843.7417.950.3313.070.0050.952.67245.212.74351.623.4428.351.5324.920.0070.595.30352.1712.59341.492.4031.436.6229.890.9875.6012.74Barabasi-Albert(n=75;=3):jEj=216,maxcliquesize=4135.134.05351.723.2811.141.165.940.0039.142.69253.425.48241.482.7316.752.3613.600.0058.885.40361.118.33241.402.4719.743.4717.390.0862.998.63459.1543.66231.411.5821.6017.0515.254.8158.1333.62 88

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CHAPTER5THEINFLUENTIALCLIQUEPROBLEM 5.1PreliminariesLetG(V;E)beasimple,undirectedgraphonjVj=nverticesandjEj=medges.AcliqueCVisawell-knowngraphstructure,whereanytwonodesi;j2Careconnectedbyanedge.TheproblemofdetectingthemaximumcliqueofagraphisaninfamousNP-hardproblemthathasattractedsignicantscienticandresearchinterestovertheyears.Further,denotetheopenneighborhoodofanodei2VasN(i)=fj:(i;j)2Eg.TheclosedneighborhoodofanodecanbesimilarlydenedasN[i]=N(i)[fig.WecanextendthesedenitionstoasetofverticesSVasN(S)=fj:(i;j)2E;forsomei2S;j=2SgandN[S]=N(S)[S.Last,wewillbeusingStosignalthecomplementofasetofnodesS,i.e.S=VnS.Wedenetheproblemofdetectingacliqueofmaximumsize,suchthatitisinuential.A(k;l)inuentialcliqueisconsideredtobeawell-connectedcliqueinsideagraphwithatleastkneighbors,andatleastlnodeswithinthecliquethatareconnectedtonodesoutsidetheclique.Informally,Inourframework,thisimpliesthatacliqueCcanreachmultiplenodeswithinonehop;andaseriesofnodeswithinthecliquehaveatleastoneoutgoingedge.Notethat,despitethesimilarities,thisproblemisinherentlydierentthanthedominatingcliqueproblem[ 35 ].Asanexample,wereferthereadertoFigure 5-1 .Inthegure,itiseasytoseethatwhilethegraphhasadominatingclique,thatsamecliquewouldnotbeconsideredwell{connected,sincethereisonlyoneoutgoingedgetooneoutsidenode.Further,itiseasytoseethatnoteverygraphcontainsadominatingclique.Onthecontrary,agraphwithadiameterbiggerthan3doesnothaveadominatingclique,andisguaranteedtohaveoneonlywhenthegraphathandisP5-andC5-free[ 35 ]. 89

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Figure5-1. Dierencebetweena(k;l)-inuentialcliqueandadominatingclique. 5.2ComplexityIntheremainderofthischapter,wewillbereferringtostandardgraphtheorynotation.AgraphwillberepresentedasG(V;E),andwhentwonodesi;j2Vthatarenotadjacent,areconnectedwithanedgeinthecomplementgraph,hence(i;j)2E.Wedenetheproblem(k;l)-INFLUENTIALCLIQUEasfollows. Denition6. (k;l)-INFLUENTIALCLIQUE:GivenagraphG(V;E)andtwointegernumbersk>0andl>0,isthereacliqueCVsuchthatjN(C)jkandjN(C)jl? 5.2.1(k;l)-InuentialCliqueWewillshowtheNP{completenessoftheproblemusingareductionfromthewell{studiedCLIQUEproblem. Theorem5.1. (k;l)-INFLUENTIALCLIQUEisNP{completeforanyk;l>0. Proof. Firstofall,observethattheproblemisinNP.GivenasetofverticesCV,itcanbeveriedinpolynomialtimewhethertheyformaclique,andifbothjN(C)jkandjN(C)jl.Now,givenagraphG(V;E),thefollowinggraphcanbeconstructedinpolynomialtime.First,foreverynodei2Vintroduceavertexset^Vi=fu(j)i;j=1;:::;kg.Overall,thisactionwilladdnknewnodesinthegraph.Then,weconnecteachnodei2Vtoeverynodethatbelongsto^Vi.Formally,weintroducetheedgeset^Ei=f(i;u(j)i;j= 90

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Figure5-2. Thegadgetofthereductionfork=3. 1 2 4 3 1 2 4 3 u(1)1 u(2)1 u(3)1 u(1)2 u(2)2 u(3)2 u(1)3 u(2)3 u(3)3 u(1)4 u(2)4 u(3)4 1;:::;kg,foreveryi2V.Hence,wecannowconstructanewgraph^G(^V;^E),where^V=V[fn[i=1^Vig;^E=E[fn[i=1^Eig:Anexamplefork=3isshowninFigure 5-2 .Therstpartofthereductionistrivial.AssumethatingraphGthereexistsacliqueCVsuchthatjCjl,thenitiseasytoseethatCisalsoacliquein^G.Further,itisadjacenttoatleastknodes.Takinganynodei2C,weimmediatelygetj^Vij=knodesthatareadjacenttotheclique.Last,thesizeofthecliqueCisatleastl,andallnodesareconnectedtoknodes,whichimpliesthatjN(C)jl.Fortheconverse,letusassumethatthereexistsnocliqueinGwithsizegreaterthanorequaltol.Foracontradiction,supposethatthereexistsacliqueC0in^GsuchthatjN(C0)jkandjN(C0)jl.ObservethatwehaveC06V,otherwiseC0wouldbeacliqueinG.Hence,thereexistsatleastonenodeu2^VnVthatispartoftheclique.However,notethatbyconstruction,thiscliquecannothavemorethantwonodes,sinceuwillonlybeconnectedtoonenodei2V.Now,byassumption,cliqueC0hasatleastknodesadjacenttoit,sinceN(C0)k.Therearek)]TJ /F1 11.955 Tf 12.43 0 Td[(1nodesremaininginthevertexset^Vi,sinceubelongstotheclique.Hence,thereexistsatleastonenodeu02Vthat 91

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Figure5-3. ExampleofthereductionfromGto^Gfor(k;0){INFLUENTIALCLIQUE. 1 2 4 3 1 2 4 3 u(1)1 ::: u(3)1 u(1)2 ::: u(3)2 u(1)3 ::: u(3)3 u(1)4 ::: u(5)4 isadjacenttoi.Observethatiandu0formacliqueCsuchthatjCj=jC0j.ThelastobservationcontradictsthefactthatthereexistsnocliqueCVofsizegreaterthanorequaltolwhenthereexistsacliqueC0^Vin^G. 5.2.2(k;0)-InuentialClique Theorem5.2. (k;0)-INFLUENTIALCLIQUEisNP{complete,forallk>0. Proof. Oncemore,theproblemisclearlyinNP,sinceitistrivialtocheckwhetherasetofverticesCVformsacliqueandisconnectedtoatleastknodesoutsidetheclique.WewillshowthattheproblemisNP{complete,reducingtheCLIQUEproblemtoit.Let,whereG(V;E)isasimple,non{emptygraph,beaninstanceofCLIQUE.Foreverynodei2V,createjVj+1=n+1nodesu(j)i;j=1;:::;n+1andconnectthemtoi.Thatcreatesthegraph^G.Now,weclaimthatgraphGhasacliqueofsizeatleastki^Ghasaclique^CsuchthatjN(C0)jk(n+1),wheren=jVj.AnexampleoftheconstructionisshowninFigure 5-3 .ObservethatGhasjVj=n=4nodes,hencewecreate5newnodesin^Gforeachnodei2V.First,assumethatGhasacliqueCVofsizeatleastk.Then,byconstruction,thesamecliqueCcoversatleastk(n+1)nodesin^G,sinceallnodesbelongingtothecliqueareadjacentton+1nodesu(j)i;j=1;:::;n+1outsidethecliqueeach.Fortheconverse, 92

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Figure5-4. ExampleofthereductionfromGto^Gforagraphwith4nodes. 1 2 4 3 1 2 4 3 u1 u2 u3 u4 assumethatGhasnocliqueofsizeatleastk,and,foracontradiction,thereexistssomecliqueC0in^GsuchthatjN(C0)jk(n+1).Observethattherearetwocases:i)C0V,butjC0j(k)-67(jC0j)(n+1)n+1,whichisacontradictionsincejN(C0)\VjjVj=n.Forthesecondcase,observethateachoftheconstructednodesu(j)iareonlyconnectedtoonenodei.Hence,acliqueconsistingofthesenodeswouldhaveasizeof(atmost)2,andjN(C0)j=n.Suchacliquewouldbeasolutiononlyfork=0,contradictingthedenitionoftheproblemthatrequiresk>0. 5.2.3(0;l)-InuentialClique Theorem5.3. (0;l){INFLUENTIALCLIQUEisNP{complete,foralll>0. Proof. Again,theproblemisinNP,sinceverifyingthatCformsacliquesuchthatjN(C)jlcanbeperformedinpolynomialtime.Followingasimilarconstructionasbefore,givenaninstanceofCLIQUE,addonenodeuiforeverynodei2V,andconnect(i;ui)withanedge.Now,assumethereexistsacliqueCVsuchthatjCjl,thenitfollowsimmediatelythatforthesamecliqueintheconstructedgraph,wehavejN(C)jl.Fortheconverse,assumeforacontradictionthatthereisnocliqueofsizeatleastlintheoriginalgraph,yetthereexistsacliquesuchthatjN(C0)jl.ItisclearthatjC0jjN(C0)jl.Thisimpliesthat 93

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C0isnotapropersubsetofV,otherwiseC0wouldformacliqueintheoriginalgraph,contradictingourassumption.Hence,thereexistsexactlyonenodeuiinC0.Thatwouldmake(byconstruction)C0=fi;uig,forsomei2V.Observe,thatifthisisthecase,thenlmusthavebeenequalto1.However,thegraphisassumedtobenonempty,whichimpliesthattheredenitelyexistsacliqueofsize1inG,nishingtheproof. 5.2.4Inapproximability Proposition5.1. (0;0){INFLUENTIALCLIQUEisequivalenttoCLIQUE,i.e.ndinga(0;0){INFLUENTIALCLIQUEofsizeatleastcisequivalenttondingaCLIQUEofsizeatleastc,andisthereforeNP{complete.Ontheotherhand,ndingafeasiblesolutionisaseasyasndingacliqueinagraph. Proposition5.2. Themaximumandminimum(k;l){INFLUENTIALCLIQUEproblemsandtheirspecialcasesfork=0;l>0orl=0;k>0areinapproximable,undertheassumptionthatP6=NP. Proof. ItfollowsimmediatelyfromtheNP{completenessofndingafeasiblesolutiontotheproblems.AssumethereexistsanalgoritheoremAthatreturnsanyapproximatesolutiontotheproblem.Hence,Aalsoreturnsafeasiblesolution.SincedetectingfeasiblesolutionsisNP{complete,thatwouldimplyP=NP. SincethefeasibilityproblemsareNP-complete,bothminimizationandmaximizationversionsoftheproblemsareNP-hard. 5.2.5AdierentcomplexityapproachConsideragainthe(k;0)-INFLUENTIALCLIQUEproblem.Wecanseetwodierentoptimizationversionsoftheproblemthatwouldbeofinterest: WhatisthebiggestsizecliqueCsuchthatjN(C)jk? WhatisthecliqueCofaspecicsizecwiththebiggestopenneighborhoodjN(C)j?Observethatbothproblemsdescribedabovehavethesamedecisionversion.Intherstproblem,weaimtomaximizethesizeofthecliqueinordertoachieveacertain 94

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inuencerequirement,whereasinthesecondonetheinuenceismaximizedsubjecttotherestrictionofthecliquesizebeingatleastn.WewillproveNP-completenessoftheproblemusingthewell-knownMAXSATproblem. Denition7. Givenasetofliteralsxi;i=1;:::;nandtheirnegations:xi;i=1;:::;n,andasetofclausesCj;j=1;:::;m,isthereavalidassignmentofliteralssuchthatatleastkclausesaresatised? Theorem5.4. MAXSATisNP-complete[ 56 ]. Theorem5.5. (k;l)-INFLUENTIALCLIQUEisNP-complete. Proof. Firstofall,itiseasytoseethatourproblemisinNP.GivenasetofnodesC,itcanbeshowninpolynomialtimewhetheritisavalidsolution,sinceweneedtoshowthatitisacliqueofsizenandhasatleastkothernodesthatitisincidentto.Now,letusconsideraninstanceofMAXSAT.Foreachoftheliteralsxi(andtheirnegations,:xi),createanode(uianduiaccordingly).Now,connectallnodeswitheveryothernode,butthenodecorrespondingtothenegationoftheirliteral.Hence,connecteachpairof(ui;uj),i6=j,and(ui;uj),i6=j.Thiswillresultinthecreationofann-partitesubgraph,withtwoelementsperpartition.Now,createanodeCjforeachoftheclauses(mnodes).Thesenodeswillbeconnectedtothenodesoftheliteralsthateachofthemconsistsof.Asanexample,ifaclausecontainsx1;x3,and:x5,thenodecorrespondingtotheclausewillbeconnectedtou1;u3,andu5.Theconstructedgraph,startingfromaMAXSATinstance,isshowninFigure 5-5 .AnexamplefortheMAXSATclausesC1=x1_x2;C2=x1_:x2;C3=:x1_x2;C4=:x1_:x2isshowninFigure 5-6 .Thisinstancecansatisfyatmost3clauses,andhencethereisacliquethatcaninuence3othernodes(butnomore).Now,weconsiderthecasewhereasolutionexistsfortheMAXSATproblem.Inthatcase,wecanhaveasetofnliteralsthatsatisfyatleastkoftheclauses.Fromtheconstructionofthegraph,selectingthenodesthatcorrespondtotheseliteralswillyield 95

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Figure5-5. ThegadgetforthereductionfromaninstanceofMAXSAT. acliqueofsizen.Now,itiseasytocheckthatthecliquewillbeconnectedtoknodes(clauses)andnnodes(otherliterals),henceoveralln+knodes.ThisimpliesthatasolutiontotheMAXSATproblemwithnliteralsandatleastkclauses,yieldsasolutiontoourproblemwithsizeofcliquenandcoveragel=n+k.LetuscheckthecasewherethereisnosolutiontotheMAXSATproblem.Foracontradiction,assumethatourproblemyieldsacliqueofsizensuchthatthenodesinthecliqueareincidentton+kothernodes.Wedistinguishtwocases: thecliqueconsistsofonlyliteralnodes(u,andv); thecliqueconsistsofliteralandclausenodes(u,v,andC). 96

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Figure5-6. TheconstructedgraphfromtheinstanceofMAXSATwith2literals(x1andx2)and4clausesC1=x1_x2;C2=x1_:x2;C3=:x1_x2;C4=:x1_:x2. Therstcasecanbedismissed,becausebyconstruction,ifsuchacliqueexists,thentheoriginalMAXSATwouldhaveatleastksatisedclauses.Letusconcentrateonthesecondone.Observe,thattherecannotexistmorethanoneclause-nodeCinanyvalidclique.Byconstruction,clause-nodesarenotconnectedtoeachother,henceasetofnodescontainingmorethanoneclause-nodeswouldviolatetheconditionofbeingaclique.Hence,thereisexactlyoneclause-nodeintheclique.Furtherobservethateachclausecanhaveatmostnliterals,andcannothavealiteralanditsnegation.Therefore,theclause-nodecanbeconnectedatmosttooneothernode.Therestofthenodesinthecliquedenitelycoverthen)]TJ /F1 11.955 Tf 12.69 0 Td[(1remainingliteral-nodesinthegraph.Thatleavesuswiththerestofthenodesintheclique(theliteralnodes)coveringatleastk)]TJ /F1 11.955 Tf 12.14 0 Td[(1clause-nodes.Now,observethatitwouldbeavalidcliqueofsizen,ifweweretodroptheclause-nodecurrentlyintheclique,andaddtheliteral-node(eithertheactualliteraloritsnegation)intheclique.Thatwaywewouldhaveacliquethatcoversatleastk)]TJ /F1 11.955 Tf 12.16 0 Td[(1clause-nodesfromtheinitiallyselectedliteral-nodesandatleast1clausenodefromthelastliteralnode.Addingtothatthenliteralnodesthatarealwayscoveredbecause 97

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oftheconstruction,andwehaveavalidcliqueofsizenthatcoversn+kothernodes.However,thatimpliesthatouroriginalMAXSATproblemalsohasasolution.Simplypickingtheliteral-nodesthatareinthecliquewouldyieldavalidassignmentthatsatisesatleastkclauses. Wewillnotbetacklingtheseproblemsherein,nonethelesswefoundthereductiontobeofinteresttotheprospectivereader. 5.3SolutionmethodLetusdenethreesetsofdecisionandauxiliaryvariables,asfollows:xi=8><>:1,ifi2Visselectedtobeintheclique0,otherwiseyi=8><>:1,ifi2Visconnectedtoatleastonenodeinsidetheclique0,otherwisezi=8><>:1,ifi2Visinthecliqueandisconnectedtoatleastonenodeoutsidetheclique0,otherwiseAlso,letkbetherequiredcoverage(inuence)weaimtohave,andlbethenumberofoutgoingnodeswithintheclique. 5.3.1FormulationTheproblemofndingthemaximum/minimum(k;l){INFLUENTIALCLIQUEcanbeeasilyformulatedasshowninFormulation 5{1 throughout 5{2 .max=minf(x)=nXi=1xi (5{1)s:t:x2Xk;lG; (5{2) 98

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whereXk;lG=fx2f0;1gn:xi+xj1;(i;j)2Exi+yi1;i2VyiPj:(i;j)2Exj;i2VnPi=1yikzixi;i2VziPj:(i;j)2E(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xj);i2VnPi=1zily2f0;1gnz2f0;1gng.LetsetXdenotetherelaxedXset,wherevariablesyandzareallowedtotakecontinuousvaluein[0;1].TheproblemcanbewrittenasR:fmin=maxf(x):x2XgThen,thefollowinglemmaistrue. Proposition5.3. IfRisfeasible,thentherealwaysexistsanoptimalsolutionofRwhereallyandzvariablestakebinaryvalues. Proof. AssumethatRhasanoptimalsolution(x;y;z),wheresomeyiandzjarein(0;1).Sincex2X,wecandeducethatxi=0,andxj=1.WealsoknowthatnPi=1yik,andnPi=1zil.Now,considerthesolution(x;^y;^z),where^y=8><>:0;ifyi=01,ifyi>0and^z=8><>:0;ifzi=01,ifzi>0: 99

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Notethat(x;^y;^z)isanalternateoptimalsolution,becausenPi=1^yi>nPi=1yik,andnPi=1^zi>nPi=1zil.Hence,whenyandzarenotrestrictedtobeintegral,butareallowedtotakecontinuousvalues,wecanconstructanoptimalsolutionwhereyandzarebinary. 5.3.2BoundsFirstofall,observethatndingafeasiblesolutioninaninstanceisNP{complete(cf. 5.2.1 , 5.2.2 , 5.2.3 ),hencewecanonlyhopetondupper/lowerboundsforthemaximization/minimizationproblemsrespectively.AssumeaniterativemethodwhereSisthesetofnodesthatthecliqueunderconsiderationconsistsof.LetLbethecandidatelist,ofallnodesthatcanbeaddedtoSsuchthatSremainsaclique.Letg(S;u)beafunctiondenedasg(S;u)=jN(S[fug)j)-222(jN(S)j.Then,thefollowingistrue. Proposition5.4. g(S;u))]TJ /F1 11.955 Tf 21.92 0 Td[(1;8u2VnS.ItiseasytoverifythatanodedoesnotdecreasethesizeoftheopenneighborhoodofSwhenaddedtoit,iN(S)\N(u)6=;.However,inthecasewhenN(S)\N(u)=;,thenaddingnodeutoSwilldecreasethesizeoftheopenneighborhoodbyexactly1.Now,letusconsiderthecaseforN(C),foracliqueCV.Firstofall,itistrivialtoseethatinthecaseof(k;l)-INFLUENTIALCLIQUEwherel>0,thesizeofthecliquethatweendupinanyfeasiblesolutionshouldbeatleastequaltol,i.e.jCjl,forany(k;l){INFLUENTIALCLIQUE.Further,consideranodeu=2C,andletS=C[fug.Then,observethatthesizeofN(S)candecreasebyatmostjCjcomparedtojN(C)j.Asanexampleofthisworst{casescenario,considerthegraphinFigure 5-7 .Inthiscase,C=f1;2;3;4gandjN(C)j=4.WhenS=C[f5g,notethatN(S)=;.AlsoobservethataddinganodetoScanatmostincreasethenumberofnodesthathaveconnectionsoutsidethecliqueby1.Leth(S;u)beafunctiondenedash(S;u)=jN( S[fug)j)-222(jN(S)j. 100

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Figure5-7. AnexampleofhowthedecreaseofthesizeofN(C)canbeasbigasjCj. 1 3 4 2 5 Proposition5.5. jSjh(S;u)1,forallu2VnS.Inthefollowingpropositions,weexploittheseobservations.AssumewehaveacliqueSandacandidatelistLofallthenodesv2VnSsuchthatS[fvgremainsaclique.Further,wecansortthenodesbelongingtoLbasedontheirgandhvalue.WewillrefertotheseorderedlistsasLgandLhrespectively.AnexampleisgiveninFigure. Proposition5.6. (a)LetLg=fv1;:::;vjLjgbetheorderedcandidatelistL,suchthatg(S;vi)g(S;vj),foranyvi;vj2L,ij,andletbetherstindexforwhichg(S;v)=0,or=jLj+1otherwise.Then,jN(S)j+)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pi=1g(S;vi)isanupperboundonthesizeoftheopenneighborhoodforS.(b)LetLh=fv1;:::;vjLjgbetheorderedcandidatelistL,suchthath(S;vi)h(S;vj),foranyvi;vj2L,ij,andletbetherstindexforwhichh(S;v)=0,or=jLj+1otherwise.Then,jN(S)j+)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pi=1h(S;vi)isanupperboundonthesizeoftheopenneighborhoodforS. Proof. (a)Itfollowsimmediatelyfromthecalculation.Eachofthenodesui2Lwillincreasethenumberofadjacentnodesbyatmostg(S;ui).Infact,fortwosetsSS0,andanodeu2VnS0wehavethatg(S0;u)g(S;u).Ifweaddonlythenodesthathaveapositiveinuenceonthenumberofadjacentnodeswehavea(loose)upperboundonthesizeoftheopenneighborhoodthatcliqueScanreach. 101

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(b)Followingasimilarlogic,eachofthenodesui2LwillincreasethesizeofN(S)byh(S;ui).Oncemore,likein(a),fortwosetsSS0,andanodeu2VnS0wehavethath(S0;u)h(S;u).Hence,themaximumsizeoftheopenneighborhoodofSthatwecanreachisjN(S)j+)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pi=1h(S;vi),sinceweareonlyaddingthenodesthatwouldincreasethesizeoftheopenneighborhood. Proposition5.7. (a)IftheupperboundonN(S)islessthank,thenthereisnofeasiblesolutioncontainingS.Otherwise,let0betherstindexforwhichjN(S)j+0Pi=1g(S;vi)
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indexforwhichjN(S)j+Pi=1g(S;vi)
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Proof. Inanyfeasible(k;l){INFLUENTIALCLIQUEC,thereexistatleastlnodes,eachwithaconnectiontosomenodeu2VnC.Thesenodeshaveadegreeofatleastl.TheremainingjCj)]TJ /F3 11.955 Tf 18.23 0 Td[(lnodes(ifany)havetobeconnectedtoatleastjCj)]TJ /F1 11.955 Tf 18.22 0 Td[(1nodeseach.NowobservethateitherjCj=lorjCj>l.Intherstcase,allnodeshaveadegreeofatleastl,whereasinthelatterallnodeshaveadegreeofatleastjCj)]TJ /F1 11.955 Tf 17.94 0 Td[(1l. Proposition 5.10 canalsoproveinfeasibilityforaproblem.Assumethereexistnomorethanl)]TJ /F1 11.955 Tf 12.03 0 Td[(1nodesthathaveadegreeofatleastl.Thisimpliesthattherecannotbeacliqueofatleastsizelsuchthatatleastlnodeswithinthecliqueareadjacenttoanodeoutsidetheclique.Theprocedureforsettingsomeofthevariablesthatcannotbelongtotheclique(preprocessing)isdescribedinAlgorithm 8 . Algorithm8Preprocessing(V) S V fori2Sdo ifdeg(i)lthen S S=fig endif endfor fori2Sdo ifdeg(i)l)]TJ /F1 11.955 Tf 11.95 0 Td[(1then S S=fig endif endfor returnS Forourbranchingscheme,wecaneitherbranchonthenodethatgivesusthemaximumupperboundforthesizeoftheclique,thek-coverage(Lglistfrombefore),orthel-coverage(Lhlistfrombefore).Theprocessofthebranchingisdescribedasfollows.Werstselectthenodewiththehighestcliquesizepotential.Ifthemaximumcliquethatcanbeobtainedissmallerthanthecurrentincumbent,wecanprunethisbranchbyoptimality.Then,weestablishthatitsupperboundonbothN(S)andN(S)coveragesatisfythekandlrequirements.Ifnot,wesaythatthetreebranchisprunedby 104

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infeasibility.Letpidenotethepotentialcliquesizeofanodei2V,whereasUBkandUBlaretheupperboundsforthek-andl-coverage,asshownintheprevioussection.Firstofall,weapplyAlgorithm 8 tothevertexsetV,leavinguswithasetof\active"nodesStochoosefrom.Ateachnodeofthebranch-and-boundtree,weselectthevertexiwiththehighestpotentialpi.Thiscreatestwobranches:onewherethecliqueCbecomesC[figandanotherwhereCstaysthesame,andxi=0.Atthatpoint,wedistinguishthefollowingcases: piincumbent:wecanprunethisbranchbyoptimality; UBki
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accordingtokandl.TherstoneensuresthatforeverysubsetofnodesSinVthatdonotsatisfyjN(S)jk,weshouldeitheraddatleastonenodetoS,orremoveatleastonenodefromS.Similarly,thelatterforcesthesameforeverysubsetofnodesSsuchthatjN(S)j
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Table5-1. Computationalresultsfordierentvaluesofkandl.Graph(k,l)ExactB&Bw.Pre.B&Bwo.Pre.ConstraintGeneration polbooks(10,3)1.480.981.211.20polbooks(10,4)1.300.761.030.99polbooks(20,3)1.390.831.191.23polbooks(20,4)1.060.530.770.78polbooks(20,5)1.571.021.121.15karate(10,3)0.020.010.010.01karate(10,4)0.150.100.110.11karate(15,3)0.030.010.020.02karate(15,4)0.040.030.040.03karate(20,3)0.150.110.130.13dolphins(10,3)0.280.150.200.19dolphins(10,4)0.350.210.280.27dolphins(10,5)0.070.040.050.06dolphins(15,3)0.250.190.230.25dolphins(15,4)0.200.130.180.14dolphins(15,5)0.050.040.060.05huck(20,5)0.130.070.100.11huck(20,6)0.140.090.110.11huck(30,5)0.210.170.190.19huck(30,6)0.230.150.170.17jean(20,5)0.120.090.120.10jean(20,6)0.130.090.120.11jean(30,5)0.130.100.120.12jean(30,6)0.140.100.130.12adjnoun(15,4)0.280.210.270.26adjnoun(15,5)0.350.230.310.31adjnoun(20,4)0.270.200.250.26adjnoun(20,5)0.350.250.310.31adjnoun(30,4)0.300.210.260.27adjnoun(30,5)0.330.240.270.28 5.6FutureworkInthischapter,weintroducedandtackledthenovelproblemofdetecting(k;l)-inuentialcliquesingraph.Weproceededtoprovidethecomplexityoftheproblem(alongwithseveralspecialcases),andformulateditmathematically.Then,weshowedmathematicalboundsonboththesizeofsuchcliquesandtheirinuencerequirementsthatweemployedinacombinatorialbranch-and-boundscheme.Thenumericalresultsinwell-known 107

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benchmarknetworksrevealthatourapproachisaviableandecientwaytosolvingtheseproblems. 108

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CHAPTER6CONCLUSIONInthisdissertation,westudiedseveralproblemsthatarisefromthewidespreaduseofsensorsinthemodernworld.Suchproblemsadmitadvancedoperationsresearchandoptimizationtechniquestoensurethelongevityandeciencyofsensornetworks.Morespecically,inChapter 2 ,wegaveanannotatedliteraturereviewoftheuseofsensorsinmoderntransportationandlogisticsnetworks.Thisreviewcanserveasastartingpointforrealizinghowmodernsensorsystemswork,andtheproblemsthatpractitionersfacetoday.Then,weproceededinChapter 3 tostatethewell-knownmulti-sensormulti-targettrackingproblem,whereasetofsensormeasurementsneedtobeclusteredtogethertomostaccuratelytrackaseriesoftargets.Westatedtheprobleminitsgraphtheoreticterms,andproposedtwonoveldecompositionschemes.Thesesubproblemswereshowntoprovideuswithvalidupperandlowerboundsfortheoriginalproblem.Further,weresearchexactandheuristicmethodsforsolvingthesubproblemsandrecombiningthesolutionstoobtainfeasible(approximate)andexactsolutionstothemasterproblem.Theseapproaches,alongwithahybridmethodthatcombineselementsfrombothpartitioningschemes,wereshowntobeveryecientforsolvingrealistic,large-scaleMAPinstances.InChapter 4 ,wedealtwiththeveryinterestingproblemofgroupcentrality.Morespecically,weextendedthedenitionsofdegree,closeness,andbetweennesscentralitytocohesivegroups(cliques).WeshowedthattheproblemsremainNP-hardwhenthegroupisrestrictedtoformaclique,andproposedmathematicalprogramingformulationsforallthreeproblems.Then,asfarasbetweennesscentralitywasconcerned,weproposednovelinterpretations,namelyprobabilisticandpessimistic,insteadofthetypicallyusedoptimisticversion.Finally,ourexperimentsdepictedthatitisnotalwaysthatbigger 109

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cliquesprovidethesamelevelofcoverage,closeness,andinformationtrackingassmaller,yet\central"cliques.Last,inChapter 5 ,theproblemofa(k;l)-Inuentialcliquewasintroducedandstudied.WeshowedthattheprobleminitsnativeformisNP-hard,alongwithseveralspecialcasesthatstemfromit.Wealsoproceededtoshowthattheproblemisinapproximable,underthelong-standingassumptionthatP6=NP.Then,weproposedamathematicalformulation,andenhanceditwithaseriesofbounds.Theseboundsactuallyhelpproposeacombinatorialbranch-and-bound,thatwasshowntooutperformCPLEXandGurobiinthecomputationalresults. 110

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REFERENCES [1] \COLOR02/03/04:GraphColoringanditsGeneralizations." http://mat.gsia.cmu.edu/COLOR03/ ,.LastaccessedSeptember9,2013. [2] Abello,J.,Pardalos,P.M.,andResende,M.G.C.\OnMaximumCliqueProblemsInVeryLargeGraphs."InExternalMemoryAlgorithms.vol.50ofDIMACSSeriesonDiscreteMathematicsandTheoreticalComputerScience.AmericanMathematicalSociety,1999,119{130. [3] Aiex,R.M.,Resende,M.G.C.,Pardalos,PM,andToraldo,G.\GRASPwithpathrelinkingforthethree-indexassignmentproblem."INFORMSJournalonComputing(2000). [4] Al-Karaki,JamalNandKamal,AhmedE.\Routingtechniquesinwirelesssensornetworks:asurvey."WirelessCommunications,IEEE11(2004).6:6{28. [5] Andrey,PhilippeandTarroux,Philippe.\UnsupervisedsegmentationofMarkovrandomeldmodeledtexturedimagesusingselectionistrelaxation."PatternAnalysisandMachineIntelligence,IEEETransactionson20(1998).3:252{262. [6] Andrijich,SharleneM.andCaccetta,Louis.\Solvingthemultisensordataassociationproblem."NonlinearAnalysis47(2001).8:5525{5536. [7] Bandelt,Hans-Jurgen,Crama,Yves,andSpieksma,FritsCR.\Approximationalgorithmsformulti-dimensionalassignmentproblemswithdecomposablecosts."DiscreteAppliedMathematics49(1994).1:25{50. [8] Bar-Shalom,Yaakov.\Multitarget-multisensortracking:advancedapplications."Norwood,MA,ArtechHouse,1990,391p.1(1990). [9] Bar-Shalom,Yaakov,Chummun,MuhammadR.,Kirubarajan,Thiagalingam,andPattipati,KrishnaR.\FastDataAssociationforMultisensor-MultitargetTrackingUsingClusteringandMultidimensionalAssignmentAlgorithms."IEEETransactionsonAerospaceandElectronicSystems37(2001).3:898{913. [10] Baskar,LakshmiDhevi,DeSchutter,Bart,andHellendoorn,Hans.\Hierarchicaltraccontrolandmanagementwithintelligentvehicles."IntelligentVehiclesSymposium,2007IEEE.IEEE,2007,834{839. [11] |||.\Optimalroutingforintelligentvehiclehighwaysystemsusingmixedintegerlinearprogramming."Proceedingsofthe12thIFACSymposiumonTransportationSystems.2009,569{575. [12] Baskar,LakshmiDhevi,DeSchutter,Bart,Hellendoorn,J,andTarau,A.\Tracmanagementforintelligentvehiclehighwaysystemsusingmodel-basedpredictivecontrol."Proceedingsofthe88thAnnualMeetingoftheTransportationResearchBoard.2009. 111

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[13] Bassett,DanielleS,Owens,EliT,Daniels,KarenE,andPorter,MasonA.\Inuenceofnetworktopologyonsoundpropagationingranularmaterials."PhysicalReviewE86(2012).4:041306. [14] Bavelas,Alex.\Amathematicalmodelforgroupstructures."Humanorganization7(1948).3:16{30. [15] |||.\Communicationpatternsintask-orientedgroups."TheJournaloftheAcousticalSocietyofAmerica22(1950).6:725{730. [16] Bemporad,AlbertoandMorari,Manfred.\Controlofsystemsintegratinglogic,dynamics,andconstraints."Automatica35(1999).3:407{427. [17] Blackman,SamuelS.Multiple-targettrackingwithradarapplications.ArtechHouse,Inc.,1986. [18] Boginski,V.,Butenko,S.,andPardalos,P.\Miningmarketdata:Anetworkapproach."Computers&OperationsResearch33(2006).11:3171{3184. [19] Boginski,VladimirL,Commander,ClaytonW,andPardalos,PanosM.Sensors:theory,algorithms,andapplications,vol.61.Springer,2012. [20] Boldi,PaoloandVigna,Sebastiano.\Axiomsforcentrality."arXivpreprintarXiv:1308.2140(2013). [21] Bomze,ImmanuelM.,Budinich,Marco,Pardalos,PanosM.,andPelillo,Marcello.\TheMaximumCliqueProblem."HandbookofCombinatorialOptimization.KluwerAcademicPublishers,1999,1{74. [22] Borgatti,StephenPandEverett,MartinG.\Agraph-theoreticperspectiveoncentrality."Socialnetworks28(2006).4:466{484. [23] Braginsky,DavidandEstrin,Deborah.\Rumorroutingalgorthimforsensornetworks."Proceedingsofthe1stACMinternationalworkshoponWirelesssensornetworksandapplications.ACM,2002,22{31. [24] Brandes,Ulrik.\Afasteralgorithmforbetweennesscentrality*."JournalofMathematicalSociology25(2001).2:163{177. [25] Buchanan,Austin,Sung,JeSang,Butenko,S,Boginski,V,andPasiliao,E.\Onconnecteddominatingsetsofrestricteddiameter."Tech.rep.,Workingpaper,2012. [26] Bullo,Francesco,Cortes,Jorge,andMartnez,Sonia.\Distributedcontrolofroboticnetworks."AppliedMathematicsSeries.PrincetonUniversityPress(2009). [27] Burkard,RainerErnstandCela,Eranda.\Quadraticandthree-dimensionalassignmentproblems:Anannotatedbibliography."AnnotatedBibliographiesinCombinatorialOptimization.eds.M.DellAmico,F.Maoli,andS.Martello,chap.21.JohnWiley&Sons,1997.373{392. 112

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[28] |||.\Linearassignmentproblemsandextensions."HandbookofCombinatorialOptimization.eds.DingzhuDuandPanosM.Pardalos,chap.21.KluwerAcademicPublishers,1999.75{149. [29] Butenko,SergiyandWilhelm,WilbertE.\Clique-detectionmodelsincomputationalbiochemistryandgenomics."EuropeanJournalofOperationalResearch173(2006).1:1{17. [30] Cela,Eranda.\AssignmentProblems."HandbookofAppliedOptimization.eds.PanosM.PardalosandMauricioG.C.Resende,chap.17.9.NewYorkNY:OxfordUniversityPress,2002.661{678. [31] Chen,Lei,Wainwright,MartinJ,Cetin,Mujdat,andWillsky,AlanS.\Multitarget-multisensordataassociationusingthetree-reweightedmax-productalgorithm."AeroSense2003.InternationalSocietyforOpticsandPhotonics,2003,127{138. [32] Chu,Maurice,Haussecker,Horst,andZhao,Feng.\Scalableinformation-drivensensorqueryingandroutingforadhocheterogeneoussensornetworks."InternationalJournalofHighPerformanceComputingApplications16(2002).3:293{313. [33] Coleri,Sinem,Cheung,SingYiu,andVaraiya,Pravin.\Sensornetworksformonitoringtrac."Allertonconferenceoncommunication,controlandcomputing.2004,32{40. [34] Costa,LdaF,Rodrigues,FranciscoA,Travieso,Gonzalo,andVillasBoas,PR.\Characterizationofcomplexnetworks:Asurveyofmeasurements."AdvancesinPhysics56(2007).1:167{242. [35] Cozzens,MargaretB.andKelleher,LauraL.\Dominatingcliquesingraphs."DiscreteMathematics86(1990).13:101{116. [36] Dangalchev,Chavdar.\Residualclosenessinnetworks."PhysicaA:StatisticalMechanicsanditsApplications365(2006).2:556{564. [37] Davis,T.A.andHu,Y.\TheUniversityofFloridaSparseMatrixCollection."ACMTransactionsonMathematicalSoftware38(2011).1:1{25. [38] DIMACS.\10thDIMACSImplementationChallenge."Availableat http://www.cc.gatech.edu/dimacs10/index.shtml ,lastaccessedSeptember,2013,2011. [39] Dixon,CoryandFrew,EricW.\Maintainingoptimalcommunicationchainsinroboticsensornetworksusingmobilitycontrol."MobileNetworksandApplications14(2009).3:281{291. [40] Dolev,Shlomi,Elovici,Yuval,Puzis,Rami,andZilberman,Polina.\Incrementaldeploymentofnetworkmonitorsbasedongroupbetweennesscentrality."InformationProcessingLetters109(2009).20:1172{1176. 113

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[41] Ercsey-Ravasz,Maria,Lichtenwalter,RyanN,Chawla,NiteshV,andToroczkai,Zoltan.\Range-limitedcentralitymeasuresincomplexnetworks."PhysicalReviewE85(2012).6:066103. [42] Erd}os,P.andRenyi,A.\Onrandomgraphs."PublicationesMathematicaeDebrecen6(1959):290{297. [43] Ergen,SinemColeriandVaraiya,Pravin.\Optimalplacementofrelaynodesforenergyeciencyinsensornetworks."Communications,2006.ICC'06.IEEEInternationalConferenceon.vol.8.IEEE,2006,3473{3479. [44] Estrada,Ernesto.\PathLaplacianmatrices:introductionandapplicationtotheanalysisofconsensusinnetworks."LinearAlgebraanditsApplications436(2012).9:3373{3391. [45] Everett,MartinGandBorgatti,StephenP.\Thecentralityofgroupsandclasses."TheJournalofmathematicalsociology23(1999).3:181{201. [46] |||.\Extendingcentrality."Modelsandmethodsinsocialnetworkanalysis35(2005).1:57{76. [47] Faruque,Jabed,Psounis,Konstantinos,andHelmy,Ahmed.\Analysisofgradient-basedroutingprotocolsinsensornetworks."DistributedComputinginSensorSystems.Springer,2005.258{275. [48] Fasolo,Elena,Rossi,Michele,Widmer,Jorg,andZorzi,Michele.\In-networkaggregationtechniquesforwirelesssensornetworks:asurvey."WirelessCommunications,IEEE14(2007).2:70{87. [49] Fink,MartinandSpoerhase,Joachim.\Maximumbetweennesscentrality:approximabilityandtractablecases."WALCOM:AlgorithmsandComputation.Springer,2011.9{20. [50] Floyd,RobertW.\Algorithm97:shortestpath."CommunicationsoftheACM5(1962).6:345. [51] Freeman,LintonC.\Asetofmeasuresofcentralitybasedonbetweenness."Sociometry(1977):35{41. [52] |||.\Centralityinsocialnetworksconceptualclarication."SocialNetworks1(1979).3:215{239.URL http://www.sciencedirect.com/science/article/pii/0378873378900217 [53] Gaonkar,Shravan,Li,Jack,Choudhury,RomitRoy,Cox,Landon,andSchmidt,Al.\Micro-blog:sharingandqueryingcontentthroughmobilephonesandsocialparticipation."Proceedingsofthe6thinternationalconferenceonMobilesystems,applications,andservices.ACM,2008,174{186. 114

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[54] Gardiner,EleanorJ,Artymiuk,PeterJ,andWillett,Peter.\Clique-detectionalgorithmsformatchingthree-dimensionalmolecularstructures."JournalofMolecularGraphicsandModelling15(1997).4:245{253. [55] Gardner,WarrenFandLawton,DarylT.\Interactivemodel-basedvehicletracking."(1995). [56] Garey,M.andJohnson,D.ComputersandIntractability:AGuidetotheTheoryofNP-completeness.FreemanandCo.,NewYork,1979. [57] Geman,StuartandGeman,Donald.\Stochasticrelaxation,Gibbsdistributions,andtheBayesianrestorationofimages."PatternAnalysisandMachineIntelligence,IEEETransactionson(1984).6:721{741. [58] Georion,ArthurM.\Generalizedbendersdecomposition."Journalofoptimizationtheoryandapplications10(1972).4:237{260. [59] Gerkey,BrianPandMataric,MajaJ.\Sold!:Auctionmethodsformultirobotcoordination."RoboticsandAutomation,IEEETransactionson18(2002).5:758{768. [60] Gilbert,KennethandHofstra,R.\MultidimensionalAssignmentModels."DecisionSciences19(1988):306{321. [61] Goldengorin,Boris,Kalyagin,ValeryA,andPardalos,PanosM.\Models,Algorithms,andTechnologiesforNetworkAnalysis."(2013). [62] Grundel,D.A.andPardalos,P.M.\Testproblemgeneratorforthemultidimensionalassignmentproblem."ComputationalOptimizationandApplications30(2005).2:133{146. [63] Hagberg,Aric,Swart,Pieter,andSChult,Daniel.\Exploringnetworkstructure,dynamics,andfunctionusingNetworkX."Tech.rep.,LosAlamosNationalLaboratory(LANL),2008. [64] Hage,PerandHarary,Frank.\Eccentricityandcentralityinnetworks."Socialnetworks17(1995).1:57{63. [65] Hirsch,MichaelJ,Pardalos,PanosM,andResende,MauricioGC.\SensorregistrationinasensornetworkbycontinuousGRASP."MilitaryCommunicationsConference,2006.MILCOM2006.IEEE.IEEE,2006,1{6. [66] Holder,LawrenceBandCook,DianeJ.\Graph-BasedDataMining."Encyclopediaofdatawarehousingandmining2(2009):943{949. [67] Hummel,B.\Mapmatchingforvehicleguidance."DynamicandMobileGIS:InvestigatingSpaceandTime(2006):437{438. 115

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[68] Inalhan,Gokhan,Stipanovic,DusanM,andTomlin,ClaireJ.\Decentralizedoptimization,withapplicationtomultipleaircraftcoordination."DecisionandControl,2002,Proceedingsofthe41stIEEEConferenceon.vol.1.IEEE,2002,1147{1155. [69] Intanagonwiwat,Chalermek,Estrin,Deborah,Govindan,Ramesh,andHeidemann,John.\Impactofnetworkdensityondataaggregationinwirelesssensornetworks."DistributedComputingSystems,2002.Proceedings.22ndInternationalConferenceon.IEEE,2002,457{458. [70] Intanagonwiwat,Chalermek,Govindan,Ramesh,andEstrin,Deborah.\Directeddiusion:ascalableandrobustcommunicationparadigmforsensornetworks."Proceedingsofthe6thannualinternationalconferenceonMobilecomputingandnetworking.ACM,2000,56{67. [71] Jacob,Riko,Koschutzki,Dirk,Lehmann,KatharinaAnna,Peeters,Leon,andTenfelde-Podehl,Dagmar.\Algorithmsforcentralityindices."NetworkAnalysis.Springer,2005.62{82. [72] Kamijo,Shunsuke,Matsushita,Yasuyuki,Ikeuchi,Katsushi,andSakauchi,Masao.\Tracmonitoringandaccidentdetectionatintersections."IntelligentTransportationSystems,IEEETransactionson1(2000).2:108{118. [73] Karp,RichardM.\On-linealgorithmsversuso-linealgorithms:Howmuchisitworthtoknowthefuture?"IFIPCongress(1).vol.12.1992,416{429. [74] Kass,Michael,Witkin,Andrew,andTerzopoulos,Demetri.\Snakes:Activecontourmodels."Internationaljournalofcomputervision1(1988).4:321{331. [75] Klavins,Eric.\Communicationcomplexityofmulti-robotsystems."AlgorithmicFoundationsofRoboticsV.Springer,2004.275{292. [76] Kolaczyk,EricD,Chua,DavidB,andBarthelemy,Marc.\Groupbetweennessandco-betweenness:Inter-relatednotionsofcoalitioncentrality."SocialNetworks31(2009).3:190{203. [77] Koschutzki,Dirk,Lehmann,KatharinaAnna,Peeters,Leon,Richter,Stefan,Tenfelde-Podehl,Dagmar,andZlotowski,Oliver.\Centralityindices."Networkanalysis.Springer,2005.16{61. [78] Kratsch,Dieter.\Findingdominatingcliqueseciently,instronglychordalgraphsandundirectedpathgraphs."DiscreteMathematics86(1990).13:225{238. [79] |||.\Algorithms."Dominationingraphs:advancedtopics.eds.HedetniemiS.T.Haynes,T.W.andP.J.Slater.1998. [80] Kratsch,DieterandLiedlo,Mathieu.\AnExactAlgorithmfortheMinimumDominatingCliqueProblem."ParameterizedandExactComputation.eds.HansL. 116

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BodlaenderandMichaelA.Langston,vol.4169ofLectureNotesinComputerScience.2006.130{141. [81] Krebs,V.\UncloakingTerroristNetworks."FirstMonday7(2002).4.Availableat http://journals.uic.edu/ojs/index.php/fm/article/view/941 ,lastaccessedSeptember9,2013. [82] |||.\Dataset:PolBooks{Krebs.". [83] Krumm,John,Letchner,Julie,andHorvitz,Eric.\Mapmatchingwithtraveltimeconstraints."SAEWorldCongress.2007. [84] Larsen,Morten.\Branchandboundsolutionofthemultidimensionalassignmentproblemformulationofdataassociation."OptimizationMethodsandSoftware27(2012).6:1101{1126. [85] Leavitt,HaroldJ.\Someeectsofcertaincommunicationpatternsongroupperformance."TheJournalofAbnormalandSocialPsychology46(1951).1:38. [86] Li,Deying,Liu,Lin,andYang,Huiqiang.\Minimumconnectedr-hopk-dominatingsetinwirelessnetworks."DiscreteMathematics,AlgorithmsandApplications1(2009).01:45{57. [87] Luce,RDuncanandPerry,AlbertD.\Amethodofmatrixanalysisofgroupstructure."Psychometrika14(1949).2:95{116. [88] Lusseau,D.,Schneider,K.,Boisseau,O.J.,Haase,P.,Slooten,E.,andDawson,S.M.\ThebottlenosedolphincommunityofDoubtfulSoundfeaturesalargeproportionoflong-lastingassociations."BehavioralEcologyandSociobiology54(2003).4:396{405.URL http://dx.doi.org/10.1007/s00265-003-0651-y [89] MacDue,JohnPaulandKrafcik,John.\Integratingtechnologyandhumanresourcesforhigh-performancemanufacturing:Evidencefromtheinternationalautoindustry."Transformingorganizations(1992):209{226. [90] Mimbela,LuzElenaYandKlein,LawrenceA.\Summaryofvehicledetectionandsurveillancetechnologiesusedinintelligenttransportationsystems."(2000). [91] Nadeem,Tamer,Dashtinezhad,Sasan,Liao,Chunyuan,andIftode,Liviu.\TracView:tracdatadisseminationusingcar-to-carcommunication."ACMSIGMOBILEMobileComputingandCommunicationsReview8(2004).3:6{19. [92] Newman,MarkEJ.\Findingcommunitystructureinnetworksusingtheeigenvectorsofmatrices."PhysicalreviewE74(2006).3:036104. 117

PAGE 118

[93] Oliveira,C.A.S.andPardalos,P.M.\Randomizedparallelalgorithmsforthemultidimensionalassignmentproblem."AppliedNumericalMathematics49(2004).1:117{133. [94] Oliveira,EduardoMR,Ramos,HeitorS,andLoureiro,AntonioAlfredoFerreira.\Centrality-basedroutingforwirelesssensornetworks."WirelessDays(WD),2010IFIP.IEEE,2010,1{5. [95] O'Toole,Randal.Gridlock:whywe'restuckintracandwhattodoaboutit.CatoInstitute,2010. [96] Pagani,GiulianoAndreaandAiello,Marco.\Thepowergridasacomplexnetwork:asurvey."PhysicaA:StatisticalMechanicsanditsApplications(2013). [97] Pardalos,PanosM.andPitsoulis,LeonidasS.,eds.NonlinearAssignmentProblems:AlgorithmsandApplications,vol.7ofCombinatorialOptimization.Dordrecht:KluwerAcademicPublishers,2000. [98] Pasiliao,EduardoL.Algorithmsformultidimensionalassignmentproblems.Ph.D.thesis,UniversityofFlorida,2003. [99] |||.\LocalNeighborhoodsfortheMultidimensionalAssignmentProblem."DynamicsofInformationSystems(2010):353{371. [100] Pasiliao,EduardoL.,Pardalos,P.M.,andPitsoulis,L.S.\Branchandboundalgorithmsforthemultidimensionalassignmentproblem."OptimizationMethodsandSoftware20(2005).1:127{143. [101] Pattillo,Jerey,Youssef,Nataly,andButenko,Sergiy.\Oncliquerelaxationmodelsinnetworkanalysis."EuropeanJournalofOperationalResearch226(2013).1:9{18. [102] Peterfreund,Natan.\RobusttrackingofpositionandvelocitywithKalmansnakes."PatternAnalysisandMachineIntelligence,IEEETransactionson21(1999).6:564{569. [103] Phillips,DonTandGarcia-Diaz,Alberto.Fundamentalsofnetworkanalysis,vol.198.Prentice-HallEnglewoodClis,NJ,1981. [104] Pierskalla,WilliamP.\Themultidimensionalassignmentproblem."OperationsResearch16(1968).2:422{431. [105] Pitsoulis,LeonidasS.andResende,MauricioG.C.\GreedyRandomizedAdaptiveSearchProcedures."HandbookofAppliedOptimization.eds.PanosM.PardalosandMauricioG.C.Resende,chap.3.6.5.NewYorkNY:OxfordUniversityPress,2002.168{183. [106] Poore,AubreyB.\Multidimensionalassignmentformulationofdataassociationproblemsarisingfrommultitargetandmultisensortracking."ComputationalOptimizationandApplications3(1994):27{57. 118

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[107] |||.\Multidimensionalassignmentformulationofdataassociationproblemsarisingfrommultitargetandmultisensortracking."ComputationalOptimizationandApplications3(1994).1:27{57. [108] Poore,AubreyB.andIII,AlexanderJ.Robertson.\ANewLagrangianrelaxationbasedalgorithmforaclassofmultidimensionalassignmentproblems."ComputationalOptimizationandApplications8(1997):129{150. [109] Puzis,Rami,Elovici,Yuval,andDolev,Shlomi.\Findingthemostprominentgroupincomplexnetworks."AIcommunications20(2007).4:287{296. [110] Puzis,Rami,Yagil,Dana,Elovici,Yuval,andBraha,Dan.\CollaborativeattackonInternetusers'anonymity."InternetResearch19(2009).1:60{77. [111] Qu,Zhihva.Cooperativecontrolofdynamicalsystems.Springer,2009. [112] Reka,A.andBarabasi,A-L.\Statisticalmechanicsofcomplexnetworks."ReviewsofModernPhysics74(2002):47{97.URL http://arxiv.org/abs/cond-mat/0106096 [113] Resende,MauricioGCandPardalos,PanosM.Handbookofappliedoptimization.Oxforduniversitypress,2002. [114] Sabidussi,Gert.\Thecentralityindexofagraph."Psychometrika31(1966).4:581{603. [115] Sadagopan,Narayanan,Krishnamachari,Bhaskar,andHelmy,Ahmed.\Activequeryforwardinginsensornetworks."AdHocNetworks3(2005).1:91{113. [116] Sharma,Vikrant,Savchenko,MichaelA,Frazzoli,Emilio,andVoulgaris,PetrosG.\TimeComplexityofSensor-BasedVehicleRouting."Robotics:ScienceandSystems.Citeseer,2005,297{304. [117] Spieksma,FritsC.R.\Multiindexassignmentproblems:Complexity,approximation,applications."NonlinearAssignmentProblems:AlgorithmsandApplications.eds.PanosM.PardalosandLeonidasS.Pitsoulis,vol.7ofCombinatorialOptimization,chap.1.Dordrecht:KluwerAcademicPublishers,2000.1{12. [118] Thiagarajan,Arvind,Ravindranath,Lenin,LaCurts,Katrina,Madden,Samuel,Balakrishnan,Hari,Toledo,Sivan,andEriksson,Jakob.\VTrack:accurate,energy-awareroadtracdelayestimationusingmobilephones."Proceedingsofthe7thACMConferenceonEmbeddedNetworkedSensorSystems.ACM,2009,85{98. [119] Vivaldini,KelenCT,Galdames,JorgePM,Bueno,ThalesS,Araujo,RobertoC,Sobral,RafaelM,Becker,Marcelo,andCaurin,GlaucoAP.\Roboticforkliftsforintelligentwarehouses:Routing,pathplanning,andauto-localization."IndustrialTechnology(ICIT),2010IEEEInternationalConferenceon.IEEE,2010,1463{1468. 119

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[120] Vogiatzis,Chrysas.\SensorsinTransportationandLogisticsNetworks."Sensors:Theory,Algorithms,andApplications.Springer,2012.145{163. [121] Wang,Jianxin,Peng,Wei,andWu,Fang-Xiang.\Computationalapproachestopredictingessentialproteins:Asurvey."PROTEOMICS-ClinicalApplications7(2013).1-2:181{192. [122] Williams,BillyMandHoel,LesterA.\Modelingandforecastingvehiculartracowasaseasonalstochastictimeseriesprocess."Tech.rep.,1999. [123] Zachary,W.W.\Aninformationowmodelforconictandssioninsmallgroups."JournalofAnthropologicalResearch33(1977):452{473. [124] Zheng,Chan,Zhang,Yiqing,andYin,Ling.\Constructing(k,r)-connecteddominatingsetsforrobustbackboneinwirelesssensornetworks."CommunicationsandInformationTechnologies(ISCIT),201111thInternationalSymposiumon.IEEE,2011,174{177. [125] Zhou,Zongheng,Das,Samir,andGupta,Himanshu.\Connectedk-coverageprobleminsensornetworks."ComputerCommunicationsandNetworks,2004.ICCCN2004.Proceedings.13thInternationalConferenceon.IEEE,2004,373{378. 120

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BIOGRAPHICALSKETCH ChrysasVogiatziswasborninAthens,Greecein1984.Hereceivedhisdiplomaofengineeringdegree(equivalenttoM.Sc.)fromtheAristotleUniversityofThessalonikiinElectricalandComputerEngineeringin2009.Histhesisoncomputationalmethodsandcomputersciencewasentitled\Iterativedistributeddecompositionalgorithmforlarge-scaletransportationproblems".InJanuary2010,ChrysasVogiatzisjoinedthegraduateprogramofIndustrialandSystemsEngineeringintheUniversityofFlorida,wherehereceivedaM.Sc.in2012andaPh.D.in2014.DuringhisstudiesintheUniversityofFlorida,hewasrecognizedforexcellenceinteachingfromtheDepartmentofIndustrialandSystemsEngineeringandtheUniversityofFlorida,whenheearnedtheprestigiousUniversityofFloridaGraduateStudentTeachingAward.Hisresearchinterestsinclude,butarenotlimitedto,mathematicalprogramingandoptimization,graphtheory,appliedmathematics,andoperationsresearch.HisworkhasbeenpublishedinjournalsofSpringer,Elsevier,ASCE,MDPI,andtheInternationalResearchCommitteeonDisastersandhehasservedasaco-editorfortwovolumesforSpringer.Further,heisatthemomentauthoringabookonMultidimensionalAssignmentProblemsandtheirapplications.HeisastudentmemberoftheInstituteofOperationsResearchandManagementScience(INFORMS)andtheSocietyofIndustrialandAppliedMathematics(SIAM). 121