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Integer Programming Models for Solving Critical Element Detection and Data Association Problems

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Title:
Integer Programming Models for Solving Critical Element Detection and Data Association Problems
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Walteros, Jose Luis
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[Gainesville, Fla.]
Florida
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University of Florida
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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
PARDALOS,PANAGOTE M
Committee Co-Chair:
BOGINSKIY,VLADIMIR L
Committee Members:
GEUNES,JOSEPH PATRICK
HAGER,WILLIAM WARD
Graduation Date:
8/9/2014

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Subjects / Keywords:
Algorithms ( jstor )
Cost allocation ( jstor )
Heuristics ( jstor )
Integers ( jstor )
Linear programming ( jstor )
Minimization of cost ( jstor )
Operations research ( jstor )
Optimal solutions ( jstor )
Vertices ( jstor )
Whips ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
assignment -- association -- cliques -- complexity -- data -- decomposition -- formulation -- graphs -- heuristics -- interdiction -- networks -- optimization
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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 dissertation we study three problems that arise in the contexts of critical element detection and data association. The principal motivation and common interest behind these problems comes from their applicability in several areas, including robust network design, evacuation planning, immunization strategies, and multi-sensor multi-target tracking; as well as from the inherent difficulty of finding practical and computationally efficient approaches to solve them. We begin by presenting a general overview of the critical element detection problems, along with a detailed description of several variations commonly found in the literature. We also survey some of the recent advances and solution techniques for these problems, such as heuristic algorithms, mathematical programming approaches, approximated algorithms, and dynamic programming schemes. We continue by presenting an exact approach for solving a critical edge detection problem that involves removing the smallest subset of edges of a graph so that the weight of all dominating sets in the remaining graph is bounded below by a given threshold. We show that the decision version of this problem is NP-hard; we present an analytical lower bound for the cardinality of an optimal solution to the problem; we propose a mixed-integer formulation and its corresponding projection onto the space of the edge-deletion variables; we study the convex hull of the feasible solutions of this problem and identify several facet-inducing inequalities for the corresponding polytope; and we develop the first exact algorithm for this problem that solves the proposed formulation by a branch-and-cut approach introducing nontrivial constraints in a lazy fashion. Furthermore, we provide the computational results obtained after solving a test-bed of randomly generated instances and real-life power-law graphs that range in size from 25 to 62 vertices. We then consider the problem of detecting critical cliques of a graph whose deletion optimally deteriorates the connectivity of the given graph. We first introduce a mathematical formulation as a mixed integer linear program. Then, we propose a two-stage decomposition strategy that first identifies a candidate clique partition and then uses this partition to transform and solve the problem as a generalized critical vertex detection problem. Furthermore, we reformulate this problem as a mixed-integer linear program with a large number of variables and constraints and solve it via branch, price, and cut. We also design a novel preprocessing algorithm for the column generation stage that uses the dual information of the master problem to prune the subproblem graph in order to produce new columns efficiently. We test our approach on a collection of randomly generated sparse graphs, following several degree distributions, and ranging in size from 30 to 100 vertices. Finally, we tackle a variant of the multidimensional assignment problem with decomposable costs for solving data association problems in which the resulting optimal assignment is described as a set of disjoint stars. To solve this problem we study two different formulations. First, we introduce a continuous nonlinear program and its linearization, along with additional valid inequalities that improve the lower bounds. Second, we reformulate this problem as a set partitioning program and solve it via branch and price. We test these approaches by solving instances ranging from tripartite to 20-partite graphs of 4 to 30 vertices per partition. The computational results show that our approaches are a viable option to solve this problem. ( 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: BOGINSKIY,VLADIMIR L.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-08-31
Statement of Responsibility:
by Jose Luis Walteros.

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8/31/2016
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969976937 ( OCLC )
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JHeuristics(2011)17:201–249 DOI10.1007/s10732-010-9133-3 Localsearchheuristicsforthemultidimensional assignmentproblemDanielKarapetyan · GregoryGutinReceived:23July2009/Revised:18December2009/Accepted:5April2010/ Publishedonline:21April2010 ©SpringerScience+BusinessMedia,LLC2010Abstract TheMultidimensionalAssignmentProblem(MAP)(abbreviated s -APin thecaseof s dimensions)isanextensionofthewell-knownassignmentproblem. ThemoststudiedcaseofMAPis3-AP,thoughtheproblemswithlargervaluesof s alsohavealargenumberofapplications.Weconsiderseveralknownneighborhoods,generalizethemandproposesomenewones.Theheuristicsareevaluated boththeoreticallyandexperimentallyanddominatingalgorithmsareselected.We alsodemonstratethatacombinationoftwoneighborhoodsmayyieldaheuristics whichissuperiortobothofitscomponents. Keywords Multidimensionalassignmentproblem · Localsearch · Neighborhood · Metaheuristics 1Introduction The MultidimensionalAssignmentProblem (MAP)(abbreviated s -APinthecase of s dimensions,alsocalled( axial )MultiIndexAssignmentProblem,MIAP,(Bandeltetal. 2004 ;PardalosandPitsoulis 2000 ))isawell-knownoptimizationproblem.Itisanextensionofthe AssignmentProblem (AP),whichisexactlythetwo 1Burkardetal.showitforaspecialcaseof3-APandsince3-APisaspecialcaseof s -APtheresultcan beextendedtothegeneralMAP. ApreliminaryversionofthispaperwaspublishedinGolumbicFestschrift,volume5420ofLect. NotesComput.Sci.,pages100–115,Springer,2009. D.Karapetyan · G.Gutin() RoyalHolloway,UniversityofLondon,EghamHill,UK e-mail: gutin@cs.rhul.ac.uk D.Karapetyan e-mail: daniel.karapetyan@gmail.com

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202D.Karapetyan,G.GutindimensionalcaseofMAP.WhileAPcanbesolvedinthepolynomialtime(Kuhn 1955 ), s -APforevery s 3isNP-hard(GareyandJohnson 1979 )andinapproximable(Burkardetal. 1996b ).1ThemoststudiedcaseofMAPisthecaseofthree dimensions(Aiexetal. 2005 ;AndrijichandCaccetta 2001 ;BalasandSaltzman 1991 ; CramaandSpieksma 1992 ;HuangandLim 2006 ;Spieksma 2000 )thoughtheproblemhasahostofapplicationsforhighernumbersofdimensions,e.g.,inmatching informationfromseveralsensors(dataassociationproblem),whicharisesinplane tracking(Murpheyetal. 1998 ;PardalosandPitsoulis 2000 ),computervision(Veenmanetal. 2003 )andsomeothers(AndrijichandCaccetta 2001 ;Bandeltetal. 2004 ; BurkardandÇela 1999 ),inroutinginmeshes(Bandeltetal. 2004 ),trackingelementaryparticles(Pusztaszerietal. 1996 ),solvingsystemsofpolynomialequations(Bekkeretal. 2005 ),imagerecognition(Grundeletal. 2004 ),resourceallocation(Grundeletal. 2004 ),etc. Foraxed s 2,theproblem s -APisstatedasfollows.Let X1= X2= ... = Xs={ 1 , 2 ,...,n } .WewillconsideronlyvectorsthatbelongtotheCartesianproduct X = X1× X2× ... × Xs.Eachvector e X isassignedanon-negativeweight w(e) .Foravector e X ,thecomponent ejdenotesits j thcoordinate,i.e., ej Xj. Acollection A of t n vectors A1,A2,...,Atisa (feasible)partialassignment if Ai j= Ak jholdsforeach i = k and j { 1 , 2 ,...,s } .The weight ofapartialassignment A is w(A) = t i = 1w(Ai) .An assignment (or fullassignment )isapartial assignmentwith n vectors.Theobjectiveof s -APistondanassignmentofminimal weight. Wealsoprovidea permutationform oftheassignmentwhichissometimesconvenientinthediscussionoftheproblemandsomeneighborhoods.Let 1,2,...,sbepermutationsof X1,X2,...,Xsrespectively.Then 12...sisanassignment withtheweight n i = 1w(1(i)2(i)...s(i)) . Itisobviousthatonepermutation,saytherstone,maybexedwithoutany lossofgenerality: 1= 1n,where1nistheidentitypermutationofsize n .Thenthe objectiveoftheproblemisasfollows: min2,...,sni = 1w(i2(i)...s(i)). Agraphformulationoftheproblemisasfollows.Havingan s -partitegraph G withparts X1, X2,..., Xs,where | Xi|= n ,ndasetof n disjointcliquesin G ofthe minimaltotalweightifeveryclique e in G isassignedaweight w(e) . Finally,anintegerprogrammingformulationoftheproblemisasfollows. min i1 X1,...,is Xsw(i1...is) · xi1...issubjectto i2 X2,...,is Xsxi1...is= 1 i1 X1, ...

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Localsearchheuristicsforthemultidimensionalassignmentproblem203i1 X1,...,is Š 1 Xs Š 1xi1...is= 1 is Xs, where xi1...is{ 0 , 1 } forall i1,..., isand | X1|= ... =| Xs|= n . Sometimestheproblemisformulatedinamoregeneralwayif | X1|= n1, | X2|= n2,..., | Xs|= nsandtherequirement n1= n2= ... = nsisomitted.Howeverthis casecanbeeasilytransformedintotheproblemdescribedabovebycomplementing theweightmatrixtoan n × n × ... × n matrixwithzeros,where n = maxini. Theproblemwasstudiedbymanyresearchers.Severalspecialcasesoftheproblemwereintensivelystudiedintheliterature(seeKurokiandMatsui 2007 andreferencesthere)andforfewclassesofinstancespolynomialtimeexactalgorithmswere found,see,e.g.,Burkardetal.( 1996a , 1996b ),Isleretal.( 2005 ).InmanycasesMAP remainshardtosolve(KurokiandMatsui 2007 ;SpieksmaandWoeginger 1996 ).For example,iftherearethreesetsofpointsofsize n onaEuclideanplainandtheobjectiveistond n triplesofpoints,onefromeachset,suchthatthetotalcircumference orareaofthecorrespondingtrianglesisminimal,thecorresponding3-APisstillNPhard(SpieksmaandWoeginger 1996 ).Theasymptoticpropertiesofsomespecial instancefamiliesarestudiedinGrundeletal.( 2004 ). Asregardsthesolutionmethods,apartfromexactandapproximationalgorithms(BalasandSaltzman 1991 ;CramaandSpieksma 1992 ;KurokiandMatsui 2007 ;Pasiliaoetal. 2005 ;Pierskalla 1968 ),severalheuristicsincludingconstructionheuristics(BalasandSaltzman 1991 ;Gutinetal. 2008 ;Karapetyanetal. 2009 ; OliveiraandPardalos 2004 ),greedyrandomizedadaptivesearchprocedures(Aiex etal. 2005 ;Murpheyetal. 1998 ;OliveiraandPardalos 2004 ;Robertson 2001 ), metaheuristics(Clemonsetal. 2004 ;HuangandLim 2006 )andparallelheuristics(OliveiraandPardalos 2004 )arepresentedintheliterature.Severallocalsearch proceduresareproposedanddiscussedinAiexetal.( 2005 ),BalasandSaltzman ( 1991 ),Bandeltetal.( 2004 ),Burkardetal.( 1996b ),Clemonsetal.( 2004 ),Huang andLim( 2006 ),OliveiraandPardalos( 2004 ),Robertson( 2001 ). Thedifferencebetweentheconstructionheuristicsandlocalsearchissometimes crucial.Whileaconstructionheuristicgeneratesasolutionfromscratchand,thus, hassomesolutionqualitylimitation,localsearchisintendedtoimproveanexisting solutionand,thus,canbeusedafteraconstructionheuristicorasapartofamore sophisticatedheuristic,socalledmetaheuristic. Thecontributionofourpaperisincollectingandgeneralizingalllocalsearch heuristicsknownfromtheliterature,proposingsomenewonesandevaluatingthem boththeoreticallyandexperimentally.Forthepurposeofexperimentalevaluationwe alsothoroughlydiscuss,classifytheexistinginstancefamiliesandproposesomenew ones. InthispaperweconsideronlythegeneralcaseofMAPand,thus,alltheheuristics whichrelyonthespecialstructuresoftheweightmatrixarenotincludedinthe comparison.Wealsoassumethatthenumberofdimensions s isasmallxedconstant whilethesize ncanbearbitrarylarge.Thiscorrespondstotherealapplications(see above)andalsofollowsfromthepreviousresearch,see,e.g.,Bekkeretal.( 2005 ), Pusztaszerietal.( 1996 ),Robertson( 2001 ).

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204D.Karapetyan,G.Gutin2Heuristics InthissectionwediscusssomewellknownandsomenewMAPlocalsearchheuristicsaswellastheircombinations. 2.1Dimensionwisevariationsheuristics TheheuristicsofthisgroupwererstintroducedbyBandeltetal.( 2004 )forMAP withdecomposablecosts.However,havingaverylargeneighborhood(seebelow), theyareveryefcienteveninthegeneralcase.Thefactthatthisapproachwas alsousedbyHuangandLimasalocalsearchprocedurefortheirmemeticalgorithm(HuangandLim 2006 )conrmsitsefciency. Theideaofthedimensionwisevariationheuristicsisasfollows.Considerthe initialassignment A inthepermutationform A = 12...s(seeSect. 1 ).Let p(A,1,2,...,s) beanassignmentobtainedfrom A byapplyingthepermutations 1,2,...,sto 1,2,...,srespectively: p(A,1,2,...,s) = 1(1)2(2)...s(s). (1) Let pD(A,) beanassignment p(A,1,2,...,s) ,where j= if j D and j= 1notherwise(1nistheidentitypermutationofsize n ): pD(A,) = p A, if1 D 1notherwise , if2 D 1notherwise ,..., if s D 1notherwise . (2) Oneveryiteration,theheuristicselectssomenonemptyset D { 1 , 2 ,...,s } ofdimensionsandsearchesforapermutation suchthat w(pD(A,)) isminimized. Foreverysubsetofdimensions D ,thereare n ! differentpermutations butthe optimalonecanbefoundinthepolynomialtime.Let swap (u,v,D) beavector whichisequaltovector u inalldimensions j { 1 , 2 ,...,s }\ D andequaltovector v inalldimensions j D : swap (u,v,D)j= ujif j/ D vjif j D for j = 1 , 2 ,...,s. (3) Letmatrix [ Mi,j]n × nbeconstructedas Mi,j= w( swap (Ai,Aj,D)). (4) Itisclearthatthesolutionofthecorresponding2-APisexactlytherequiredpermutation .Indeed,assumethereexistssomepermutation suchthat w(pD(A,))< w(pD(A,)) .Observethat pD(A,) ={ swap (Ai,A(i),D) : i { 1 , 2 ,...,n }} . Thenwehaveni = 1w( swap (Ai,A(i),D))
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Localsearchheuristicsforthemultidimensionalassignmentproblem205Since w( swap (Ai,A(i),D)) = Mi,(i),thesum n i = 1w( swap (Ai,A(i),D)) isalreadyminimizedtotheoptimumandno canexist. Theneighborhoodofadimensionwiseheuristicisasfollows: NDV(A) = pD(A,) : D D and isapermutation , (5) where D includesalldimensionsubsetsacceptablebyacertainheuristic.Observe that pD(A,) = p D(A,Š 1), (6) where Š 1() = (Š 1) = 1nand D ={ 1 , 2 ,...,s }\ D ,and,hence, pD(A,) : isapermutation = p D(A,) : isapermutation (7) forany D .FromEq. 7 andtheobviousfactthat p(A,) = p{ 1 , 2 ,...,s }(A,) = A for any weintroducethefollowingrestrictionsfor D : D D D/ D and , { 1 , 2 ,...,s } / D . (8) Withtheserestrictions,onecanseethatforanypairofdistinctsets D1,D2 D the equation pD1(A,1) = pD2(A,2) holdsifandonlyif 1= 2= 1n.Hence,thesize oftheneighborhood NDV(A) is | NDV(A) |=| D |· (n !Š 1 ) + 1 . (9) InBandeltetal.( 2004 )itisdecidedthatthenumberofiterationsshouldnotbe exponentialwithregardstoneither n nor s whilethesizeofthemaximum D is | D |= 2s Š 1Š 1.Thereforetwoheuristics,LS1andLS2,areevaluatedinBandeltet al.( 2004 ).LS1includesonlysingletonvaluesof D ,i.e., D ={ D :| D |= 1 } ;LS2 includesonlydoubletonvaluesof D ,i.e., D ={ D :| D |= 2 } .Itissurprisingbutaccordingtoboth(Bandeltetal. 2004 )andourcomputationalexperience,theheuristic LS2producesworsesolutionsthanLS1thoughitobviouslyhaslargerneighborhoodandlargerrunningtimes.Weimprovetheheuristicbyallowing | D | 2,i.e., D ={ D :| D | 2 } .Thisdoesnotchangethetheoreticaltimecomplexityofthealgorithmbutimprovesitsperformance.TheheuristicLS1iscalled1DVinourpaper; LS2with | D | 2iscalled2DV.Wealsoassume(seeSect. 1 )thatthevalueof s isa smallxedconstantand,thus,introduceaheuristicsDVwhichenumeratesallfeasible (recall( 8 )) D { 1 , 2 ,...,s } . Theorderinwhichtheheuristicstakethevalues D D inourimplementationsis asfollows.For1DVitis { 1 } , { 2 } ,..., { s } .2DVbeginsas1DVandthentakesallpairsof dimensions: { 1 , 2 } , { 1 , 3 } ,..., { 1 ,s } , { 2 , 3 } ,..., { s Š 1 ,s } .Notethatbecauseof( 8 ) itenumeratesnopairsofvectorsfor s = 3,andfor s = 4itonlytakesthefollowing pairs: { 2 , 3 } , { 2 , 4 } and { 3 , 4 } .sDVtakesrstallsets D ofsize1,thenallsets D ofsize 2andsoonupto | D |= s/ 2 .If s iseventhenweshouldtakeonlyhalfofthesets D ofsize s/ 2(recall( 7 ));forthispurposewetakeallthesubsetsof D { 2 , 3 ,...,s } , | D |= s/ 2inthesimilarorderasbefore. Itisobviousthat N1DV(A) N2DV(A) Ns DV(A) forany s ;howeverfor s = 3 alltheneighborhoodsareequalandfor s = 42DVandsDValsocoincide.

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206D.Karapetyan,G.GutinAccordingto( 8 )and( 9 ),theneighborhoodsizeof1DVis | N1DV(A) |= s · (n !Š 1 ) + 1 , of2DVis | N2DV(A) |= ( 2s Š 1Š 1 ) · (n !Š 1 ) + 1if s { 3 , 4 } s 2 + s · (n !Š 1 ) + 1if s 5 , andofsDVis | Ns DV(A) |= ( 2s Š 1Š 1 ) · (n !Š 1 ) + 1 . ThetimecomplexityofeveryrunofDVis O( | D |· n3) asevery2-APtakes O(n3) and,hence,thetimecomplexityof1DVis O(s · n3) ,of2DVis O(s2· n3) andofMDV is O( 2s Š 1· n3) . 2.2 k -opt Thek-optheuristicfor3-APfor k = 2and k = 3wasrstintroducedbyBalasand Saltzman( 1991 )asa pairwise and tripleinterchangeheuristic .2-optaswellasits variationswerealsodiscussedinAiexetal.( 2005 ),Clemonsetal.( 2004 ),Murphey etal.( 1998 ),OliveiraandPardalos( 2004 ),Pasiliaoetal.( 2005 ),Robertson( 2001 ) andsomeotherpapers.Wegeneralizetheheuristicforarbitraryvaluesof k and s . Theheuristicproceedsasfollows.Foreverysubsetof k vectorstakenintheassignment A itremovesallthesevectorsfrom A andinsertssomenew k vectors suchthattheassignmentfeasibilityispreservedanditsweightisminimized.Anotherdenitionisasfollows:foreverysetofdistinctvectors e1,e2,...,ek A let X j={ e1 j,e2 j,...,ek j} for j = 1 , 2 ,...,k .Let A={ e 1,e 2,...,e k} bethesolution ofthis s -APofsize k .Replacethevectors e1,e2,...,ekintheinitialassignment A with e 1,e 2,...,e k. Thetimecomplexityofk-optisobviously O( n k · k !s Š 1) ;for k n itcanbe replacedwith O(nk· k !s Š 1) .Itisanaturalquestionifonecanusesomefastersolver oneveryiteration.Indeed,accordingtoSect. 1 itispossibletosolve s -APofsize k in O(k !s Š 2· k3) .However,itiseasytoseethat k !s Š 1
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Localsearchheuristicsforthemultidimensionalassignmentproblem207Recallthat1nistheidentitypermutationofsize n and p(A,1,2,...,s) isdened by( 1 ). Theneighborhood Nk -opt(A) isdenedasfollows: Nk -opt(A) = E A, | E |= kW(A,E). (10) Let Y,Z A suchthat | Y |=| Z |= k .Observethat W(A,Y) W(A,Z) isnonemptyandapartfromtheinitialassignment A thisintersectionmaycontainassignmentswhicharemodiedonlyinthecommonvectors Y Z .Tocalculatethesize oftheneighborhoodofk-optletusintroduce W(A,E) asasetofallassignmentsin W(A,E) suchthateveryvectorin E ismodiedinatleastonedimension,where E A isthesetof k selectedvectorsintheassignment A : W(A,E) = A W(A,E) :| A A|= n Š k . Thentheneighborhood Nk -opt(A) ofk-optis Nk -opt(A) = E A, | E | kW(A,E) (11) andsince W(A,Y) W(A,Z) = if Y = Z wehave | Nk -opt(A) |= E A, | E | k| W(A,E) |=ki = 0 n i Ni, (12) where Ni=| W(A,E) | forany E with | E |= i .Observethat W(A,E) = W(A,E) \ E EW(A,E) and | W(A,E) |= k !s Š 1for | E |= k and,hence, Nk= k !s Š 1Šk Š 1i = 0 k i Ni. (13) Itisobviousthat N0= 1sinceonecanobtainexactlyoneassignment(thegivenone) bychangingnovectors.Fromthisand( 13 )wehave N1= 0, N2= 2s Š 1Š 1and N3= 6s Š 1Š 3 · 2s Š 1+ 2.Fromthisand( 12 )itfollows | N2-opt(A) |= 1 + n 2 ( 2s Š 1Š 1 ), (14) | N3-opt(A) |= 1 + n 2 ( 2s Š 1Š 1 ) + n 3 ( 6s Š 1Š 3 · 2s Š 1+ 2 ). (15) Inourimplementation,weskipaniterationifthecorrespondingsetofvectors E eitherconsistsofthevectorsoftheminimalweight( w(e) = mine Xw(e) forevery e E )orallthesevectorshaveremainedunchangedduringthepreviousrunof k -opt.

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208D.Karapetyan,G.GutinItisassumedintheliterature(BalasandSaltzman 1991 ;Pasiliaoetal. 2005 ; Robertson 2001 )thatk-optfor k> 2istooslowtobeappliedinpractice.However, theneighborhood Nk -optdoesnotonlyincludestheneighborhood N(k Š 1 ) -optbutalso growsexponentiallywiththegrowthof k and,thus,becomesverypowerful.Wedecidedtoinclude2-optand3-optinourresearch.Greatervaluesof k arenotconsidered inthispaperbecauseofnonpracticaltimecomplexity(observethatthetimecomplexityof4-optis O(n4· 24s Š 1) )andeven3-optwithalltheimprovementsdescribed abovestilltakesalotoftimetoproceed.However,3-optismorerobustwhenusedin acombinationwithsomeotherheuristic(seeSect. 2.4 ). Itisworthnotingthatourextensionofthepairwise(triple)interchangeheuristic(BalasandSaltzman 1991 )isnottypical.Manypapers(Aiexetal. 2005 ; Clemonsetal. 2004 ;Murpheyetal. 1998 ;Pasiliaoetal. 2005 ;Robertson 2001 ) consideranotherneighborhood: Nk -opt*(A) = pD(A,) : D { 1 , 2 ,...,s } , | D |= 1and movesatmost k elements , where pDisdenedin( 2 ).Thesizeofsuchneighborhoodis | Nk -opt*(A) |= s · n k · (k !Š 1 ) + 1andthetimecomplexityofonerunofk-opt*intheassumption k n is O(s · nk· k ! ) ,i.e.,unlikek-opt,itisnotexponentialwithrespecttothenumberof dimensions s whichisconsideredtobeimportantbymanyresearchers.However,as itisstatedinSect. 1 ,weassumethat s isasmallxedconstantand,thus,thetime complexityofk-optisstillreasonable.Atthesametime,observethat Nk -opt*(A) N1-DV(A) forany k n ,i.e.,1DVperformsasgoodasn-opt*withthetimecomplexity of3-opt*.Onlyinthecaseof k = 2theheuristic2-opt*isfasterintheoryhowever itisknown(BurkardandÇela 1999 )thattheexpectedtimecomplexityofAPis signicantlylessthan O(n3) and,thus,therunningtimesof2-opt*and1DVaresimilar while1DVisdenitelymorepowerful.Becauseofthiswedonotconsider2-opt*in ourcomparison. 2.3VariableDepthInterchange(v-opt) The VariableDepthInterchange (VDI)wasrstintroducedbyBalasandSaltzman for3-APasaheuristicbasedonthewellknownLin-Kernighanheuristicforthe travelingsalesmanproblem(BalasandSaltzman 1991 ).Weprovidehereanatural extensionv-optoftheVDIheuristicforthe s -dimensionalcase, s 3,andthenimprovethisextension.Ourcomputationalexperimentsshowthattheimprovedversion ofv-optissuperiortothenaturalextensionofVDIwithrespecttosolutionqualityat thecostofareasonableincreaseinrunningtime.Inwhatfollows,v-optreferstothe improvedversionoftheheuristicunlessotherwisespecied. InBalasandSaltzman( 1991 ),theheuristicisdescribedquitebriey.Ourcontributionisnotonlyintheextending,improvingandanalyzingitbutalsoinamore detailedand,webelieve,clearerexplanationofit.Wedescribetheheuristicina differentwaytothedescriptionprovidedinBalasandSaltzman( 1991 ),however, bothversionsofouralgorithmareequaltoVDIincaseof s = 3.Thisfactwasalso checkedbyreproductionofthecomputationalevaluationresultsreportedinBalas andSaltzman( 1991 ).

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Localsearchheuristicsforthemultidimensionalassignmentproblem209Furtherwewillusefunction U(u,v) whichreturnsasetofswapsbetweenvectors u and v .Thedifferencebetweenthetwoversionsofv-optisonlyinthe U(u,v) denition.ForthenaturalextensionofVDI,let U(u,v) beasetofallthepossible swaps(see( 3 ))inatmostonedimensionbetweenthevectors u and v ,wherethe coordinatesinatmostonedimensionareswapped: U(u,v) = swap (u,v,D) : D { 1 , 2 ,...,s } and | D | 1 . Fortheimprovedversionofv-opt,let U(u,v) beasetofallthepossibleswapsin atmost s/ 2 dimensionsbetweenthevectors v and w : U(u,v) = swap (u,v,D) : D { 1 , 2 ,...,s } and | D | s/ 2 . Theconstraint | D | s/ 2guaranteesthatatleasthalfofthecoordinatesofevery swapareequaltotherstvectorcoordinates.Thecomputationalexperimentsshow thatremovingthisconstraintincreasestherunningtimeanddecreasestheaverage solutionquality. Letvector µ(u,v) betheminimumweightswapbetweenvectors u and v : µ(u,v) = argmine U(u,v)w(e). Let A beaninitialassignment. 1.Foreveryvector c A dotherestofthealgorithm. 2.Initializethe totalgain G = 0,the bestassignment Abest= A ,andasetofavailablevectors L = A \{ c } . 3.Findvector m L suchthat w(µ(c,m)) isminimized.Set v = µ(c,m) and vj= { cj,mj}\{ vj} forevery1 j s .Now v U(c,m) istheminimumweight swapof c withsomeothervector m intheassignment,and v isthecomplementary vector. 4.Set G = G + w(c) Š w(v) .Ifnow G 0,set A = Abestandgotothenextiteration (Step 1 ). 5.Mark m asanunavailableforthefurtherswaps: L = L \{ m } .Notethat c isalready markedunavailable: c/ L . 6.Replace m and c with v and v .Set c = v . 7.If w(A)
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210D.Karapetyan,G.Gutinassignmentevenifthecorrespondingswapisin U(c,m) .Thus,wedonotprovide anyresultsfortheneighborhoodofv-opt. 2.4Combinedneighborhood Wehavealreadypresentedtwotypesofneighborhoodsinthispaper,letussay dimensionwise (Sect. 2.1 )and vectorwise (Sects. 2.2 and 2.3 ).Theideaofthecombinedheuristicistousethedimensionwiseandthevectorwiseneighborhoodstogether,combiningthemintosocalledVariableNeighborhoodSearch(Talbi 2009 ). Thecombinedheuristicimprovestheassignmentbymovingitintothelocaloptimum withrespecttothedimensionwiseneighborhood,thenitimprovesitbymovingitto thelocalminimumwithrespecttothevectorwiseneighborhood.Theprocedureis repeateduntiltheassignmentoccursinthelocalminimumwithrespecttoboththe dimensionwiseandthevectorwiseneighborhoods. Moreformally,thecombinedheuristicDVoptconsistsofadimensionwiseheuristic DV (either1DV,2DVorsDV)andavectorwiseheuristic opt (either2-opt,3-optorv-opt).DVoptproceedsasfollows. 1.Applythedimensionwiseheuristic A = DV(A) . 2.Repeat: (a)Savetheassignmentweight x = w(A) andapplythevectorwiseheuristic A = opt(A) . (b)If w(A) = x stopthealgorithm. (c)Savetheassignmentweight x = w(A) andapplythedimensionwiseheuristic A = DV(A) . (d)If w(A) = x stopthealgorithm. Step 1 ofthecombinedheuristicisthehardestone.Indeed,itistypicalthatit takesalotofiterationstomoveabadsolutiontoalocalminimumwhileforagood solutionittakesjustafewiterations.Hence,therstofthetwoheuristicsshouldbe themostefcientone,i.e.,itshouldperformquicklyandproduceagoodsolution. Inthiscasethedimensionwiseheuristicsaremoreefcientbecause,havingapproximatelythesameasvectorwiseheuristicstimecomplexity,theysearchmuchlarger neighborhoods.Thefactthatthedimensionwiseheuristicsaremoreefcientthanthe vectorwiseonesisalsoconrmedbyexperimentalevaluation(seeSect. 4 ). Itisclearthattheneighborhoodofacombinedheuristicisdenedasfollows: NDVopt(A) = NDV(A) Nopt(A), (16) where NDV(A) and Nopt(A) areneighborhoodsofthecorrespondingdimensionwise andvectorwiseheuristicsrespectively.Tocalculatethesizeoftheneighborhood NDVopt(A) weneedtondthesizeoftheintersectionoftheseneighborhoods.Observethat NDV(A) Nk -opt(A) = pD(A,) : D D and movesatmost k elements , (17) where pD(A,) isdenedby( 2 ).Thismeansthat,if rkisthenumberofpermutations on n elementswhichmoveatmost k elements,theintersection( 17 )hassize | NDV(A) Nk -opt(A) |=| D |· (rkŠ 1 ) + 1 . (18)

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Localsearchheuristicsforthemultidimensionalassignmentproblem211Thenumber rkcanbecalculatedas rk=ki = 0 n i · di, (19) where diisthenumberofderangementson i elements,i.e.,permutationson i elementssuchthatnoneoftheelementsappearontheirplaces; di= i !· i m = 0( Š 1 )m/m ! (Harrisetal. 2008 ).For k = 2, r2= 1 + n 2 ;for k = 3, r3= 1 + n 2 + 2 n 3 .From( 9 ),( 12 ),( 16 )and( 18 )weimmediatelyhave NDVk -opt(A) = 1 +| D |· (n !Š 1 ) + ki = 2 n i Ni Š| D |· (rkŠ 1 ), (20) where Niand rkarecalculatedaccordingto( 13 )and( 19 )respectively.Substituting thevalueof k ,wehave: NDV2-opt(A) = 1 +| D |· (n !Š 1 ) + n 2 ( 2s Š 1Š 1 ) Š| D |· n 2 and(21) NDV3-opt(A) = 1 +| D |· (n !Š 1 ) + n 2 ( 2s Š 1Š 1 ) + n 3 ( 6s Š 1Š 3 · 2s Š 1+ 2 ) Š| D |· n 2 + 2 n 3 (22) Onecaneasilysubstitute | D |= s , | D |= s 2 and | D |= 2s Š 1Š 1to( 21 )or( 22 )to gettheneighborhoodsizesof1DV2,2DV2,sDV2,1DV3,2DV3andsDV3.Wewillonly showtheresultsforsDV2: Ns DV2(A) = 1 + ( 2s Š 1Š 1 ) · (n !Š 1 ) + n 2 ( 2s Š 1Š 1 ) Š ( 2s Š 1Š 1 ) · n 2 = 1 + ( 2s Š 1Š 1 ) · (n !Š 1 ), (23) i.e., | Ns DV2(A) |=| Ns DV(A) | .Since Ns DV(A) Ns DV2(A) (see( 16 )),wecanconcludethat Ns DV2(A) = Ns DV(A) .Indeed,theneighborhoodof2-optcanbedened asfollows: N2-opt= pD(A,) : D { 2 , 3 ,...,s } and swapsatmosttwoelements , whichisobviouslyasubsetof Ns DV(A) (see( 5 )).Hence,thecombinedheuristicsDV2isofnointerest. Forothercombinationstheintersection( 17 )issignicantlysmallerthanboth neighborhoods NDV(A) and Nk-opt(A) (recallthattheneighborhood Nv-opthasavariablestructure).Indeed, | NDV(A) || NDV(A) Nk -opt(A) | because | D |· (n !Š 1 ) | D |· (rkŠ 1 ) for k n .Similarly, | N2-opt(A) || NDV(A) Nk -opt(A) | because n 2 ( 2s Š 1Š 1 ) | D |· n 2 if | D | 2s Š 1,whichisthecasefor1DVand2DVif s is largeenough.Finally, | N3-opt(A) || NDV(A) Nk -opt(A) | because n 2 ( 2s Š 1Š 1 ) +

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212D.Karapetyan,G.Gutinn 3 ( 6s Š 1Š 3 · 2s Š 1+ 2 ) | D |· n 2 + 2 n 3 ,whichistrueevenfor | D |= 2s Š 1,i.e., forsDV. Thetimecomplexityofthecombinedheuristicis O(nk· k !s Š 1+| D |· n3) incase of opt = k -optand O(n3· ( 2s Š 1+| D | )) if opt = v-opt.Theparticularformulasare providedinthefollowingtable. 2-opt3-optv-opt 1DV O( 2s Š 1· n2+ s · n3)O( 6s Š 1· n3)O( 2s· n3) 2DV O( 2s Š 1· n2+ s2· n3)O( 6s Š 1· n3)O( 2s· n3) s DV(nointerest) O( 6s Š 1· n3)O( 2s· n3) Notethatallthecombinationswith3-optandv-opthaveequaltimecomplexities; thisisbecausethetimecomplexitiesof3-optandv-optaredominant.Ourexperiments showthattheactualrunningtimesof3-optandv-optarereallymuchhighertheneven thesDVrunningtime.ThismeansthatthecombinationsoftheseheuristicswithsDVareapproximatelyasfastasthecombinationsoftheseheuristicswithlightdimensionwiseheuristics1DVand2DV.Moreover,asitwasnoticedaboveinthissection, thedimensionwiseheuristic,beingexecutedrst,simpliesthejobforthevectorwise heuristicand,hence,theincreaseofthedimensionwiseheuristicpowermaydecrease therunningtimeofthewholecombinedheuristic.Atthesametime,theneighborhoodsofthecombinationswithsDVaresignicantlylargerthantheneighborhoods ofthecombinationswith1DVand2DV.Wecanconcludethatthe‘light’heuristics1DV3,2DV3,1DVvand2DVvareofnointerestbecausethe‘heavy’heuristicssDV3andsDVv,havingthesametheoreticaltimecomplexity,aremorepowerfuland,moreover, outperformedthe‘light’heuristicsinourexperimentswithrespecttobothsolution qualityandrunningtimeonaverageandinmostofsingleexperiments. 2.5Otheralgorithms HereweprovidealistofsomeotherMAPalgorithmspresentedintheliterature. –Ahostoflocalsearchproceduresandconstructionheuristicswhichoftenhave someapproximationguarantee(Bandeltetal. 2004 ;Burkardetal. 1996b ;Crama andSpieksma 1992 ;Isleretal. 2005 ;KurokiandMatsui 2007 ;Murpheyetal. 1998 andsomeothers)areproposedforspecialcasesofMAP(usuallywithdecomposableweights,seeSect. 3.2 )andexploitthespecicsoftheseinstances.However, asitwasstatedinSect. 1 ,weconsideronlythegeneralcaseofMAP,i.e.,allthe algorithmsincludedinthispaperdonotrelyonanyspecialstructureoftheweight matrix. –Anumberofconstructionheuristicsareintendedtogeneratesolutionsforgeneral caseMAP(BalasandSaltzman 1991 ;Gutinetal. 2008 ;Karapetyanetal. 2009 ; OliveiraandPardalos 2004 ).Whilesomeofthemarefastandlowquality,likeGreedy,some,likeMax-Regret,aresignicantlyslowerbutproducemuchbetter solutions.Aspecialclassofconstructionheuristics,GreedyRandomizedAdaptive SearchProcedure(GRASP),wasalsoinvestigatedbymanyresearchers(Aiexet al. 2005 ;Murpheyetal. 1998 ;OliveiraandPardalos 2004 ;Robertson 2001 ).

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Localsearchheuristicsforthemultidimensionalassignmentproblem213–Severalmetaheuristics,includingasimulatedannealingprocedure(Clemonsetal. 2004 )andamemeticalgorithm(HuangandLim 2006 ),wereproposedintheliterature.Metaheuristicsaresophisticatedalgorithmsintendedtosearchforthenear optimalsolutionsinareasonablylargetime.Proceedingformuchlongerthanlocal searchandbeinghardfortheoreticalanalysisoftherunningtimeortheneighborhood,metaheuristicscannotbecomparedstraightforwardlytolocalsearchprocedures. –Someweakvariationsof2-optareconsideredinAiexetal.( 2005 ),Murpheyetal. ( 1998 ),Pasiliaoetal.( 2005 ),Robertson( 2001 ).Whileourheuristic2-opttriesall possiblerecombinationsofapairofassignmentvectors,i.e.,2s Š 1combinations, thesevariationsonlytrytheswapsinonedimensionatatime,i.e., s combinations foreverypairofvectors.Wehavealreadydecidedthatthesevariationshaveno practicalinterest,fordetailsseeSect. 2.2 . 3Testbed Whilethetheoreticalanalysiscanhelpinheuristicdesign,selectionofthebestapproachesrequiresempiricalevaluation(GutinandKarapetyan 2009 ;RardinandUzsoy 2001 ).InthissectionwediscussthetestbedandinSect. 4 theexperimental resultsarereportedanddiscussed. Thequestionofselectingpropertestbedisoneofthemostimportantquestionsinheuristicexperimentalevaluation(RardinandUzsoy 2001 ).Whilemanyresearchersfocusedoninstanceswithrandomindependentweights(AndrijichandCaccetta 2001 ;BalasandSaltzman 1991 ;Krokhmaletal. 2007 ;Pasiliaoetal. 2005 and someothers)andrandominstanceswithpredenedsolutions(Clemonsetal. 2004 ; GrundelandPardalos 2005 ;Karapetyanetal. 2009 ),severalmoresophisticated modelsareofgreaterpracticalinterest(Bandeltetal. 2004 ;Burkardetal. 1996b ; CramaandSpieksma 1992 ;FriezeandYadegar 1981 ;KurokiandMatsui 2007 ). Thereisalsoanumberofpaperswhichconsiderreal-worldandpseudoreal-world instances(Bekkeretal. 2005 ;Murpheyetal. 1998 ;PardalosandPitsoulis 2000 )but theauthorsofthispapersupposethattheseinstancesdonotwellrepresentallthe instanceclassesandbuildingaproperbenchmarkwiththereal-worldinstancesisa subjectforanotherresearch. Inthispaperwegroupalltheinstancefamiliesintotwoclasses:instanceswith independentweights(Sect. 3.1 )andinstanceswithdecomposableweights(Sect. 3.2 ). Laterweshowthattheheuristicsperformdifferentlyontheinstancesoftheseclasses and,thus,thisdevisionhelpsusincorrectexperimentalanalysisofthelocalsearch algorithms. 3.1Instanceswithindependentweights OneofthemoststudiedclassofinstancesforMAPis RandomInstanceFamily .InRandom,theweightassignedtoavectorisarandomuniformlydistributedintegral valueintheinterval [ a,b Š 1 ] .RandominstanceswereusedinAiexetal.( 2005 ), AndrijichandCaccetta( 2001 ),BalasandSaltzman( 1991 ),Pierskalla( 1968 )and someothers.

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214D.Karapetyan,G.GutinSincetheinstancesarerandomandquitelarge,itispossibletoestimatetheaveragesolutionvaluefortheRandomInstanceFamily.Thepreviousresearchinthis area(Krokhmaletal. 2007 )showthatif n tendstoinnitythantheproblemsolution approachesthebound an ,i.e.,theminimalpossibleassignmentweight(observethat theminimalassignmentincludes n vectorsofweight a ).Moreover,anestimation ofthemeanoptimalsolutionisprovidedinGrundeletal.( 2004 )butthisestimationisnotaccurateenoughforourexperiments.InGutinandKarapetyan( 2009 )we provethatitisverylikelythateverybigenoughRandominstancehasatleastone an -assignment,where x -assignment meansanassignmentofweight x . Let bethenumberofassignmentsofweight an andlet c = b Š a .Wewouldlike tohaveanupperboundontheprobabilityPr ( = 0 ) .Suchanupperboundisgiven inthefollowingtheoremwhoseproof(seeGutinandKarapetyan 2009 )isbasedon theExtendedJansenInequality(seeTheorem8.1.2ofAlonandSpencer 2000 ). Theorem1 Forany n suchthat n 3 and n Š 1 e s Š 1 c · 21 n Š 1, (24) wehave Pr ( = 0 ) eŠ1 2 , where = n Š 2 k = 1(n k)· ck [ n · (n Š 1 ) ··· (n Š k + 1 ) ]s Š 1. ThelowerboundsofPr (> 0 ) fordifferentvaluesof s and n andfor b Š a = 100, arereportedbelow. s = 4 s = 5 s = 6 s = 7 n Pr (> 0 )n Pr (> 0 )n Pr (> 0 )n Pr (> 0 ) 150.575100.99181.00071.000 200.823110.998 250.943121.000 300.986 350.997 401.000 Onecanseethata4-APRandominstancehasan (an) -assignmentwiththeprobabilitywhichisverycloseto1if n 40;a5-APinstancehasan (an) -assignment withprobabilityverycloseto1for n 12,etc.;so,theoptimalsolutionsforall theRandominstancesusedinourexperiments(seeSect. 4 )areverylikelytobeof weight an .For s = 3Theorem 1 doesnotprovideagoodupperbound,butweare abletousetheresultsfromTableIIinBalasandSaltzman( 1991 )instead.Balasand Saltzmanreportthatintheirexperimentstheaverageoptimalsolutionof3-APforRandominstancesreducesveryquicklyandhasasmallvalueevenfor n = 26.Since thesmallestRandominstanceweuseinourexperimentshassize n = 150,weassume thatall3-APRandominstancesconsideredinourexperimentareverylikelytohave an an -assignment. Usefulresultscanalsobeobtainedfrom(11)inGrundeletal.( 2004 )whichis anupperboundfortheaverageoptimalsolution.Grundeletal.( 2004 )considerthe

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Localsearchheuristicsforthemultidimensionalassignmentproblem215sameinstancefamilyexcepttheweightsofthevectorsarerealnumbersuniformly distributedintheinterval [ a,b ] .HowevertheresultsfromGrundeletal.( 2004 )can beextendedtoourdiscretecase.Let w(e) bearealweightofthevector e ina continuousinstance.Consideradiscreteinstancewith w(e) = w(e) (if w(e) = b , set w(e) = b Š 1).Notethattheweight w(e) isauniformlydistributedintegerinthe interval [ a,b Š 1 ] .Theoptimalassignmentweightofthisinstanceisnotlargerthan theoptimalassignmentweightofthecontinuousinstanceand,thus,theupperbound fortheaverageoptimalsolutionforthediscretecaseiscorrect. Infact,theupperbound ¯ z u(seeGrundeletal. 2004 )fortheaverageoptimalsolutionisnotaccurateenough.Forexample, ¯ z u an + 6 . 9for s = 3, n = 100and b Š a = 100,and ¯ z u an + 3 . 6for s = 3, n = 200and b Š a = 100,soitcannotbe usedfor s = 3inourexperiments.Theupperbound ¯ z ugivesabetterapproximation forlargervaluesof s ,e.g., ¯ z u an + 1 . 0for s = 4, n = 40and b Š a = 100,however, Theorem 1 providesstrongerresults(Pr (> 0 ) 1 . 000forthiscase). Anotherclassofinstanceswithalmostindependentweightsis GPInstanceFamily whichcontainspseudo-randominstanceswithpredenedoptimalsolutions.GPinstancesaregeneratedbyanalgorithmproducedbyGrundelandPardalos( 2005 ). Thegeneratorisnaturallydesignedfor s -APforarbitrarylargevaluesof s and n . However,itisrelativelyslowand,thus,itwasimpossibletogeneratelargeGPinstances.Nevertheless,thisiswhatweneedsincenallywehavebothsmall(GP)and large(Random)instanceswithindependentweightswithknownoptimalsolutions. 3.2Instanceswithdecomposableweights Inmanycasesitisnoteasytodeneaweightforan s -tupleofobjectsbutitispossible todenearelationbetweeneverypairofobjectsfromdifferentsets.Inthiscaseone shoulduse decomposableweights (Spieksma 2000 )(or decomposablecosts ),i.e.,the weightofavector e shouldbedenedasfollows: w(e) = f d1 , 2 e1,e2,d1 , 3 e1,e3,...,ds Š 1 ,s es Š 1,es , (25) where di,jisadistancematrixbetweenthesets Xiand Xjand f issomefunction. ThemostnaturalinstancefamilywithdecomposableweightsisClique,whichdenesthefunction f asthesumofallarguments: wc(e) =n Š 1i = 1 nj = i + 1di,j ei,ej. (26) TheCliqueinstancefamilywasinvestigatedinBandeltetal.( 2004 ),Cramaand Spieksma( 1992 ),FriezeandYadegar( 1981 )andsomeothers.Itwasproven(Crama andSpieksma 1992 )thatMAPrestrictedtoCliqueinstancesremainsNP-hard. AspecialcaseofCliqueis GeometricInstanceFamily .InGeometric,thesets X1, X2,..., XscorrespondtosetsofpointsinEuclideanspace,andthedistancebetween twopoints u Xiand v XjisdenedasEuclideandistance;weconsiderthetwo dimensionalEuclideanspace: dg(u,v) = (uxŠ vx)2+ (uyŠ vy)2.

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216D.Karapetyan,G.GutinItisproven(SpieksmaandWoeginger 1996 )thattheGeometricinstancesareNP-hard tosolvefor s = 3and,thus,GeometricisNP-hardforevery s 3. Inthispaper,weproposeanewspecialcaseofthedecomposableweights,SquareRoot.ItisamodicationoftheCliqueinstancefamily.Assumewehave s radarsand n planesandeachradarobservesalltheplanes.Theproblemistoassignsignalswhich comefromdifferentradarstoeachother.Itisquitenaturaltodeneadistancefunctionbetweeneachpairofsignalsfromdifferentradars,andforasetofsignalswhich correspondtooneplanethesumofthedistancesshouldbesmallso( 26 )isagood choice.However,itisnotactuallycorrecttominimizethetotaldistancebetweenthe assigningsignalsbutoneshouldalsoensurethatnoneofthesedistancesistoolarge. Samerequirementsappearinanumberofotherapplications.Weproposeaweight functionwhichleadstobothsmalltotaldistancebetweentheassignedsignalsand smalldispersionofthedistances: wsq(e) = n Š 1i = 1 nj = i + 1 di,j ei,ej2. (27) SimilarapproachisusedinKurokiandMatsui( 2007 )thoughtheydonotusesquare root,i.e.,avectorweightisjustasumofsquaresoftheedgeweightsinaclique.In addition,theedgeweightsinKurokiandMatsui( 2007 )arecalculatedasdistances betweensomenodesinaEuclideanspace. Anotherspecialcaseofthedecomposableweights,Product,isstudiedinBurkard etal.( 1996b ).Burkardetal.consider3-APanddenetheweight w(e) as w(e) = a1 e1· a2 e2· a3 e3,where a1, a2and a3arerandomvectorsofpositivenumbers.Itiseasyto showthattheProductweightfunctioncanberepresentedintheform( 25 ).Itisproven thattheminimizationproblemfortheProductinstancesisNP-hardincase s = 3and, thus,itisNP-hardforevery s 3. 4Computationalexperimentation Inthissection,theresultsofempiricalevaluationarereportedanddiscussed.Theexperimentswereconductedforthefollowinginstances(forinstancefamilydenitions seeSect. 3 ): –Randominstanceswhereeachweightwasrandomlychosenin { 1 , 2 ,..., 100 } ,i.e., a = 1and b = 101.AccordingtoSect. 3.1 ,theoptimalsolutionsofalltheconsideredRandominstancesareverylikelytobe an = n . –GPinstanceswithpredenedoptimalsolutions(seeSect. 3.1 ). –CliqueandSquareRootinstances,wheretheweightofeachedgeinthegraphwas randomlyselectedfrom { 1 , 2 ,..., 100 } .Insteadoftheoptimalsolutionvaluewe usethebestknownsolutionvalue. –Geometricinstances,wherebothcoordinatesofeverypointwererandomlyselected from { 1 , 2 ,..., 100 } .Thedistancesbetweenthepointsarecalculatedprecisely whiletheweightofavectorisroundedtothenearestinteger.Insteadoftheoptimal solutionvalueweusethebestknownsolutionvalue.

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Localsearchheuristicsforthemultidimensionalassignmentproblem217–Productinstances,whereeveryvalue aj iwasrandomlyselectedfrom { 1 , 2 ,..., 10 } . Insteadoftheoptimalsolutionvalueweusethebestknownsolutionvalue. Aninstancenameconsistsofthreeparts:thenumber s ofdimensions,thetypeof theinstance(‘gp’forGP,‘r’forRandom,‘c’forClique,‘g’forGeometric,‘p’forProductand’sr’forSquareRoot),andthesize n oftheinstance.Forexample, 5r40 means avedimensionalRandominstanceofsize40.Foreverycombinationofinstancesize andtypewegenerated10instances,usingthenumber seed = s + n + i asaseedof therandomnumbersequences,where i isanindexoftheinstanceofthistypeand size, i { 1 , 2 ,..., 10 } .Thereby,everyexperimentisconductedfor10differentinstancesofsomexedtypeandsize,i.e.,everynumberreportedinthetablesbelowis averagefor10runsfor10differentinstances. ThesizesofallbutGPinstancesareselectedsuchthateveryalgorithmcould processthemallinapproximatelythesametime.TheGPinstancesareincludedin ordertoexaminethebehavioroftheheuristicsonsmallerinstances(recallthatGPis theonlyinstancesetforwhichweknowtheexactsolutionsforsmallinstances). AlltheheuristicsareimplementedinVisualC++.Theevaluationplatformisbased onAMDAthlon64X23.0GHzprocessor. Further,theresultsoftheexperimentsofthreedifferenttypesareprovidedand discussed: –InSect. 4.1 ,thelocalsearchheuristicsareappliedtotheassignmentsgeneratedby someconstructionheuristic.Theseexperimentsallowustoexcludeseverallocal searchesfromtherestoftheexperiments,however,thecomparisonoftheresults iscomplicatedbecauseofthesignicantdifferenceinboththesolutionqualityand therunningtime. –InSect. 4.2 ,twosimplemetaheuristicsareusedtoequatetherunningtimesof differentheuristics.Thisisdonebyvaryingofnumberofiterationsofthemetaheuristics. –InSect. 4.3 ,theresultsofallthediscussedapproachesaregatheredintwotables tondthemostsuccessfulsolversfortheinstancewithindependentanddecomposableweightsforeveryparticularrunningtime. 4.1Purelocalsearchexperiments First,weruneverylocalsearchheuristicforeveryinstanceexactlyonce.Thelocalsearchisappliedtosolutionsgeneratedwithoneofthefollowingconstruction heuristics: 1.Trivial,whichwasrstmentionedinBalasandSaltzman( 1991 )as Diagonal .Trivialconstructionheuristicsimplyassigns Ai j= i forevery i = 1 , 2 ,...,n and j = 1 , 2 ,...,s . 2.Greedyheuristicwasdiscussedinmanypapers,see,e.g.BalasandSaltzman ( 1991 ),Burkardetal.( 1996b ),Gutinetal.( 2008 ),GutinandKarapetyan( 2008 , 2009 ),Karapetyanetal.( 2009 ).Itwasproven(Gutinetal. 2008 )thatintheworst caseGreedyproducestheuniqueworstsolution;however,itwasshown(Gutinand Karapetyan 2008 )thatinsomecasesGreedymaybeagoodselectionasafastand simpleheuristic.

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218D.Karapetyan,G.Gutin3.Max-Regretwasdiscussedinanumberofpapers,see,e.g.,BalasandSaltzman ( 1991 ),Burkardetal.( 1996b ),Gutinetal.( 2008 ),Karapetyanetal.( 2009 ), Robertson( 2001 ).AsforGreedy,itisproven(Gutinetal. 2008 )thatinthe worstcaseMax-Regretproducestheuniqueworstsolutionhowevermanyresearchers(BalasandSaltzman 1991 ;Karapetyanetal. 2009 )notedthatMax-Regretisquitepowerfulinpractice. 4.ROMwasrstintroducedinGutinetal.( 2008 )asaheuristicofalargedominationnumber.Oneveryiteration,theheuristiccalculatesthetotalweightforevery setofvectorswiththexedrsttwocoordinates: Mi,j= e X,e1= i,e2= jw(e) . Thenitsolvesa2-APfortheweightmatrix M andreorderstheseconddimensionoftheassignmentaccordingtothissolutionandtherstdimensionofthe assignment.Theprocedureisrepeatedrecursivelyforthesubproblemwherethe rstdimensionisexcluded.FordetailsseeGutinetal.( 2008 ),Karapetyanetal. ( 2009 ). Wewillbeginourdiscussionfromtheexperimentsstartedfromtrivialassignments.TheresultsreportedinTables 2 and 3 areaveragesfor10experimentssince everyrowofthesetablescorrespondsto10instancesofsomexedtypeandsize butofdifferentseedvalues(seeabove).Thetablesaresplitintotwoparts;therst partcontainsonlytheinstanceswithindependentweights(GPandRandom)whilethe secondpartcontainsonlytheinstanceswithdecomposableweights(Clique,Geometric,ProductandSquareRoot).Theaveragevaluesfordifferentinstancefamiliesand numbersofdimensionsareprovidedatthebottomofeachpartofeachtable.The tablesarealsosplitverticallyaccordingtotheclassesofheuristics.Thewinnerin everyrowandeveryclassofheuristicsisunderlined. Thevalueofthesolutionerroriscalculatedas (w(A)/w(Abest) Š 1 ) · 100%,where A istheobtainedassignmentand Abestistheoptimalassignment(orthebestknown one,seeabove). Inthegroupofthevectorwiseheuristicsthemostpowerfuloneisdenitely3-opt.v-optoutperformsitonlyinafewexperiments,mostlythreedimensionalones(recall thattheneighborhoodofk-optincreasesexponentiallywiththeincreaseofthenumber ofdimensions s ).Asitwasexpected,2-optneveroutperforms3-optsince N2-opt N3-opt(seeSect. 2.2 ).Thetendenciesfortheindependentweightinstancesandfor thedecomposableweightinstancesaresimilar;theonlydifferencewhichisworthto noteisthatallbutv-optheuristicsofthisgroupsolvetheProductinstancesverywell. NotethatthedispersionoftheweightsinProductinstancesisreallyhighand,thus,v-opt,whichminimizestheweightofonlyonevectorineverypairofvectorswhile theweightofthecomplementaryvectormayincreasearbitrary,cannotbeefcient forthem. Asonecanexpect,sDVismoresuccessfulthan2DVand2DVismoresuccessful than1DVwithrespecttothesolutionquality(obviously,alltheheuristicsofthisgroup performequallyfor3-AP,and2DVandsDVarealsoequalfor4-AP,seeSect. 2.1 ). However,fortheinstanceswithdecomposableweightsallthedimensionwiseheuristicsperformverysimilarlyandevenforthelarge s ,sDVisnotsignicantlymore powerfulthan1DVor2DVwhichmeansthatincaseofdecomposableinstancesthe mostefcientiterationsarewhen | D |= 1.Wecanassumethatif c isthenumber ofedgesconnectingthexedandunxedpartsoftheclique,thenaniterationof

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Localsearchheuristicsforthemultidimensionalassignmentproblem219adimensionwiseheuristicisratherefcientwhen c issmall.Observethat,e.g.,forCliquethediversityofvaluesintheweightmatrix [ Mi,j]n × n(see( 4 ))decreaseswith theincreaseofthenumber c and,hence,thespaceforoptimizationoneveryiteration isdecreasing.Observealsothatinthecase c = 1theiterationleadstotheoptimal matchbetweenthexedandunxedpartsoftheassignmentvectors. Allthecombinedheuristicsshowimprovementsinthesolutionqualityovereach oftheircomponents,i.e.,overbothcorrespondingvectorswiseanddimensionwise localsearches.Inparticular,1DV2outperformsboth2-optand1DV,2DV2outperforms both2-optand2DV,sDV3outperformsboth3-optandsDVandsDVvoutperformsbothv-optandsDV.Moreover,sDV3issignicantlyfasterthan3-optandsDVvissignicantlyfasterthanv-opt.Hence,wewillnotdiscussthesingleheuristics3-optandv-optintherestofthepaper.Theheuristics1DV2and2DV2,obviously,performequallyfor 3-APinstances. Whilefortheinstanceswithindependentweightsthecombinationofthedimensionwiseheuristicswiththevectorwiseonessignicantlyimprovesthesolutionquality,itisnotthecasefortheinstanceswithdecomposableweights(observethat1DVperformsalmostaswellasthemostpowerfulheuristicsDV3)whichshowstheimportanceoftheinstancesdivision.Weconcludethatthevectorwiseneighborhoods arenotefcientfortheinstanceswithdecomposableweights. Nextweconductedtheexperimentsstartingfromtheotherconstructionheuristics.Butrstwecomparedtheconstructionheuristicsthemselves,seeTable 1 .Itis notsurprisingthatTrivialproducestheworstsolutions.However,onecanseethatTrivialoutperformsGreedyandMax-RegretforeveryProductinstance.Thereasonisinthe extremelyhighdispersionoftheweightsinProduct.BothGreedyandMax-Regretconstructtheassignmentsbyaddingnewvectorstoit.Thedecisionwhichvectorshould beaddeddoesnotdepend(ordoesnotdependenoughincaseofMax-Regret)onthe restofthevectorsand,thus,attheendoftheprocedureonlythevectorswithhuge weightsareavailable.Forotherinstancefamilies,Greedy,Max-RegretandROMperformsimilarlythoughtherunningtimeoftheheuristicsisverydifferent.Max-Regretisdenitelytheslowestconstructionheuristic;GreedyisveryfastfortheRandominstances(thisisbecauseofthelargenumberofvectorsoftheweight a andtheimplementationfeatures,seeKarapetyanetal. 2009 fordetails)andrelativelyslowfor therestoftheinstances;ROM’srunningtimealmostdoesnotdependontheinstance andisconstantlymoderate. StartingfromGreedy(Table 4 )signicantlyimprovesthesolutionquality.This mostlyinuencestheweakestheuristics,e.g.,2-optaverageerrordecreasedinour experimentsfrom59%and20%to15%and6%forindependentanddecomposable weights,respectively,though,e.g.,themostpowerfulheuristicsDV3erroralsonoticeablydecreased(from2.8%and5.8%to2.0%and2.5%).Asregardstherunning time,Greedyisslowerthanmostofthelocalsearchheuristicsand,thus,therunning timesofallbutsDV3andsDVvheuristicsareverysimilar.Thebestoftherestofthe heuristicsinthisexperimentissDVthough1DV2and2DV2performsimilarly. StartingfromMax-Regretimprovesthesolutionqualityevenmorebutatthecost ofverylargerunningtimes.Inthiscasethedifferenceintherunningtimeofthelocal searchheuristicsalmostdisappearsandsDV3,thebestone,reachestheaverageerrorvalues1.3%and2.2%forindependentanddecomposableweights,respectively.

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220D.Karapetyan,G.Gutin Table1 Constructionheuristicscomparison Inst.BestSolutionerror,%Runningtimes,ms TrivialGreedyMax-RegretROMTrivialGreedyMax-RegretROM 3gp100504 . 41576 6100 407999 3r150150 . 049975429 340 14425326 4gp30145 . 2158992 0 352067 4r8080 . 049857449 760 1227285278 5gp1266 . 2147139 90 6362 5r4040 . 04911159116 1690 637214686 6gp841 . 8143251 140 5332 6r2222 . 05180295218 3100 624750861 7gp525 . 6157276 200 181 7r1414 . 05116377 4543960 217032805 8gp419 . 2113217 280 181 8r99 . 05262579514 5430 25604342 Allavg.2610137118 1340 119769252 GPavg.146176 140 151824 Rand.avg.5075256230 2550 719356500 3-APavg.25773017 220 27252617 4-APavg.25714129 390 2313745142 5-APavg.25298662 890 618625344 6-APavg.2662160110 1620 512391432 7-APavg.2637202 2302080 28520403 8-APavg.2687300261 2860 12806171 3cq1501738 . 512194120 370 56438827 3g1501552 . 086519273 0 53422628 3p15014437 . 2762151227 0 580431837 3sr1501077 . 812504221 430 60436329 4cq503034 . 84002722 320 1563713161 4g501705 . 249221292 0 2173828148 4p5020096 . 81034842788 0 10303725151 4sr501496 . 63672520 320 1933847150 5cq304727 . 12182017 240 6409636583 5g302321 . 834026333 0 9369650604 5p3055628 . 513710176468 0 27119536619 5sr301842 . 01961613 280 6669627615 6cq185765 . 51421515 180 4266758267 6g182536 . 026026273 0 5636802262 6p18135515 . 3163211812638 0 10986758323 6sr181856 . 31211313 190 4206775261 7cq126663 . 7911411 150 10376653924 7g123267 . 215619232 0 12176614944

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Localsearchheuristicsforthemultidimensionalassignmentproblem221 Table1 ( Continued ) Inst.BestSolutionerror,%Runningtimes,ms TrivialGreedyMax-RegretROMTrivialGreedyMax-RegretROM 7p12558611 . 7346316219949 0 18726463335 7sr121795 . 77899 150 9806510268 8cq87004 . 962101010 0 4652416130 8g83679 . 510515211 0 5692446120 8p82233760 . 0177360523099 0 7102413140 8sr81622 . 1527 7100 4742448132 Allavg.30945729014 0 7145580302 Cliqueavg.3552116 230 4635594349 Geom.avg.37021272 0 5935594351 Productavg.167176711028 0 13345536268 SRavg.3441914 240 4655595242 3-APavg.853794722 0 187432430 4-APavg.3401398719 0 3993778152 5-APavg.22327017715 0 12389612605 6-APavg.17154332912 0 6276773278 7-APavg.16880150910 0 12766560618 8-APavg.999095878 0 5552431131 StartingfromROMimprovesthequalityonlyfortheworstheuristics.Thisisprobably becauseallthebestheuristicscontainsDVwhichdoesagoodvectorwiseoptimization(recallthatROMexploitsasimilartothedimensionwiseneighborhoodidea).At thesametime,startingfromROMincreasestherunningtimeoftheheuristicssignicantly;theresultsforbothMax-RegretandROMareexcludedfromthepaper;onecan ndthemontheweb(Karapetyan 2009 ). Itisclearthattheconstructionheuristicsarequiteslowcomparingtothelocal searchandweshouldanswerthefollowingquestion:isitworthtospendsomuch timeontheinitialsolutionconstructionorthereissomewaytoapplylocalsearch severaltimesinordertoimprovetheassignmentsiteratively?Itisknownthatthe algorithmswhichapplylocalsearchseveraltimesarecalledmetaheuristics.There isanumberofdifferentmetaheuristicapproachessuchastabusearchormemetic algorithms,butthisisnotthesubjectofthispaper.Inwhatfollows,wearegoingto usetwosimplemetaheuristics,ChainandMultichain. 4.2Experimentswithmetaheuristics Itisobviousthatthereisnosenseinapplyingalocalsearchproceduretoonesolutionseveraltimesbecausethelocalsearchmovesthesolutiontoalocalminimum withrespecttoitsneighborhood,i.e.,thesecondexplorationofthisneighborhoodis useless.Inordertoapplythelocalsearchseveraltimes,oneshouldperturbthesolutionobtainedonthepreviousiteration.Thisideaimmediatelybringsustotherst metaheuristic,letussayChain:

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222D.Karapetyan,G.Gutin Table2 LocalsearchheuristicsstartedfromTrivial Inst.BestSolutionerror,% 2-opt3-optv-opt1DV2DV s DV1DV22DV2s DV3s DVv 3gp100504 . 419.610.0 19.84.9 4.9 4.9 4.94.94.6 4.9 3r150150 . 0134.516.01.5 2.4 2.4 2.4 2.42.42.10.7 4gp30145 . 217.44.2 13.411.17.9 7.9 10.77.94.2 7.5 4r8080 . 0115.07.32.0 20.511.5 11.5 18.911.54.11.6 5gp1266 . 210.62.1 8.512.56.9 6.9 10.16.91.8 6.9 5r4040 . 0104.54.33.8 63.034.3 34.3 47.334.33.5 5.3 6gp841 . 86.72.4 5.312.45.75.0 6.55.52.4 4.8 6r2222 . 0105.50.9 8.6125.062.354.5 80.955.51.8 9.1 7gp525 . 66.33.9 10.221.59.05.9 5.95.13.9 5.5 7r1414 . 095.70.0 36.4244.3111.472.1 92.170.00.7 16.4 8gp419 . 26.85.2 10.917.29.46.2 7.86.85.2 6.2 8r99 . 081.10.0 67.8323.3173.360.0 73.377.80.0 40.0 Allavg.58.64.7 15.771.536.622.6 30.124.02.9 9.1 GPavg.11.24.6 11.313.37.36.1 7.66.23.7 6.0 Rand.avg.106.14.7 20.0129.865.939.1 52.541.92.0 12.2 3-APavg.77.113.010.6 3.6 3.6 3.6 3.63.63.32.8 4-APavg.66.25.7 7.715.89.7 9.7 14.89.74.2 4.6 5-APavg.57.53.2 6.137.820.6 20.6 28.720.62.7 6.1 6-APavg.56.11.7 6.968.734.029.8 43.730.52.1 6.9 7-APavg.51.02.0 23.3132.960.239.0 49.037.52.3 10.9 8-APavg.43.92.6 39.4170.391.433.1 40.642.32.6 23.1 3cq1501738 . 5125.149.922.8 20.1 20.1 20.1 20.120.119.918.9 3g1501552 . 00.0 0.0 5.90.0 0.0 0.0 0.0 0.0 0.0 0.0 3p15014437 . 20.10.0 15.00.0 0.0 0.0 0.00.00.0 0.0 3sr1501077 . 8144.264.028.0 22.0 22.0 22.0 22.022.021.821.3 4cq503034 . 852.531.330.3 23.323.1 23.1 23.223.121.420.1 4g501705 . 20.0 0.0 11.10.20.0 0.0 0.0 0.0 0.0 0.0 4p5020096 . 80.00.0 49.60.10.0 0.0 0.10.00.0 0.0 4sr501496 . 656.830.6 31.927.224.8 24.8 27.224.823.4 23.9 5cq304727 . 130.918.7 21.416.916.6 16.6 16.816.615.5 16.1 5g302321 . 80.0 0.0 9.20.20.0 0.0 0.0 0.0 0.0 0.0 5p3055628 . 50.00.0 53.20.10.0 0.0 0.00.00.0 0.0 5sr301842 . 038.319.0 23.921.720.4 20.4 21.120.417.6 18.3 6cq185765 . 517.612.2 16.111.510.3 11.611.310.310.1 11.1 6g182536 . 00.0 0.0 15.40.50.0 0.0 0.0 0.0 0.0 0.0 6p18135515 . 30.00.0 98.30.20.00.0 0.00.00.0 0.0 6sr181856 . 320.911.9 17.412.7 13.913.612.713.911.5 12.6 7cq126663 . 711.95.3 10.48.07.05.9 7.16.95.7 5.8 7g123267 . 20.0 0.0 9.90.10.00.0 0.0 0.0 0.0 0.0

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Localsearchheuristicsforthemultidimensionalassignmentproblem223 Table2 ( Continued ) Inst.BestSolutionerror,% 2-opt3-optv-opt1DV2DV s DV1DV22DV2s DV3s DVv 7p12558611 . 70.00.0 123.60.20.00.0 0.00.00.0 0.0 7sr121795 . 712.17.6 11.08.510.17.1 8.310.15.9 7.0 8cq87004 . 96.43.0 8.56.44.4 4.85.34.12.2 4.7 8g83679 . 50.0 0.0 9.10.20.00.0 0.0 0.0 0.0 0.0 8p82233760 . 00.00.0 143.80.10.00.0 0.00.00.0 0.0 8sr81622 . 16.62.6 7.45.75.04.7 4.94.43.5 4.7 Allavg.21.810.7 32.27.87.47.3 7.57.46.6 6.9 Cliqueavg.40.720.018.2 14.413.6 13.714.013.512.5 12.8 Geom.avg.0.0 0.0 10.10.20.00.0 0.0 0.0 0.0 0.0 Productavg.0.00.0 80.60.10.00.0 0.00.00.0 0.0 SRavg.46.522.619.9 16.316.115.4 16.016.013.9 14.7 3-APavg.67.328.517.9 10.5 10.5 10.5 10.510.510.410.1 4-APavg.27.315.5 30.712.712.0 12.0 12.612.011.211.0 5-APavg.17.39.4 26.99.79.3 9.3 9.59.38.3 8.6 6-APavg.9.66.0 36.86.26.1 6.36.06.15.4 6.0 7-APavg.6.03.2 38.74.24.33.2 3.84.32.9 3.2 8-APavg.3.21.4 42.23.12.4 2.42.62.11.4 2.4 1.Initializeanassignment A ; 2.Set Abest= A ; 3.Repeat: (a)Applylocalsearch A = LS(A) ; (b)If w(A)
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224D.Karapetyan,G.Gutin Table3 LocalsearchheuristicsstartedfromTrivial Inst.Runningtime,ms 2-opt3-optv-opt1DV2DV s DV1DV22DV2s DV3s DVv 3gp1006.2 820.6181.814.314.0 16.518.416.7 430.679.0 3r15019.8 1737.965.717.618.717.1 22.718.9 147.845.4 4gp301.5 150.345.00.7 1.21.11.4 1.4116.917.5 4r8010.5 987.564.57.9 18.015.311.2 18.4344.898.2 5gp120.3 38.53.60.2 0.40.50.5 0.530.61.6 5r4016.9 425.934.32.3 7.26.34.6 8.6386.935.3 6gp80.2 57.22.50.2 0.30.40.4 0.542.01.3 6r222.2 218.916.70.9 2.63.91.9 4.3259.022.7 7gp50.1 48.90.90.1 0.20.30.1 0.340.00.9 7r141.4 237.112.00.4 1.62.91.8 3.0210.915.5 8gp40.1 117.50.80.2 0.30.60.2 0.372.30.9 8r90.9 191.96.70.3 1.12.31.0 3.1177.77.1 Allavg.5.0 419.436.23.8 5.55.65.3 6.3188.327.1 GPavg.1.4 205.539.12.6 2.73.23.53.3 122.116.9 Rand.avg.8.6 633.233.34.9 8.27.97.2 9.4254.537.4 3-APavg.13.0 1279.2123.816.0 16.416.820.517.8 289.262.2 4-APavg.6.0 568.954.74.3 9.68.26.3 9.9230.857.8 5-APavg.8.6 232.219.01.3 3.83.42.5 4.5208.718.5 6-APavg.1.2 138.19.60.5 1.52.11.1 2.4150.512.0 7-APavg.0.7 143.06.50.3 0.91.61.0 1.6125.58.2 8-APavg.0.5 154.73.80.3 0.71.40.6 1.7125.04.0 3cq15022.1 4366.51388.442.139.334.9 41.0 46.01503.9497.6 3g15019.0 2229.3780.026.228.125.5 37.233.0 1299.5201.2 3p15015.4 2149.7847.182.0 89.889.796.0 101.91730.1458.6 3sr15021.7 3949.91157.536.0 37.537.941.2 47.11400.9469.6 4cq506.1 872.0308.93.8 8.57.36.1 10.8468.0167.2 4g505.3 542.9251.23.7 5.95.96.76.6 273.087.3 4p505.7 586.6251.27.3 14.213.613.4 15.7441.595.5 4sr505.6 1009.3296.43.3 7.46.26.0 7.9424.3111.6 5cq304.6 1087.3177.72.0 5.25.53.3 6.0560.063.5 5g303.7 673.9182.51.8 4.14.03.6 5.7319.841.8 5p304.5 762.8103.62.7 10.19.56.1 12.2580.344.1 5sr304.8 1115.4163.51.9 4.74.53.6 6.3667.763.2 6cq183.5 1205.963.41.0 2.73.71.5 3.1630.226.6 6g182.0 731.655.20.9 1.82.71.9 2.4346.318.1 6p183.1 929.831.11.3 3.85.42.5 5.2658.319.9 6sr182.3 1369.759.90.9 2.93.01.5 3.4778.434.4 7cq121.7 1658.331.70.6 2.03.41.2 2.9728.512.6

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Localsearchheuristicsforthemultidimensionalassignmentproblem225 Table3 ( Continued ) Inst.Runningtime,ms 2-opt3-optv-opt1DV2DV s DV1DV22DV2s DV3s DVv 7g121.4 1048.328.20.6 1.32.41.1 2.0555.411.1 7p122.1 1324.417.50.8 2.46.41.8 3.91088.914.6 7sr121.9 1622.440.90.7 2.03.51.1 2.5965.611.0 8cq81.1 2112.313.30.5 1.52.81.0 2.01909.58.5 8g81.0 1675.515.60.4 0.82.10.8 1.2728.57.2 8p81.7 2051.47.60.4 1.23.10.9 1.81492.97.9 8sr81.3 2439.916.40.3 1.32.91.0 1.81252.78.1 Allavg.5.9 1563.1262.09.2 11.611.911.7 13.8866.8103.4 Cliqueavg.6.5 1883.7330.68.3 9.99.69.0 11.8966.7129.4 Geom.avg.5.4 1150.2218.85.6 7.07.18.58.5 587.161.1 Productavg.5.4 1300.8209.715.8 20.221.320.1 23.4998.7106.8 SRavg.6.3 1917.8289.17.2 9.39.79.1 11.5914.9116.3 3-APavg.19.5 3173.81043.346.6 48.747.053.8 57.01483.6406.8 4-APavg.5.7 752.7276.94.5 9.08.28.0 10.2401.7115.4 5-APavg.4.4 909.9156.82.1 6.05.94.2 7.5532.053.2 6-APavg.2.7 1059.252.41.0 2.83.71.9 3.5603.324.8 7-APavg.1.7 1413.429.60.7 1.93.91.3 2.8834.612.3 8-APavg.1.2 2069.713.20.4 1.22.70.9 1.71345.97.9 3.Repeat: (a)Savethebest c assignmentsfrom P into C1,C2,...,Ccsuchthat w(Ci) w(Ci + 1) ; (b)If w(C1)
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226D.Karapetyan,G.GutinTable4 LocalsearchheuristicsstartedfromGreedy Inst.Solutionerror,%Runningtimes,ms 2-opt1DV2DV s DV1DV22DV2s DV3s DVv2-opt1DV2DV s DV1DV22DV2s DV3s DVv 3gp1004.33.43.43.43.43.43.3 3.40.04 0.040.040.040.050.050.360.09 3r15016.71.21.21.21.21.20.80.7 0.02 0.020.020.030.020.030.110.05 4gp304.53.73.63.63.63.62.6 3.60.040.03 0.040.040.040.040.110.05 4r8015.87.96.16.17.96.12.61.5 0.01 0.020.020.020.020.020.210.08 5gp125.46.34.54.55.34.51.8 4.50.01 0.010.010.010.010.010.030.01 5r4018.519.813.513.515.013.52.3 3.50.01 0.010.010.010.010.010.180.04 6gp84.18.95.54.36.04.52.4 3.80.01 0.010.010.010.010.010.040.01 6r2225.944.128.626.426.827.32.7 8.60.01 0.010.010.010.010.010.210.02 7gp55.511.37.05.96.65.93.5 5.10.000.00 0.000.000.000.000.040.00 7r1437.988.655.733.651.444.30.0 15.00.000.00 0.000.000.000.000.140.01 8gp44.211.55.23.64.23.63.1 3.60.00 0.000.000.000.000.000.070.00 8r940.0158.9107.854.465.665.60.0 30.00.000.00 0.000.000.000.000.130.01 Allavg.15.230.520.213.416.415.32.1 6.90.01 0.010.010.010.010.010.140.03 GPavg.4.77.54.94.24.84.32.8 4.00.02 0.020.020.020.020.020.110.03 Rand.avg.25.853.435.522.528.026.31.4 9.90.01 0.010.010.010.010.010.160.03 3-APavg.10.52.32.32.32.32.32.12.0 0.03 0.030.030.040.040.040.240.07 4-APavg.10.15.84.94.95.74.92.62.5 0.02 0.030.030.030.030.030.160.06 5-APavg.12.013.09.09.010.19.02.0 4.00.01 0.010.010.010.010.010.110.02 6-APavg.15.026.517.115.316.415.92.6 6.20.01 0.010.010.010.010.010.130.01 7-APavg.21.749.931.419.729.025.11.8 10.00.000.00 0.000.000.000.000.090.01 8-APavg.22.185.256.529.034.934.61.6 16.80.000.00 0.000.000.000.000.100.01

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Localsearchheuristicsforthemultidimensionalassignmentproblem227Table4 ( Continued ) Inst.Solutionerror,%Runningtimes,ms 2-opt1DV2DV s DV1DV22DV2s DV3s DVv2-opt1DV2DV s DV1DV22DV2s DV3s DVv 3cq15026.88.18.18.18.18.18.08.0 0.07 0.070.070.070.080.081.180.26 3g1500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.07 0.070.070.070.080.081.090.22 3p1500.20.00.00.00.00.00.0 0.00.61 0.650.660.660.660.661.920.96 3sr15029.99.89.89.89.89.89.49.1 0.07 0.070.070.070.080.091.490.26 4cq5019.011.611.611.611.611.611.3 11.60.160.16 0.160.160.160.160.440.21 4g500.0 0.30.0 0.0 0.0 0.0 0.0 0.0 0.220.22 0.220.220.220.220.430.29 4p500.10.20.10.10.10.10.0 0.11.041.04 1.041.041.041.051.391.12 4sr5020.010.911.311.310.911.310.3 11.00.19 0.190.200.200.200.200.470.25 5cq3014.29.69.59.59.69.59.3 9.40.640.64 0.640.640.640.641.030.68 5g300.0 0.40.0 0.0 0.0 0.0 0.0 0.0 0.940.94 0.940.940.940.941.260.97 5p300.00.20.00.00.00.00.0 0.02.722.71 2.722.722.722.723.232.76 5sr3011.78.98.58.58.38.57.1 8.50.670.67 0.670.670.670.671.230.69 6cq189.88.27.87.57.97.86.3 7.30.430.43 0.430.430.430.431.080.44 6g180.0 0.50.0 0.0 0.0 0.0 0.0 0.0 0.560.56 0.560.570.560.570.900.58 6p180.00.20.00.00.00.00.0 0.01.101.10 1.101.101.101.101.691.12 6sr189.78.68.28.28.58.26.5 7.80.420.42 0.420.420.420.421.150.44 7cq127.15.75.05.15.15.04.0 4.91.041.04 1.041.041.041.042.201.05 7g120.0 0.50.10.0 0.0 0.0 0.0 0.0 1.221.22 1.221.221.221.221.771.23 7p120.00.40.00.00.00.00.0 0.01.881.87 1.871.881.871.882.901.89 7sr126.55.75.15.25.65.14.0 5.00.980.98 0.980.980.980.982.150.99 8cq84.74.13.12.83.72.72.2 2.60.470.47 0.470.470.470.471.970.47 8g80.0 0.70.00.0 0.0 0.0 0.0 0.0 0.570.57 0.570.570.570.571.380.58

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228D.Karapetyan,G.GutinTable4 ( Continued ) Inst.Solutionerror,%Runningtimes,ms 2-opt1DV2DV s DV1DV22DV2s DV3s DVv2-opt1DV2DV s DV1DV22DV2s DV3s DVv 8p80.00.20.00.00.00.00.0 0.00.710.71 0.710.710.710.712.110.72 8sr83.23.72.82.62.62.52.1 2.40.470.47 0.470.480.470.481.720.48 Allavg.6.84.13.83.83.83.83.4 3.70.72 0.720.720.720.720.721.510.78 Cliqueavg.13.67.97.57.47.67.56.8 7.30.47 0.470.470.470.470.471.320.52 Geom.avg.0.0 0.40.00.0 0.0 0.0 0.0 0.0 0.600.60 0.600.600.600.601.140.65 Productavg.0.10.20.00.00.00.00.0 0.01.34 1.351.351.351.351.352.211.43 SRavg.13.57.97.67.67.67.66.6 7.30.47 0.470.470.470.470.471.370.52 3-APavg.14.24.54.54.54.54.54.44.3 0.20 0.220.220.220.220.231.420.43 4-APavg.9.85.75.75.75.65.75.4 5.70.400.40 0.410.410.400.410.680.47 5-APavg.6.54.84.54.54.54.54.1 4.51.241.24 1.241.241.241.241.691.27 6-APavg.4.94.44.03.94.14.03.2 3.80.630.63 0.630.630.630.631.210.64 7-APavg.3.43.12.62.62.72.52.0 2.51.281.28 1.281.281.281.282.261.29 8-APavg.2.02.21.51.41.61.31.1 1.30.560.55 0.560.560.560.561.800.56

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Localsearchheuristicsforthemultidimensionalassignmentproblem229Table5 ChainmetaheuristicstartedfromTrivial,GreedyandROM.5secondsgiven.1—2-opt,2—1DV,3—2DV,4— s DV,5—1DV2,6—2DV2,7— s DV3,8— s DVv Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3gp10015.31.8 1.8 1.8 1.81.82.82.55.31.71.7 1.7 1.81.82.92.39.81.9 1.9 1.9 1.91.92.62.3 3r15077.70.0 0.0 0.0 0.0 0.0 0.10.0 41.40.0 0.0 0.0 0.0 0.0 0.0 0.0 33.60.0 0.0 0.0 0.0 0.0 0.0 0.0 4gp307.01.61.11.11.81.10.8 1.46.31.90.80.8 1.80.90.8 1.42.21.70.90.91.71.00.8 1.4 4r8055.04.41.91.94.12.30.40.0 41.64.61.61.64.51.80.80.0 57.04.32.02.04.32.00.90.0 5gp121.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 5r4040.818.58.08.016.38.00.0 0.0 34.019.38.08.013.58.50.0 0.0 40.319.38.08.015.88.80.50.0 6gp82.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 6r2220.530.010.96.415.58.20.0 0.0 19.127.711.85.515.59.10.0 0.0 15.532.713.68.615.09.50.0 0.0 7gp53.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.1 3.53.53.53.53.1 3.53.53.5 3.93.93.93.93.93.93.9 7r142.933.611.42.16.43.60.0 0.0 3.633.610.72.15.72.10.0 0.0 4.335.77.90.72.13.60.0 0.0 8gp42.1 5.24.74.22.1 3.65.25.20.5 3.13.13.12.61.63.12.61.0 4.74.73.62.64.24.74.7 8r90.0 25.64.40.0 0.0 0.0 0.0 0.0 0.0 22.22.20.0 0.0 0.0 0.0 0.0 0.0 26.74.40.0 0.0 0.0 0.0 0.0 Allavg.19.110.74.32.84.63.01.41.4 13.210.14.02.54.42.71.21.1 14.311.24.32.84.33.21.41.3 GPavg.5.42.72.62.52.2 2.42.82.83.22.42.22.22.31.9 2.42.33.42.72.52.42.3 2.52.62.7 Rand.avg.32.818.76.13.17.03.70.10.0 23.317.95.72.96.53.60.10.0 25.119.86.03.26.24.00.20.0 3-APavg.46.50.9 0.9 0.9 0.90.91.41.223.30.90.9 0.9 0.90.91.41.121.70.9 0.9 0.9 1.01.01.31.1 4-APavg.31.03.01.51.53.01.70.6 0.723.93.31.21.23.11.30.80.7 29.63.01.41.43.01.50.80.7 5-APavg.21.110.04.84.88.94.80.8 0.8 17.810.44.84.87.55.00.8 0.8 20.910.44.84.88.65.11.00.8 6-APavg.11.416.26.74.48.95.31.2 1.2 10.715.17.13.98.95.71.2 1.2 8.917.68.05.58.76.01.2 1.2 7-APavg.3.418.77.73.05.23.72.0 2.0 3.318.57.12.84.62.61.8 1.8 3.919.85.92.33.03.72.0 2.0 8-APavg.1.0 15.44.62.11.0 1.82.62.60.3 12.72.71.61.30.81.61.30.5 15.74.61.81.32.12.32.3

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230D.Karapetyan,G.GutinTable5 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3cq15080.76.2 6.2 6.2 6.76.717.09.838.26.0 6.0 6.0 6.16.08.46.336.86.4 6.4 6.4 6.56.515.811.3 3g1500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3p1500.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 3sr15096.07.0 7.0 7.17.67.618.311.841.07.4 7.4 7.4 7.97.99.17.4 42.86.7 6.7 6.87.07.217.811.4 4cq5027.75.4 5.85.85.66.112.79.522.55.4 5.75.76.15.89.87.426.45.15.25.0 5.45.613.08.0 4g500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4p500.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 4sr5031.96.4 7.17.17.37.414.48.823.36.6 7.27.27.67.49.27.630.06.5 7.17.17.37.313.510.4 5cq3011.62.72.52.4 2.72.58.34.411.82.3 2.72.62.92.85.63.911.92.6 2.82.62.93.19.04.8 5g300.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5p300.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 5sr3015.34.14.13.8 5.14.210.56.613.54.24.0 4.0 4.94.27.46.014.94.3 4.74.54.64.79.85.9 6cq183.20.30.2 0.40.50.35.91.43.30.30.3 0.40.40.54.41.52.70.40.30.40.60.2 6.51.3 6g180.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6p180.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 6sr184.10.70.5 0.81.21.17.61.94.01.00.90.7 0.90.95.72.54.21.10.7 1.01.20.77.12.4 7cq120.50.0 0.00.0 0.0 0.03.80.30.40.0 0.00.00.0 0.02.70.40.40.0 0.00.10.00.04.30.2 7g120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7p120.00.00.00.00.0 0.00.00.00.00.00.00.00.00.00.0 0.00.0 0.00.00.0 0.0 0.00.00.0 7sr120.60.00.0 0.00.10.0 4.60.40.70.0 0.0 0.10.00.13.40.60.40.00.00.10.0 0.15.10.3 8cq80.0 0.0 0.0 0.0 0.0 0.0 2.00.00.0 0.0 0.0 0.0 0.0 0.0 2.50.00.0 0.0 0.0 0.0 0.0 0.0 1.90.0 8g80.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Localsearchheuristicsforthemultidimensionalassignmentproblem231Table5 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 8p80.00.00.00.00.0 0.0 0.00.00.0 0.00.00.0 0.0 0.0 0.00.00.00.00.00.00.0 0.0 0.00.0 8sr80.0 0.0 0.0 0.0 0.0 0.0 2.00.00.0 0.0 0.0 0.0 0.0 0.0 1.70.0 0.0 0.0 0.0 0.0 0.0 0.0 2.40.0 Allavg.11.31.4 1.41.41.51.54.52.36.61.4 1.41.41.51.52.91.87.11.4 1.41.41.51.54.42.3 Cliqueavg.20.62.4 2.42.52.62.68.34.212.72.3 2.42.52.62.55.63.213.02.42.42.4 2.62.68.44.3 Geom.avg.0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Productavg.0.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 SRavg.24.63.0 3.13.13.53.49.64.913.73.2 3.23.23.53.46.14.015.43.1 3.23.23.43.39.35.0 3-APavg.44.23.3 3.3 3.33.63.68.85.419.83.4 3.4 3.4 3.53.54.43.419.93.3 3.3 3.33.43.48.45.7 4-APavg.14.93.0 3.23.23.23.46.84.611.43.0 3.23.23.43.34.83.814.12.9 3.13.03.23.26.64.6 5-APavg.6.71.71.71.6 1.91.74.72.76.31.6 1.71.62.01.83.22.56.71.8 1.91.81.91.94.72.7 6-APavg.1.80.30.2 0.30.40.33.40.81.80.30.30.3 0.30.32.51.01.70.40.30.40.50.2 3.40.9 7-APavg.0.30.00.0 0.00.00.02.10.20.30.00.0 0.00.00.01.50.20.20.00.00.00.0 0.02.40.1 8-APavg.0.00.00.00.00.0 0.0 1.00.00.0 0.00.00.0 0.0 0.0 1.10.00.00.00.00.00.0 0.0 1.10.0

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232D.Karapetyan,G.GutinTable6 ChainmetaheuristicstartedfromTrivial,GreedyandROM.10secondsgiven.1—2-opt,2—1DV,3—2DV,4— s DV,5—1DV2,6—2DV2,7— s DV3,8— s DVv Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3gp10015.11.6 1.6 1.6 1.71.72.32.25.31.61.61.6 1.61.62.52.19.81.6 1.6 1.6 1.81.72.22.1 3r15075.30.0 0.0 0.0 0.0 0.0 0.0 0.0 41.40.0 0.0 0.0 0.0 0.0 0.0 0.0 33.60.0 0.0 0.0 0.0 0.0 0.0 0.0 4gp306.51.40.80.81.31.00.7 1.36.21.70.80.81.50.80.7 1.12.21.40.80.81.40.80.7 1.2 4r8052.13.91.11.03.61.10.10.0 41.43.91.01.04.31.10.40.0 55.04.01.11.13.41.30.40.0 5gp121.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 5r4036.516.36.55.813.06.80.0 0.0 32.318.87.07.013.07.00.0 0.0 36.816.56.86.813.87.30.0 0.0 6gp82.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 6r2216.827.79.15.012.37.70.0 0.0 15.526.811.44.513.28.20.0 0.0 14.130.010.55.912.38.60.0 0.0 7gp53.5 3.93.93.93.93.93.93.93.1 3.53.53.53.53.1 3.53.52.7 3.93.93.93.93.53.93.9 7r141.429.37.11.42.92.90.0 0.0 0.731.46.40.74.30.0 0.0 0.0 2.929.35.70.70.0 2.10.0 0.0 8gp41.65.24.73.61.0 2.15.23.60.5 3.12.12.61.61.62.62.61.0 4.74.73.61.0 2.14.24.2 8r90.0 23.31.10.0 0.0 0.0 0.0 0.0 0.0 15.61.10.0 0.0 0.0 0.0 0.0 0.0 22.24.40.0 0.0 0.0 0.0 0.0 Allavg.17.79.73.32.33.62.61.31.2 12.59.23.22.13.92.31.11.1 13.59.83.62.43.52.61.31.3 GPavg.5.12.72.52.32.0 2.12.72.53.22.32.02.12.01.8 2.22.23.32.62.52.32.02.0 2.52.5 Rand.avg.30.416.74.22.25.33.10.00.0 21.916.14.52.25.82.70.10.0 23.717.04.72.44.93.20.10.0 3-APavg.45.20.8 0.8 0.8 0.90.91.11.123.30.80.80.8 0.80.81.31.021.70.8 0.8 0.8 0.90.91.11.1 4-APavg.29.32.71.00.92.51.00.4 0.723.82.80.90.92.90.90.5 0.628.62.71.01.02.41.00.5 0.6 5-APavg.19.08.94.03.67.34.10.8 0.8 16.910.14.34.37.34.30.8 0.8 19.19.04.14.17.64.40.8 0.8 6-APavg.9.615.15.73.77.35.11.2 1.2 8.914.66.93.57.85.31.2 1.2 8.216.26.44.27.35.51.2 1.2

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Localsearchheuristicsforthemultidimensionalassignmentproblem233Table6 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 7-APavg.2.516.65.52.73.43.42.0 2.0 1.917.55.02.13.91.6 1.81.82.816.64.82.32.0 2.82.0 2.0 8-APavg.0.814.32.91.80.5 1.02.61.80.3 9.31.61.30.80.81.31.30.5 13.54.61.80.5 1.02.12.1 3cq15079.85.4 5.4 5.4 5.95.913.38.338.25.6 5.6 5.6 5.75.87.96.136.86.15.9 5.9 6.36.212.78.2 3g1500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3p1500.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 3sr15093.96.3 6.3 6.3 6.76.715.810.241.06.2 6.2 6.2 6.66.68.37.242.86.66.5 6.5 6.76.714.59.1 4cq5026.25.05.04.9 5.25.39.96.522.44.9 5.35.25.45.58.97.025.54.6 4.84.85.24.911.37.1 4g500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4p500.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 4sr5030.85.8 6.66.66.56.911.28.223.36.4 6.56.56.76.68.96.729.36.26.1 6.1 6.96.711.39.2 5cq3010.92.21.9 2.02.02.16.94.211.01.9 2.22.22.32.55.13.411.42.42.3 2.32.42.47.33.7 5g300.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5p300.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 5sr3013.83.73.53.2 3.93.58.95.012.23.93.5 3.5 4.03.76.34.914.04.04.04.03.8 4.28.64.8 6cq182.50.20.1 0.30.40.24.10.82.70.30.20.0 0.20.43.50.82.30.20.1 0.20.20.24.81.0 6g180.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6p180.00.00.00.0 0.00.0 0.0 0.00.00.00.00.00.00.0 0.0 0.00.00.00.00.00.00.00.0 0.0 6sr183.40.40.40.4 0.70.85.11.03.40.70.70.3 0.60.64.82.03.80.5 0.60.50.80.65.51.8

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234D.Karapetyan,G.GutinTable6 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 7cq120.20.0 0.00.00.00.0 2.70.10.20.0 0.00.00.0 0.02.10.10.20.0 0.0 0.0 0.00.0 3.40.1 7g120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8cq80.0 0.0 0.0 0.0 0.0 0.0 1.40.00.0 0.0 0.0 0.0 0.0 0.0 1.40.00.0 0.0 0.0 0.0 0.0 0.0 1.40.0 8g80.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8p80.00.00.00.0 0.0 0.0 0.0 0.00.0 0.00.00.0 0.0 0.0 0.00.0 0.00.00.00.00.00.00.0 0.0 8sr80.0 0.0 0.0 0.0 0.0 0.0 1.40.0 0.0 0.0 0.0 0.0 0.0 0.0 1.40.0 0.0 0.0 0.0 0.0 0.0 0.0 2.10.0 Allavg.10.91.21.21.2 1.31.33.51.96.51.31.31.2 1.31.32.51.66.91.31.3 1.31.41.33.61.9 Cliqueavg.19.92.12.1 2.12.22.36.43.312.42.1 2.22.22.32.44.82.912.72.22.2 2.22.42.36.83.3 Geom.avg.0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Productavg.0.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.0 SRavg.23.72.7 2.82.73.03.07.74.113.42.92.82.8 3.02.95.33.515.02.92.92.9 3.03.07.74.2 3-APavg.43.42.9 2.9 2.9 3.13.17.34.619.83.0 3.0 3.0 3.13.14.03.319.93.23.1 3.13.23.26.84.3 4-APavg.14.22.7 2.92.92.93.15.33.711.42.8 2.92.93.03.04.43.413.72.7 2.72.73.02.95.74.1 5-APavg.6.21.51.31.3 1.51.43.92.35.81.51.41.4 1.61.62.82.16.41.61.61.6 1.61.74.02.1 6-APavg.1.50.10.1 0.20.30.32.30.51.50.20.20.1 0.20.32.10.71.50.20.20.2 0.30.22.60.7 7-APavg.0.10.00.00.00.00.0 1.50.10.20.00.0 0.00.00.01.10.10.10.00.00.00.00.0 1.90.1 8-APavg.0.00.00.00.0 0.0 0.0 0.70.00.0 0.00.00.0 0.0 0.0 0.70.00.00.00.00.00.0 0.0 0.90.0

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Localsearchheuristicsforthemultidimensionalassignmentproblem235thantry | D | > 1.FortheexplanationofthisphenomenonseeSect. 4.1 .Thesuccess of1DV2and2DV2forGPmeansexistenceofacertainstructureintheweightmatrices oftheseinstances. Onecanseethattheinitializationoftheassignmentisnotcrucialforthenalsolutionquality.However,usingGreedyinsteadofTrivialclearlyimprovesthesolutions foralmosteveryinstanceandlocalsearchheuristic.IncontrasttoGreedy,usingofROMusuallydoesnotimprovethesolutionquality.Itonlyinuences2-optwhichis theonlypurevectorwiselocalsearchinthecomparison(recallthatROMhasadimensionwisestructureand,thus,itisgoodincombinationwithvectorwiseheuristics). TheMultichainmetaheuristic,giventhesametime,obtainsbetterresultsthanChain. However,Multichainfailsforsomecombinationsofslowlocalsearchandhardinstancebecauseitisnotabletocompleteeventherstiterationinthegiventime.Chain,havingmucheasieriterations,donothavethisdisadvantage. Givingmoretimetoametaheuristicalsoimprovesthesolutionquality.Therefore, oneisabletoobtainhighqualitysolutionsusingmetaheuristicswithlargerunning times. 4.3Solverscomparison Tocomparealltheheuristicsandmetaheuristicsdiscussedinthispaperweproduced Tables 9 and 10 .Thesetablesindicatewhichheuristicsshouldbechosentosolve particularinstancesinthegiventimelimitations.Severalbestheuristicsareselected foreverycombinationoftheinstanceandthegiventime.Aheuristicisincludedin thetableifitwasabletosolvetheprobleminthegiventime,andifitssolutionquality isnotworsethan1 . 1 · w(Abest) anditsrunningtimeisnotlargerthan1 . 1 · tbest,where Abestisthebestassignmentproducedbytheconsideredheuristicsand tbestisthetime spenttoproduce Abest. ThefollowinginformationisprovidedforeverysolverinTables 9 and 10 : –Metaheuristictype( C forChain, MC forMultichainoremptyiftheexperimentis single). –Localsearchprocedure(2-opt,1DV,2DV,sDV,1DV2,2DV2,sDV3,sDVvoremptyif nolocalsearchwasappliedtotheinitialsolution). –Constructionheuristictheexperimentwasstartedwith(Gr,M-Roremptyifthe assignmentwasinitializedbyTrivial). –Thesolutionerrorinpercent. Thefollowingsolverswereincludedinthisexperiment: –ConstructionheuristicsGreedy,Max-RegretandROM. –Singleheuristics2-opt,1DV,2DV,sDV,1DV2,2DV2,sDV3andsDVvstartedfrom eitherTrivial,Greedy,Max-RegretorROM. –ChainandMultichainmetaheuristicsforeither2-opt,1DV,2DV,sDV,1DV2,2DV2,sDV3orsDVvandstartedfromeitherTrivial,Greedy,Max-RegretorROM.Themetaheuristicsproceededuntilthegiventimelimitations. Notethatforcertaininstancesweexcludeduplicatingsolvers(recallthatallthe dimensionwiseheuristicsperformequallyfor3-APaswellas2DVandsDVperform

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236D.Karapetyan,G.GutinTable7 MultichainmetaheuristicstartedfromTrivial,GreedyandROM.5secondsgiven.1—2-opt,2—1DV,3—2DV,4— s DV,5—1DV2,6—2DV2,7— s DV3,8— s DVv Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3gp10011.81.1 1.21.1 1.31.3156.92.05.31.21.21.1 1.41.35.62.29.71.2 1.2 1.2 1.31.39.82.0 3r15068.10.0 0.0 0.0 0.0 0.0 0.0 0.0 41.40.0 0.0 0.0 0.0 0.0 0.0 0.0 33.60.0 0.0 0.0 0.0 0.0 0.0 0.0 4gp303.00.90.7 0.7 0.80.7 0.7 1.13.10.80.7 0.7 1.00.7 0.7 1.02.10.7 0.7 0.7 0.7 0.7 0.81.0 4r8045.33.41.51.42.31.50.40.0 38.93.30.90.93.10.90.80.0 44.62.41.01.02.61.145.30.0 5gp121.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 5r4026.015.35.35.510.86.3516.80.0 26.814.55.35.810.05.00.0 0.0 28.315.36.86.811.07.0152.50.0 6gp82.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 6r228.620.06.84.57.75.90.0 0.0 9.120.97.72.77.34.50.0 0.0 6.420.58.65.59.55.030.00.0 7gp53.93.93.5 3.93.93.93.93.93.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.93.93.93.93.93.5 3.93.9 7r140.0 18.67.10.70.0 2.10.0 0.0 0.0 23.68.61.40.72.90.0 0.0 0.0 21.47.12.10.0 2.10.0 0.0 8gp42.1 5.24.74.22.1 4.23.64.21.6 3.62.62.62.11.6 3.63.10.5 4.75.24.22.64.74.73.6 8r90.0 14.42.20.0 0.0 0.0 0.0 0.0 0.0 17.82.20.0 0.0 0.0 0.0 0.0 0.0 14.42.20.0 0.0 0.0 0.0 0.0 Allavg.14.47.23.12.22.72.557.21.3 11.17.83.01.92.72.01.51.1 11.17.43.42.43.02.520.91.2 GPavg.4.12.52.32.32.0 2.328.22.52.92.22.02.02.01.8 2.92.33.42.42.52.32.1 2.43.82.4 Rand.avg.24.711.93.82.03.52.686.20.0 19.413.34.11.83.52.20.10.0 18.812.34.32.63.92.538.00.0 3-APavg.40.00.6 0.60.6 0.60.678.51.023.30.60.60.6 0.70.72.81.121.70.6 0.6 0.6 0.70.74.91.0 4-APavg.24.12.11.11.01.51.10.5 0.621.02.00.80.82.00.80.70.5 23.31.50.80.81.70.923.00.5 5-APavg.13.88.43.43.56.13.9259.10.8 14.18.03.43.65.83.30.8 0.8 14.98.44.14.16.34.377.00.8 6-APavg.5.511.24.63.55.14.21.2 1.2 5.711.75.12.64.83.51.2 1.2 4.411.45.53.96.03.716.21.2 7-APavg.2.0 11.25.32.32.0 3.02.0 2.0 1.8 13.56.02.52.13.21.8 1.8 2.0 12.75.53.02.0 2.82.0 2.0 8-APavg.1.0 9.83.52.11.0 2.11.82.10.8 10.72.41.31.00.8 1.81.60.3 9.63.72.11.32.32.31.8

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Localsearchheuristicsforthemultidimensionalassignmentproblem237Table7 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3cq15075.23.93.83.7 4.44.31219.1491.938.22.52.5 2.5 3.13.041.120.936.83.93.83.7 4.84.836.824.2 3g1500.0 0.0 0.0 0.0 0.0 0.0 865.30.0 0.0 0.0 0.0 0.0 0.0 0.0 19.50.0 0.0 0.0 0.0 0.0 0.0 0.0 2.90.0 3p1500.00.0 0.0 0.0 0.0 0.0 76.376.30.00.00.00.0 0.00.0215.3215.30.00.0 0.00.00.00.07.27.2 3sr15085.84.54.3 4.55.65.51249.7630.541.03.2 3.23.2 3.43.341.97.442.84.03.9 4.04.94.942.832.7 4cq5012.72.8 4.44.23.64.8283.69.710.61.9 2.92.92.53.513.96.613.53.83.2 3.2 3.83.628.49.6 4g500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4p500.00.00.00.00.0 0.0102.70.00.00.00.00.00.0 0.0484.20.00.00.00.00.00.0 0.08.30.0 4sr5016.43.0 3.43.53.13.9155.410.713.72.1 3.03.12.23.419.57.115.93.8 4.24.03.94.829.210.5 5cq303.42.11.41.4 3.01.6154.54.43.62.0 2.22.22.32.220.23.24.12.22.2 2.22.72.321.24.6 5g300.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.30.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5p300.00.00.0 0.0 0.00.0 137.20.00.00.00.0 0.0 0.00.0 1016.70.00.00.00.0 0.00.00.07.60.0 5sr306.02.4 3.43.42.73.6195.65.34.62.32.32.22.2 2.415.84.24.73.43.1 3.23.93.427.66.4 6cq182.82.12.01.4 1.81.8141.93.01.91.61.51.51.71.2 15.42.32.72.21.91.81.4 2.118.12.3 6g180.0 0.0 0.0 0.0 0.0 0.0 260.10.0 0.0 0.0 0.0 0.0 0.0 0.0 26.30.0 0.0 0.0 0.0 0.0 0.0 0.0 2.60.0 6p180.0 0.00.00.00.0 0.0 162.90.00.00.00.00.00.0 0.0 2117.70.00.0 0.00.00.00.0 0.07.80.0 6sr183.81.9 2.42.22.32.3120.73.53.02.02.12.0 2.12.113.22.63.92.31.82.12.71.7 19.13.0 7cq120.91.01.01.00.5 0.991.51.20.70.60.70.60.80.4 13.80.61.10.80.2 0.90.80.314.81.1 7g120.0 0.0 0.0 0.0 0.0 0.0 156.40.0 0.0 0.0 0.0 0.0 0.0 0.0 18.90.0 0.0 0.0 0.0 0.0 0.0 0.0 2.40.0 7p120.00.00.00.00.0 0.0346.10.00.00.00.00.00.0 0.03161.50.00.00.00.00.00.0 0.09.20.0 7sr121.11.40.91.21.01.177.70.9 1.41.01.11.10.6 0.79.41.21.81.10.7 0.81.01.014.91.2

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238D.Karapetyan,G.GutinTable7 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 8cq80.10.30.20.20.1 0.462.30.20.20.20.10.20.10.1 10.40.30.20.30.20.30.20.2 9.90.3 8g80.0 0.0 0.0 0.0 0.0 0.0 104.50.0 0.0 0.0 0.0 0.0 0.0 0.0 14.90.0 0.0 0.0 0.0 0.0 0.0 0.0 1.40.0 8p80.0 0.00.00.0 0.0 0.0 176.70.00.0 0.00.00.0 0.0 0.0 3604.60.0 0.00.00.00.00.0 0.09.00.0 8sr80.50.20.30.40.1 0.251.90.30.20.50.40.40.2 0.36.60.60.30.50.30.60.2 0.39.80.4 Allavg.8.71.1 1.11.11.21.3258.051.65.00.8 0.90.90.90.9454.311.35.31.21.1 1.11.31.213.84.3 Cliqueavg.15.92.02.12.0 2.22.3325.585.19.21.5 1.71.71.81.719.15.69.72.21.9 2.02.32.221.57.0 Geom.avg.0.0 0.0 0.0 0.0 0.0 0.0 231.10.0 0.0 0.0 0.0 0.0 0.0 0.0 13.60.0 0.0 0.0 0.0 0.0 0.0 0.0 1.50.0 Productavg.0.00.00.00.00.0 0.0167.012.70.00.00.00.00.0 0.01766.735.90.00.00.00.00.0 0.08.21.2 SRavg.18.92.3 2.52.52.52.8308.5108.510.71.82.02.01.8 2.017.73.911.62.52.3 2.42.82.723.99.0 3-APavg.40.22.12.0 2.12.52.4852.6299.719.81.41.41.4 1.61.679.460.919.92.01.9 1.92.42.422.416.0 4-APavg.7.31.5 2.01.91.72.2135.45.16.11.0 1.51.51.21.7129.43.47.41.91.81.8 1.92.116.55.0 5-APavg.2.31.2 1.21.21.41.3121.82.42.11.1 1.11.11.11.2263.81.82.21.41.3 1.41.61.414.12.8 6-APavg.1.61.01.10.9 1.01.0171.41.61.20.90.90.91.00.8 543.11.21.71.10.9 1.01.01.011.91.3 7-APavg.0.50.60.50.60.4 0.5167.90.50.50.40.40.40.40.3 800.90.50.70.50.2 0.40.50.310.30.6 8-APavg.0.20.10.10.10.0 0.298.90.10.10.20.10.20.1 0.1909.10.20.10.20.10.20.1 0.17.50.2

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Localsearchheuristicsforthemultidimensionalassignmentproblem239Table8 MultichainmetaheuristicstartedfromTrivial,GreedyandROM.10secondsgiven.1—2-opt,2—1DV,3—2DV,4— s DV,5—1DV2,6—2DV2,7— s DV3,8— s DVv Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3gp10011.21.0 1.0 1.0 1.11.12.51.75.31.0 1.0 1.0 1.11.12.81.89.71.00.9 1.01.21.12.31.7 3r15065.30.0 0.0 0.0 0.0 0.0 0.0 0.0 41.40.0 0.0 0.0 0.0 0.0 0.0 0.0 33.60.0 0.0 0.0 0.0 0.0 0.0 0.0 4gp302.90.7 0.7 0.7 0.7 0.7 0.7 1.02.60.80.7 0.7 0.7 0.7 0.7 0.92.10.7 0.7 0.7 0.7 0.7 0.7 0.8 4r8043.82.50.90.92.11.00.20.0 38.12.30.60.62.30.60.50.0 42.42.00.80.82.00.80.50.0 5gp121.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 5r4025.812.54.04.09.04.00.0 0.0 23.513.34.84.59.04.80.0 0.0 25.814.35.55.59.85.50.0 0.0 6gp82.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 6r226.418.25.92.76.45.00.0 0.0 6.418.26.42.35.02.70.0 0.0 5.917.36.83.65.54.50.0 0.0 7gp53.93.93.53.93.93.93.1 3.93.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.93.93.5 3.93.93.5 3.93.9 7r140.0 16.44.30.0 0.0 0.0 0.0 0.0 0.0 19.35.00.0 0.0 1.40.0 0.0 0.0 17.95.00.0 0.0 0.0 0.0 0.0 8gp40.0 5.24.73.61.62.63.64.21.0 3.12.62.11.0 1.63.62.10.0 4.75.23.61.63.64.22.6 8r90.0 13.30.0 0.0 0.0 0.0 0.0 0.0 0.0 13.30.0 0.0 0.0 0.0 0.0 0.0 0.0 10.01.10.0 0.0 0.0 0.0 0.0 Allavg.13.66.52.41.72.41.91.2 1.210.56.62.41.62.21.71.31.0 10.66.32.81.92.42.01.31.1 GPavg.3.72.42.32.21.9 2.02.32.42.72.12.01.91.7 1.82.42.03.32.42.42.21.9 2.12.52.2 Rand.avg.23.510.52.51.32.91.70.00.0 18.211.12.81.22.71.60.10.0 17.910.23.21.62.91.80.10.0 3-APavg.38.20.5 0.5 0.5 0.60.61.30.923.30.5 0.5 0.5 0.60.51.40.921.60.50.5 0.50.60.51.20.9 4-APavg.23.31.60.80.81.40.80.5 0.520.41.50.70.71.50.70.60.4 22.21.30.70.71.30.70.60.4 5-APavg.13.67.02.82.85.32.80.8 0.8 12.57.43.13.05.33.10.8 0.8 13.67.93.53.55.63.50.8 0.8 6-APavg.4.410.34.22.64.43.71.2 1.2 4.410.34.42.33.72.61.2 1.2 4.29.84.63.03.93.51.2 1.2 7-APavg.2.010.23.92.02.02.01.6 2.01.8 11.44.31.8 1.8 2.51.8 1.8 2.010.94.32.02.01.8 2.02.0 8-APavg.0.0 9.32.31.80.81.31.82.10.5 8.21.31.00.5 0.81.81.00.0 7.33.21.80.81.82.11.3

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240D.Karapetyan,G.GutinTable8 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 3cq15071.41.91.7 1.83.02.91219.18.838.21.3 1.3 1.3 2.02.041.16.036.82.42.2 2.33.02.936.810.1 3g1500.0 0.0 0.0 0.0 0.0 0.0 865.30.0 0.0 0.0 0.0 0.0 0.0 0.0 19.50.0 0.0 0.0 0.0 0.0 0.0 0.0 2.90.0 3p1500.00.0 0.0 0.0 0.00.076.30.00.00.0 0.0 0.0 0.0 0.0 215.30.00.00.0 0.0 0.0 0.00.0 7.20.0 3sr15082.13.03.12.9 4.03.91249.710.541.01.9 1.91.92.82.841.96.242.82.92.82.8 3.73.742.810.3 4cq5011.12.6 3.43.33.23.811.37.810.31.5 2.72.72.42.88.64.611.12.82.8 2.8 3.53.112.37.3 4g500.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4p500.00.00.00.00.0 0.00.00.00.00.00.00.00.0 0.00.00.00.00.00.0 0.00.0 0.0 0.00.0 4sr5014.62.0 2.72.62.63.212.98.112.52.12.32.32.0 2.89.15.014.73.63.43.43.3 3.512.67.8 5cq302.22.11.31.3 2.41.38.03.52.81.5 2.22.21.82.25.02.63.02.22.0 2.22.52.28.04.2 5g300.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5p300.00.00.00.00.0 0.00.0 0.00.00.00.00.00.0 0.0809.70.00.00.00.00.00.00.00.0 0.0 5sr304.62.2 3.33.32.63.49.64.33.61.91.9 2.02.02.06.63.23.53.12.9 2.9 3.62.9 13.74.6 6cq182.82.12.01.4 1.81.75.62.41.91.61.51.51.51.2 4.21.92.02.21.91.81.4 1.95.82.3 6g180.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6p180.0 0.00.00.0 0.0 0.0 0.0 0.00.0 0.00.00.00.00.01038.30.00.00.00.00.00.00.00.0 0.0 6sr183.51.8 2.42.02.32.36.53.23.02.02.11.9 2.12.15.32.33.92.31.82.12.61.7 6.72.6 7cq120.91.01.01.00.5 0.938.61.20.70.60.70.60.80.4 7.70.41.00.80.2 0.80.80.39.20.9 7g120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Localsearchheuristicsforthemultidimensionalassignmentproblem241Table8 ( Continued ) Inst.Solutionerror,% TrivialGreedyROM 123456781234567812345678 7p120.00.00.00.00.0 0.0346.10.00.00.00.00.00.0 0.03161.50.00.00.00.00.00.0 0.09.20.0 7sr121.01.40.91.11.01.162.40.9 1.01.01.11.10.6 0.79.41.21.61.10.7 0.81.00.913.01.1 8cq80.10.20.20.20.0 0.462.30.20.20.20.10.20.10.1 10.40.30.20.30.20.30.20.2 9.90.3 8g80.0 0.0 0.0 0.0 0.0 0.0 104.50.0 0.0 0.0 0.0 0.0 0.0 0.0 14.90.0 0.0 0.0 0.0 0.0 0.0 0.0 1.40.0 8p80.00.00.00.00.0 0.0176.70.00.0 0.00.00.0 0.0 0.0 3604.60.0 0.00.00.00.00.0 0.09.00.0 8sr80.50.20.30.40.1 0.251.90.20.20.50.40.40.2 0.36.60.60.30.50.30.60.1 0.39.80.4 Allavg.8.10.9 0.90.91.01.0179.42.14.80.7 0.80.80.80.8375.81.45.01.00.9 0.91.11.08.82.2 Cliqueavg.14.81.71.61.5 1.81.8224.14.09.01.1 1.41.41.41.512.82.69.01.81.6 1.71.91.813.74.2 Geom.avg.0.0 0.0 0.0 0.0 0.0 0.0 161.60.0 0.0 0.0 0.0 0.0 0.0 0.0 5.70.0 0.0 0.0 0.0 0.0 0.0 0.0 0.70.0 Productavg.0.00.00.00.00.0 0.099.80.00.00.00.00.00.0 0.01471.60.00.00.00.00.00.00.0 4.20.0 SRavg.17.71.8 2.12.02.12.4232.24.510.21.6 1.61.61.61.813.23.111.12.22.0 2.12.42.216.44.5 3-APavg.38.41.21.21.2 1.71.7852.64.819.80.8 0.80.81.21.279.43.119.91.31.3 1.31.71.622.45.1 4-APavg.6.41.2 1.51.51.41.76.14.05.70.9 1.21.21.11.44.42.46.41.61.61.6 1.71.76.23.8 5-APavg.1.71.1 1.11.11.31.24.41.91.60.8 1.01.11.01.1205.31.51.61.31.2 1.31.51.35.42.2 6-APavg.1.61.01.10.9 1.01.03.01.41.20.90.90.90.90.8 262.01.01.51.10.91.01.00.9 3.11.2 7-APavg.0.50.60.50.50.4 0.5111.80.50.40.40.40.40.40.3 794.70.40.60.50.2 0.40.50.37.80.5 8-APavg.0.20.10.10.10.0 0.298.90.10.10.20.10.20.1 0.1909.10.20.10.20.10.20.1 0.17.50.2

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242D.Karapetyan,G.GutinTable9 Heuristicscomparisonfortheinstanceswithindependentweights Inst. < 10ms < 30ms < 100ms < 300ms < 1000ms 3r150– C C C s DV s DV 2DV2Gr1.4 1.5 1.5 C s DVv0.3 C C C C C C C s DV 2DV2s DVvs DVvs DV 2DV2s DVvGr R R R 0.0 0.0 0.0 0.0 0.0 0.0 0.0 (nobettersolutions) 4r80 C 1DV25.8 s DV 2DV2Gr Gr 6.1 6.1 s DVvGr1.5 C s DVvGr0.3 C C C s DVvs DVvs DVvGr R 0.0 0.0 0.0 5r40 1DV2Gr15.0 2DV s DV 2DV2Gr Gr Gr 13.5 13.5 13.5 C s DVv1.2 C s DVv0.0 (nobettersolutions) 6r22 C C 2DV s DV 46.4 47.3 2-optGr25.9 C s DVvGr1.4 C s DVvGr0.0 (nobettersolutions) 7r14 C 2-optGr28.6 C s DVvGr13.6 C s DVv1.4 C MC C s DVvs DVvs DVvGr 0.0 0.0 0.0 (nobettersolutions)

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Localsearchheuristicsforthemultidimensionalassignmentproblem243Table9 ( Continued ) Inst. < 10ms < 30ms < 100ms < 300ms < 1000ms 8r9 C C 2-opt 2-opt Gr 22.2 24.4 C s DVv12.2 C s DVv0.0 (nobettersolutions)(nobettersolutions) Total— C C C s DV 2DV2s DV Gr Gr 18.6 19.3 20.2 C s DVvGr4.8 C s DVvGr0.1 C C s DVvs DVvGr 0.0 0.0

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244D.Karapetyan,G.GutinTable10 Heuristicscomparisonfortheinstanceswithdecomposableweights Inst. < 100ms < 300ms < 1000ms < 3000ms < 10000ms 3cq150 s DVGr8.1 C C s DV 2DV2Gr Gr 7.8 7.8 MC MC s DV 2DV2Gr Gr 6.6 7.1 MC MC s DV 2DV2Gr Gr 3.1 3.4 MC s DVGr1.3 3sr150 C C s DV s DV 1DV22DV2Gr Gr Gr Gr 9.6 9.8 9.8 10.2 C C s DV 2DV2Gr Gr 8.4 8.4 MC s DVGr6.6 MC s DVGr3.5 MC s DVGr2.0 4cq50 C MC C 1DV 1DV 1DV29.7 10.0 10.3 MC MC 1DV 1DV Gr6.4 6.9 MC MC MC MC MC 1DV 1DV21DV s DV 1DV2Gr Gr R R 4.7 4.9 5.0 5.1 5.1 MC 1DVGr2.7 MC 1DVGr1.5 4sr50 C MC 1DV 1DV 11.7 12.2 MC MC 1DV 1DV Gr7.0 7.7 MC MC 1DV 1DV2Gr Gr 4.7 5.0 MC MC 1DV 1DV2Gr Gr 2.6 2.7 MC MC MC MC 1DV21DV 1DV 1DV Gr Gr M-R 2.0 2.0 2.1 2.1 5cq30 C MC 1DV 1DV 6.3 6.4 MC 1DV3.2 MC MC MC 2DV 1DV s DV 2.6 2.6 2.7 MC MC 2DV s DV 1.7 1.7 MC MC MC s DV 2DV 2DV21.3 1.3 1.3

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Localsearchheuristicsforthemultidimensionalassignmentproblem245Table10 ( Continued ) Inst. < 100ms < 300ms < 1000ms < 3000ms < 10000ms 5sr30 MC C 1DV 1DV 7.9 8.3 MC 1DV3.9 MC 1DV3.2 MC MC MC MC MC 1DV22DV s DV 1DV 2DV2Gr Gr Gr Gr 2.4 2.5 2.5 2.5 2.6 MC MC MC MC MC 2DV 1DV s DV 2DV21DV2Gr Gr Gr Gr Gr 1.9 1.9 2.0 2.0 2.0 6cq18 C 1DV2.1 C 1DV1.0 C 1DV0.7 C 2DVGr0.3 C s DVGr0.0 6sr18 MC C 1DV 1DV 3.8 3.8 MC C 1DV 1DV 2.1 2.1 C C 2DV 2DV2R 1.4 1.5 C 1DV0.8 C s DVGr0.3 7cq12 C 1DV0.7 C 1DV0.2 C 1DV20.1 C 1DV0.0 C C C C C C C C 1DV 2DV21DV 1DV21DV 2DV s DV 2DV2Gr Gr R R R R 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7sr12 C 1DV1.2 C C 1DV 1DV20.5 0.5 C 1DVR0.1 C 2DV0.0 (nobettersolutions)

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246D.Karapetyan,G.GutinTable10 ( Continued ) Inst. < 100ms < 300ms < 1000ms < 3000ms < 10000ms 8cq8 C 1DV0.0 C 1DV0.0 (nobettersolutions)(nobettersolutions)(nobettersolutions) 8sr8 C 1DV0.3 C C 1DV 2DV 0.0 0.0 C C C C C C C C C C C C 1DV 2DV 1DV22DV21DV 1DV22DV22-opt 1DV 2DV 1DV22DV2Gr Gr Gr R R R R R 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 (nobettersolutions)(nobettersolutions) Total C 1DV6.4 C C 1DV 2DV 4.5 5.0 MC MC C MC 1DV 2DV 2DV 1DVR 3.5 3.7 3.7 3.8 MC MC MC MC 1DV 2DV s DV 1DV2Gr Gr Gr Gr 1.9 2.1 2.1 2.1 MC 1DVGr1.3

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Localsearchheuristicsforthemultidimensionalassignmentproblem247equallyfor4-AP,seeSect. 2.1 ).ThecommonruleisthatweleavesDVratherthan2DVand2DVratherthan1DV.Forexample,ifthelistofsuccessfulsolversforsome 3-APinstancecontains C 1DVGr, C 2DVGrand C s DVGr,thenonly C s DVGr willbeincludedinthetable.Thisisalsoapplicabletothecombinedheuristics,e.g, having1DV2Rand2DV2Rfora3-APinstance,weincludeonly2DV2Rinthenal results. Thelastrowineverytableindicatestheheuristicswhicharethemostsuccessful onaverage,i.e.,theheuristicswhichcansolvealltheinstanceswiththebestaverage results. Singleconstructionheuristicsarenotpresentedinthetables;singlelocalsearch proceduresappearonlyforthesmallallowedtimeswhenallotherheuristicstake moretimetorun;themostofthebestsolversarethemetaheuristics.Multichainseems tobemoresuitablethanChainforlargerunningtimes;however,Multichaindoesnot appearfortheinstanceswithsmall n .Thisisprobablybecausethepoweroftheperturbationdegreeincreaseswiththedecreaseoftheinstancesize(notethat perturb (A) perturbsatleasttwovectorsinspiteof n ). ThemostsuccessfulheuristicsfortheassignmentinitializationareTrivialandGreedy;Trivialisusefulratherforsmallrunningtimes.Max-RegretandROMappear onlyafewtimesinthetables. Thesuccessofalocalsearchdependsontheinstancetype.Themostsuccessful localsearchheuristicfortheinstancewithindependentweightsisdenitelysDVv. ThesDVheuristicalsoappearsseveraltimesinTable 9 ,especiallyforthesmallrunningtimes.Fortheinstanceswithdecomposableweights,themostsuccessfularethe dimensionwiseheuristicsand,inparticular,1DV. 5Conclusions Severalneighborhoodsaregeneralizedanddiscussedinthispaper.Anefcientapproachofjoiningdifferentneighborhoodsissuccessfullyapplied;theyieldedheuristicsshowedthattheycombinethestrengthsoftheircomponents.Theexperimental evaluationforasetofinstancesofdifferenttypesshowthatthereareseveralsuperiorheuristicapproachessuitablefordifferentkindsofinstancesandrunningtimes. Twokindsofinstancesareselected:instanceswithindependentweightsandinstances withdecomposableweights.TherstonesarebettersolvablebyacombinedheuristicsDVv;thesecondonesarebettersolvableby1DV.Inbothcases,itisgoodtoinitialize theassignmentwiththeGreedyconstructionheuristicifthereisenoughtime;otherwiseoneshoulduseatrivialassignmentastheinitialone.Theresultscanalsobe signicantlyimprovedbyapplyingmetaheuristicapproachesforaslongaspossible. Thereby,itisshowninthepaperthatmetaheuristicsappliedtothefastheuristicsdominateslowheuristicsand,thus,furtherresearchofsomemoresophisticated metaheuristicssuchasmemeticalgorithmsisofinterest. ReferencesAiex,R.M.,Resende,M.G.C.,Pardalos,P.M.,Toraldo,G.:Graspwithpathrelinkingforthree-indexassignment.INFORMSJ.Comput. 17 (2),224–247(2005)

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248D.Karapetyan,G.Gutin Alon,N.,Spencer,J.:TheProbabilisticMethod,2ndedn.Wiley,NewYork(2000) Andrijich,S.M.,Caccetta,L.:Solvingthemultisensordataassociationproblem.NonlinearAnal. 47 (8), 5525–5536(2001) Balas,E.,Saltzman,M.J.:Analgorithmforthethree-indexassignmentproblem.Oper.Res. 39 (1),150– 161(1991) Bandelt,H.J.,Maas,A.,Spieksma,F.C.R.:Localsearchheuristicsformulti-indexassignmentproblems withdecomposablecosts.J.Oper.Res.Soc. 55 (7),694–704(2004) Bekker,H.,Braad,E.P.,Goldengorin,B.:Usingbipartiteandmultidimensionalmatchingtoselectthe rootsofasystemofpolynomialequations.In:ComputationalScienceandItsApplications—ICCSA 2005.LectureNotesComp.Sci.,vol.3483,pp.397–406.Springer,Berlin(2005) Burkard,R.E.,Çela,E.:Linearassignmentproblemsandextensions.In:Du,Z.,Pardalos,P.(eds.)HandbookofCombinatorialOptimization,pp.75–149.KluwerAcademic,Dordrecht(1999) Burkard,R.E.,Klinz,B.,Rudolf,R.:PerspectivesofMongepropertiesinoptimization.DiscreteAppl. Math. 70 (2),95–161(1996a) Burkard,R.E.,Rudolf,R.,Woeginger,G.J.:Three-dimensionalaxialassignmentproblemswithdecomposablecostcoefcients.TechnicalReport238,Graz(1996b) Clemons,W.K.,Grundel,D.A.,Jeffcoat,D.E.:In:TheoryandAlgorithmsforCooperativeSystems.ApplyingSimulatedAnnealingtotheMultidimensionalAssignmentProblem,pp.45–61.WorldScientic,Singapore(2004) Crama,Y.,Spieksma,F.C.R.:Approximationalgorithmsforthree-dimensionalassignmentproblemswith triangleinequalities.Eur.J.Oper.Res. 60 (3),273–279(1992) Frieze,A.M.,Yadegar,J.:Analgorithmforsolving3-dimensionalassignmentproblemswithapplication toschedulingateachingpractice.J.Oper.Res.Soc. 32 ,989–995(1981) Garey,M.R.,Johnson,D.S.:ComputersandIntractability:AGuidetotheTheoryofNP-Completeness. SeriesofBooksintheMathematicalSciences.Freeman,NewYork(1979) Grundel,D.A.,Pardalos,P.M.:Testproblemgeneratorforthemultidimensionalassignmentproblem. Comput.Optim.Appl. 30 (2),133–146(2005) Grundel,D.,Oliveira,C.,Pardalos,P.:Asymptoticpropertiesofrandommultidimensionalassignment problems.J.Optim.TheoryAppl. 122 (3),33–46(2004) Gutin,G.,Karapetyan,D.:Greedylikealgorithmsforthetravelingsalesmanproblemandmultidimensionalassignmentproblem.In:AdvancesinGreedyAlgorithms.I-Tech(2008) Gutin,G.,Karapetyan,D.:Aselectionofusefultheoreticaltoolsforthedesignandanalysisofoptimization heuristics.MemeticComput. 1 (1),25–34(2009) Gutin,G.,Goldengorin,B.,Huang,J.:Worstcaseanalysisofmax-regret,greedyandotherheuristicsfor multidimensionalassignmentandtravelingsalesmanproblems.J.Heuristics 14 (2),169–181(2008) Harris,J.M.,Hirst,J.L.,Mossinghoff,M.J.:CombinatoricsandGraphTheory.Mathematics(2008) Huang,G.,Lim,A.:Ahybridgeneticalgorithmforthethree-indexassignmentproblem.Eur.J.Oper.Res. 172 (1),249–257(2006) Isler,V.,Khanna,S.,Spletzer,J.,Taylor,C.J.:Targettrackingwithdistributedsensors:Thefocusofattentionproblem.Comput.Vis.ImageUnderst.J. 100 (1–2),225–247(2005).SpecialIssueonAttention andPerformanceinComputerVision Karapetyan,D.: http://www.cs.rhul.ac.uk/Research/ToC/publications/Karapetyan/ (2009) Karapetyan,D.,Gutin,G.,Goldengorin,B.:Empiricalevaluationofconstructionheuristicsforthemultidimensionalassignmentproblem.In:Chan,J.,Daykin,J.W.,Rahman,M.S.(eds.)LondonAlgorithmics2008:TheoryandPractice.TextsinAlgorithmics,pp.107–122.CollegeSci.Publ.,State College(2009) Krokhmal,P.,Grundel,D.,Pardalos,P.:Asymptoticbehavioroftheexpectedoptimalvalueofthemultidimensionalassignmentproblem.Math.Program. 109 (2–3),525–551(2007) Kuhn,H.W.:Thehungarianmethodfortheassignmentproblem.Nav.Res.Logist.Q. 2 ,83–97(1955) Kuroki,Y.,Matsui,T.:Anapproximationalgorithmformultidimensionalassignmentproblemsminimizingthesumofsquarederrors.DiscreteAppl.Math. 157 (9),2124–2135(2007) Murphey,R.,Pardalos,P.,Pitsoulis,L.:AGRASPforthemultitargetmultisensortrackingproblem.NetworksDiscreteMath.Theor.Comput.Sci.Ser. 40 ,277–302(1998) Oliveira,C.A.S.,Pardalos,P.M.:Randomizedparallelalgorithmsforthemultidimensionalassignment problem.Appl.Numer.Math. 49 ,117–133(2004) Pardalos,P.M.,Pitsoulis,L.S.:NonlinearAssignmentProblems.Springer,Berlin(2000) Pardalos,P.M.,Pitsoulis,L.S.:In:QuadraticandMultidimensionalAssignmentProblems.NonlinearOptimizationandApplications,vol.2,pp.235–276.KluwerAcademic,Dordrecht(2000)

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Localsearchheuristicsforthemultidimensionalassignmentproblem249 Pasiliao,E.L.,Pardalos,P.M.,Pitsoulis,L.S.:Branchandboundalgorithmsforthemultidimensionalassignmentproblem.Optim.MethodsSoftw. 20 (1),127–143(2005) Pierskalla,W.P.:Themultidimensionalassignmentproblem.Oper.Res. 16 ,422–431(1968) Pusztaszeri,J.,Rensing,P.,Liebling,Th.M.:Trackingelementaryparticlesneartheirprimaryvertex: acombinatorialapproach.J.Glob.Optim. 9 ,41–64(1996) Rardin,R.L.,Uzsoy,R.:Experimentalevaluationofheuristicoptimizationalgorithms:Atutorial. J.Heuristics 7 (3),261–304(2001) Robertson,A.J.:Asetofgreedyrandomizedadaptivelocalsearchprocedure(GRASP)implementations forthemultidimensionalassignmentproblem.Comput.Optim.Appl. 19 (2),145–164(2001) Spieksma,F.C.R.:In:MultiIndexAssignmentProblems:Complexity,Approximation,Applications.NonlinearAssignmentProblems,AlgorithmsandApplication,pp.1–12.KluwerAcademic,Norwell (2000) Spieksma,F.,Woeginger,G.:Geometricthree-dimensionalassignmentproblems.Eur.J.Oper.Res. 91 , 611–618(1996) Talbi,E.-G.:Metaheuristics:FromDesigntoImplementation.Wiley,NewYork(2009) Veenman,C.J.,Reinders,M.J.T.,Backer,E.:Establishingmotioncorrespondenceusingextendedtemporal scope.Artif.Intell. 145 (1–2),227–243(2003)



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INTEGERPROGRAMMINGMODELSFORSOLVINGCRITICALELEMENTDETECTIONANDDATAASSOCIATIONPROBLEMSByJOSEL.WALTEROSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014JoseL.Walteros

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DedicatedtomybabyNicolas

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ACKNOWLEDGMENTS IwouldliketotakethisopportunitytothankmycommitteechairDr.PanosM.PardalosforhissupportduringmyPh.D.studies.Iamextremelygratefulforhisenthusiasmandmotivation,bothproventobeindispensableforthenalcompletionofthisthesis.Hiscarefulguidancehasbeenoneofthefundamentalpillarsofmyearlyacademiccareer.IseeDr.PardalosnotonlyasagreatmentorbutalsoasadearfriendwhohashelpedmefullloneofthemostimportantacademicdreamsIeverhad.Itisagreathonorformetobecomepartofwhatwecordiallyrefertoasthe\PardalosAcademicFamily",whichisfullwithextraordinary,talented,andcaringmembers.IhopethatinthefutureIcanbewithmystudentsashelpfulandvaluableastheyhavebeenwithme.Iwishtoexpressmysinceregratitudetomywonderfulcommitteemembers:Dr.JosephGeunes,Dr.WilliamHager,andDr.VladimirBoginskifortheirtime,support,andhelpfulsuggestionsthatsignicantlyhelpedimprovingthisdissertation.Mydeepthanksgotomy\academicbrother"ChrysasVogiatzisforsharingalltheseamazingexperiencesduringourPh.D.studies.Icouldnotaskforabetterfriendandcolleague.IalsowanttoextendmygratitudetothecoauthorsofthepapersIhaveworkedonduringmyPh.D.studies|inadditiontotheonesIhavealreadymentioned:AustinBuchanan,Dr.SergiyButenko,Dr.FoadMahdaviPajouh,andDr.EduardoL.Pasiliao.Iconsidermyselfextremelyluckytohavetheopportunitytoreceiveguidanceandsupportfromsuchamazingindividuals.TomyfellowPh.D.studentsfromtheCenterforAppliedOptimization(CAO)andtheIndustrialandSystemsEngineeringDepartment(ISE)attheUniversityofFlorida,Iowethemmymostsinceregratitude.Thelastfouryearshavebeenanamazingexperiencethankstothem.IwanttogivemydeepestthankstomyM.Sc.advisorDr.AndresL.Medagliaforopeningthedooroftheacademicworldtome.Besidesofbeingoneofmyrolemodels, 4

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Dr.Medagliahasbeenatruementor.Hehasalwayssupportedmeandbelievedinme.WithouthisknowledgeandhelpIcouldnothaveaccomplishedallmygoalssofar.Forthat,Iwillalwaysbegrateful.Finally,Iwouldliketoclosetheacknowledgmentswiththemostimportantgroupofall:myfamily.Icannotexpresswithwordsallthelove,respect,andgratitudethatIhaveforbothofmyparentsMyriamSilvaandJoseL.WalterosM.,aswellasformysisterLauraWalteros.Iwillalwaysbeindebtwithyou.IamwhatIambecauseofyou.TotheloveofmylifeLilianaandNicolas,Imustsaythatthishasbeenagreatjourney.Iamgratefulthatitisnallycomingtoahappyending.Youbotharewhatkeepsmemotivated.ThisresearchwaspartiallysupportedbyDTRA,NSF,AFRLMathematicalModelingandOptimizationInstitute,andDURIPgrants.Thenancialsupportprovidedbytheseinstitutionisgreatlyappreciated. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTIONANDMOTIVATION ...................... 12 2CRITICALELEMENTDETECTIONPROBLEMS ................ 17 2.1ProblemDenition:TheBigPicture ..................... 17 2.2GraphStructuralPropertiesandCommonDeteriorationMeasures ..... 19 2.3DetectingCriticalVertices ........................... 21 2.3.1ComputationalComplexity ....................... 22 2.3.2Mixed-integerFormulations ....................... 23 2.3.3HeuristicsApproaches .......................... 26 2.3.4ApproximationAlgorithms ....................... 27 2.3.5Dynamicprogramming ......................... 28 2.4DetectingCriticalEdges ............................ 28 2.4.1Path-EnumerationApproaches ..................... 29 2.4.2Mixed-integerFormulations ....................... 29 2.4.3ApproximationAlgorithms ....................... 32 3IDENTIFYINGCRITICALEDGESFORBLOCKINGDOMINATINGSETS . 33 3.1ProblemDenition ............................... 33 3.1.1RelatedProblems ............................ 34 3.1.2ApplicationsinWirelessNetworksandRelatedAreas ........ 35 3.2ComputationalComplexity ........................... 37 3.3FormulationsfortheUnweightedCase .................... 38 3.3.1InitialFormulation ........................... 38 3.3.2ProjectedFormulation ......................... 41 3.4FormulationsfortheWeightedCase ...................... 45 3.4.1ProjectedFormulationfortheWeightedcase ............. 46 3.4.2AGeneralizationoftheProjectedFormulation ............ 46 3.5PolyhedralAnalysisoftheProposedFormulation .............. 53 3.6RowGenerationBasedExactAlgorithm ................... 59 3.7NumericalResults ................................ 60 3.7.1ResultsfortheUniformRandomGraph(URG)Instances ...... 62 3.7.2ResultsforthePower-LawInstances .................. 64 6

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4IDENTIFYINGCRITICALCLIQUESINSPARSEGRAPHS .......... 68 4.1ProblemDenition ............................... 68 4.2ComputationalComplexity ........................... 70 4.3AggregatedFormulations ............................ 73 4.4DecompositionApproachforSolvingtheCCP ................ 76 4.4.1ConstructingCliquePartitions ..................... 77 4.4.2CliqueCollapsing ............................ 78 4.4.3CNPGeneralizationforSolvingtheCCP ............... 78 4.5DisaggregatedFormulations .......................... 80 4.5.1NewCandidateCliquesGeneration .................. 83 4.5.2UsingtheDualInformationtoImprovetheSearchforNewColumns 85 4.5.3UpdatingtheSetofPathConstraints ................. 87 4.5.4BranchingRule ............................. 88 4.6ComputationalExperiments .......................... 89 5MULTIDIMENSIONALASSIGNMENTFORMULATIONSFORSOLVINGDATAASSOCIATIONPROBLEMS ........................ 94 5.1ProblemDenition ............................... 94 5.1.1AssignmentCosts ............................ 95 5.1.2PreviousWorks ............................. 96 5.1.3DataAssociationApplicationsfortheMSAP ............. 98 5.2AggregatedFormulationsforSolvingtheMSAP ............... 101 5.2.1ContinuousNonlinearFormulation ................... 102 5.2.2Linearization ............................... 109 5.2.3ValidInequalities ............................ 110 5.3ASetPartitioningFormulationforSolvingtheMSAP ............ 116 5.3.1GeneratingtheInitialSetofColumns ................. 117 5.3.2OptimalityConditionsandNewCandidateStarsGeneration .... 120 5.3.3BranchingRule ............................. 122 5.3.4StarGenerationasaShortestPathProblemwithSideConstraints . 123 5.3.5DynamicProgrammingAlgorithmforFindingCandidateStars ... 126 5.3.6StabilizationTechniques ........................ 130 5.4ComputationalExperiments .......................... 131 6CONCLUDINGREMARKS ............................. 136 APPENDIXPROOFOFCLAIM 1 ............................ 139 REFERENCES ....................................... 141 BIOGRAPHICALSKETCH ................................ 150 7

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LISTOFTABLES Table page 3-1StatisticsfortheURGinstances. .......................... 63 3-2StatisticsfortheURGinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . ................ 64 3-3Statisticsfortheunweightedpower-lawinstances. ................. 65 3-4Statisticsfortheunweightedpower-lawinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . ...... 66 3-5Statisticsfortheweightedpower-lawinstances. .................. 66 3-6Statisticsfortheweightedpower-lawinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . ...... 67 4-1Computationaltimesandoptimalitygapsforthe30-and40-vertexinstances .. 90 4-2Computationaltimesandoptimalitygapsforthe60,70-and100-vertexinstances 91 5-1Computationaltimes,optimalitygapsandthenumberofsolvedinstances. ... 132 5-2Numberofnodesandcutsgenerated. ........................ 134 5-3StatisticsoftheSPformulation. .......................... 135 8

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LISTOFFIGURES Figure page 2-1Optimalsolutionsfordierentstructuralproperties ................ 22 2-2Suboptimalsolutionfoundbygreedyapproachesvstheoptimalsolution .... 26 3-1ComparisonbetweentheoptimalsolutionoftheEBDPandthesolutionobtainedbyconsecutivecalculationsofthebondagenumberinagraph .......... 36 3-2ExampleofdierentwhipsfordominatingsetD=f1;2g ............. 42 4-1Examplefora9-nodegraph ............................. 69 4-2Exampleoftheproposedtransformation ...................... 72 4-3Exampleoftheclique-collapsealgorithm ...................... 79 5-1Exampleofthreeestimatedlocations ........................ 100 5-2Exampleofa3-sensor2-targettrackingproblem .................. 102 5-3Avalidstarassignmentforagraphwithn=4andm=3 ............ 103 5-4Exampleofafractionalsolution ........................... 108 5-5ExampleofafractionalsolutionofthelinearrelaxationofMIPa ........ 114 5-6CyclecutexampleforProposition 5.4 ........................ 114 5-7TripletcutexampleforProposition 5.5 ....................... 115 5-8Subproblemnetwork ................................. 125 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyINTEGERPROGRAMMINGMODELSFORSOLVINGCRITICALELEMENTDETECTIONANDDATAASSOCIATIONPROBLEMSByJoseL.WalterosAugust2014Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineeringInthisdissertationwestudythreeproblemsthatariseinthecontextsofcriticalelementdetectionanddataassociation.Theprincipalmotivationandcommoninterestbehindtheseproblemscomesfromtheirapplicabilityinseveralareas,includingrobustnetworkdesign,evacuationplanning,immunizationstrategies,andmulti-sensormulti-targettracking;aswellasfromtheinherentdicultyofndingpracticalandcomputationallyecientapproachestosolvethem.Webeginbypresentingageneraloverviewofthecriticalelementdetectionproblems,alongwithadetaileddescriptionofseveralvariationscommonlyfoundintheliterature.Wealsosurveysomeoftherecentadvancesandsolutiontechniquesfortheseproblems,suchasheuristicalgorithms,mathematicalprogrammingapproaches,approximatedalgorithms,anddynamicprogrammingschemes.Wecontinuebypresentinganexactapproachforsolvingacriticaledgedetectionproblemthatinvolvesremovingthesmallestsubsetofedgesofagraphsothattheweightofalldominatingsetsintheremaininggraphisboundedbelowbyagiventhreshold.WeshowthatthedecisionversionofthisproblemisNP-hard;wepresentananalyticallowerboundforthecardinalityofanoptimalsolutiontotheproblem;weproposeamixed-integerformulationanditscorrespondingprojectionontothespaceoftheedge-deletionvariables;westudytheconvexhullofthefeasiblesolutionsofthisproblemandidentifyseveralfacet-inducinginequalitiesforthecorrespondingpolytope;andwedeveloptherstexactalgorithmforthisproblemthatsolvestheproposed 10

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formulationbyabranch-and-cutapproachintroducingnontrivialconstraintsinalazyfashion.Furthermore,weprovidethecomputationalresultsobtainedaftersolvingatest-bedofrandomlygeneratedinstancesandreal-lifepower-lawgraphsthatrangeinsizefrom25to62vertices.Wethenconsidertheproblemofdetectingcriticalcliquesofagraphwhosedeletionoptimallydeterioratestheconnectivityofthegivengraph.Werstintroduceamathematicalformulationasamixedintegerlinearprogram.Then,weproposeatwo-stagedecompositionstrategythatrstidentiesacandidatecliquepartitionandthenusesthispartitiontotransformandsolvetheproblemasageneralizedcriticalvertexdetectionproblem.Furthermore,wereformulatethisproblemasamixed-integerlinearprogramwithalargenumberofvariablesandconstraintsandsolveitviabranch,price,andcut.Wealsodesignanovelpreprocessingalgorithmforthecolumngenerationstagethatusesthedualinformationofthemasterproblemtoprunethesubproblemgraphinordertoproducenewcolumnseciently.Wetestourapproachonacollectionofrandomlygeneratedsparsegraphs,followingseveraldegreedistributions,andranginginsizefrom30to100vertices.Finally,wetackleavariantofthemultidimensionalassignmentproblemwithdecomposablecostsforsolvingdataassociationproblemsinwhichtheresultingoptimalassignmentisdescribedasasetofdisjointstars.Tosolvethisproblemwestudytwodierentformulations.First,weintroduceacontinuousnonlinearprogramanditslinearization,alongwithadditionalvalidinequalitiesthatimprovethelowerbounds.Second,wereformulatethisproblemasasetpartitioningprogramandsolveitviabranchandprice.Wetesttheseapproachesbysolvinginstancesrangingfromtripartiteto20-partitegraphsof4to30verticesperpartition.Thecomputationalresultsshowthatourapproachesareaviableoptiontosolvethisproblem. 11

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CHAPTER1INTRODUCTIONANDMOTIVATIONInrecentyears,therehasbeenadramaticincreaseintheuseofoptimizationtechniquesforsolvingproblemsthatariseinthecontextsofnetworkanalysisanddataassociation.Thisisingeneralaconsequenceoftheconstantdevelopmentofnewsolutiontechniques,theadvancementofcomputinginfrastructures,andtheecientgatheringofrelevantdata.Inthiswork,wetakeadvantageofsuchtechnologiesinordertoanalyzeandsolveseveralpracticalproblemsintheareasofcriticalelementdetectionanddataassociation.Intherstpartofthisdissertation,wedealwiththeproblemofdetectingsubsetsofelementsofagraphthatareimportantfortheintegrityofeither,thephysicalnetwork(e.g.,theelectricalnetworkofacountry),thetechnicalprocess(e.g.,thecross-docktransportationnetworkofasupplychain),orthesocialstructure(e.g.,aninternetsocialnetwork)thatisbeingrepresentedbythegivengraph.Dependingonthecontext,identifyingthevertices,edges,paths,clusters,cliques,etc.,thatareresponsibleforthegraph'scohesioncanbecrucialforstudyingmanyfundamentalproperties,suchasattacktolerance,robustness,andvulnerability;itcanalsohelpforclassifyingmembersbasedontheircentrality,prestige,andreputation;anditcanbeusetodeterminedominantclustersandpartitions.Fromthepointofviewofrobustnessandvulnerabilityanalysis,evaluatinghowwellagraphwillperformundercertaindisruptiveeventsplaysavitalroleinthedesignandoperationofsuchagraph.Todetectvulnerabilityissues,itisofparticularimportancetoanalyzehowwellconnectedagraphwillremainafteradisruptiveeventtakesplacedestroyingorimpairingasetofitselements.Themainstrategyistoidentifythesetofcriticalelementsthatmustbeprotectedorreinforcedinordertomitigatethenegativeimpacttheabsenceofsuchelementsmayproduceinthegraph.Applicationsofthiskindaretypicallyfoundinhomelandsecurity Grubesicetal. ( 2008 ); Houcketal. ( 2004 ), 12

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evacuationplanning MatisziwandMurray ( 2009 ),immunizationstrategies Taoetal. ( 2005 ),energy Salmeronetal. ( 2004 ),andtransportation Jeneliusetal. ( 2006 ).Fromthemember-classicationperspective,identifyingmemberswithahighreputationandinuentialpowerwithinasocialnetworkcouldbeofgreatimportancewhendesigningamarketingstrategy.Positioningaproduct,spreadingarumor,ordevelopingacampaignagainstdrugsandalcoholabusemayhaveagreatimpactoversocietyifthestrategyisproperlytargetedamongthemostinuentialandrecognizedmembersofacommunity.Therecentemergenceofsocialnetworks,suchasFacebook,Twitter,LinkedIn,etc.providescountlessapplicationsforproblemsofcritical-elementdetection.Furthermore,determiningdominantcliquesorclustersoverdierentindustriesandmarketsviacriticalcliquedetectionmaybecrucialforanalyzingmarketshareconcentration,debtconcentration,orspottingpossiblecollusiveactions.Thesecondpartofthisdissertationisinspiredbythecontextofdataassociation.Inparticular,weareinterestedinapplicationsforsolvingmulti-sensormulti-targettrackingproblemsthatinvolveassociatingaseriesofsensorobservationswithasetofdierenttargets.Generally,thiskindofproblemsareoftenformulatedasmultidimensionalassignmentproblems(MAP's).Therelationshipbetweenthemulti-sensormulti-targettrackingproblemsandtheMAPhasbeenstatedandstudiedbymanyauthorsincluding Bandeltetal. ( 2004 ); Chummunetal. ( 2001 ); Debetal. ( 1993 , 1997 ); Moreeld ( 1977 ); Murpheyetal. ( 1999 ); Poore ( 1994 ),and Pusztaszerietal. ( 1996 ),amongothers.Inthiswork,weareinterestedinavariationoftheMAPinwhichtheresultingoptimalassignmentisdescribedasasetofdisjointstars.Thereareseveralcontextswhereusingstarcostscanprovebenecialwhensolvingmulti-sensormulti-targettrackingproblems.Inparticular,whentheassignmentcostshavemetricproperties(i.e,nonnegativity,symmetry,andsubadditivity),itisinterestingtoseethat,insomecases,consideringallpairwisecostswithintheassignmentsisnotnecessarytoobtainavalidsolution.We 13

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providetwoexampleswithcoststhatsatisfythosepropertiesandshowthatthestarcostvariationisavaluabletooltosolvethoseproblems.Intherstexample,assumeweaimtoidentifyasetoflandminesthatareplantedonaeld.Tondthelocationofthemines,adroneissenttoyovertheeldemittingasignal.Oncethesignalreacheseachofthemines,itbouncesbackandisanalyzedbythesensorsofthedrone.Afterthedronehasownovertheeldanumberoftimesanditssensorshavecollectedthesetofdierentsignals(severalofthoseassociatedwitheachofthemines),itispossibletocalculateasetofestimatedlocationswheretheminescouldbelocated.TheideabehindsolvingaMAPistoassociatethelocationsthatareclosetoeachother,whichwouldhelppinpointtheactualpositionsofthemines.Inthiscontext,theassignmentcostsrepresenttheEuclideandistancesbetweentheestimatedlocationsand,sincethecostssatisfythetriangleinequality,notallthecostsneedtobeconsideredtoobtainavalidassociation.ThesecondexampleisacasewherethecostsdonotrepresentEuclideandistancesandhence,theconceptofthegeometriccenterdoesnothaveaproperinterpretation.Assumewehaveagroupofantennasthatareplacedtointerceptacollectionofencryptedmessagesthatwewanttodecode.Becauseoftheinterferenceandthenoiseoftheenvironment,whenthemessagesarereceivedbytheantennastheyaresomehowdisruptedandthusnotperfectfordecoding.Eachoftheantennascanpotentiallyreceiveadierentversionofeachofthemessages.Theideaistoidentifywhichofthereceivedversionsaretheonesthatmostcloselyresemblethecorrect(transmitted)onessotheycanbesentfordecoding.Here,sincethereisnowaytocomparethedisruptedversionswiththecorrectmessages,thedecisionofwhichversionsarenallysenttodecodeshouldbemadebyanalyzingthedissimilaritiesbetweenthereceivedmessages.Inmanycontextslikethis,thedecodingprocesscouldberatherdicult.Hence,itisdesiredtoselectonlyoneofthedisruptedversionsofeachofthemessages.Assumingthatthemessagesareencryptedinsomekindofalphabet(e.g.,binarycodes),usingthecentroidofthedisruptedversions|as 14

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isgenerallyusedbyothercostvariations|isnotavalidapproachhere.Inthisframework,theuseofthestarcostscouldbeofbenetbecausethemessagesassociatedwiththecentersofthestarscanbetheonesselectedfordecoding(representative).Forthisproblem,thecostscanbeassumedtobethenumberofdierentcharactersbetweenthemessages,thesumofhowfarapartinthealphabetthedierentcharactersofthemessagesare,oranydesiredcorrelationmetric.Theoutlineofthisdissertationisasfollows.InChapter 2 ,wepresentandglobaldescriptionofthecriticalelementdetectionproblems,includingacomprehensivedescriptionofseveralvariationsthatcanbefoundintheliterature.Wealsoprovideashortdescriptionofsomeoftherecentdevelopmentsandsolutionapproachesforsolvingtheseproblems.Weputourattentiononheuristicalgorithms,mathematicalprogrammingapproaches,approximatedalgorithms,anddynamicprogrammingschemes.InChapter 3 ,westudyacriticaledgedetectionproblemthatwenametheminimumedgeblockerdominatingsetproblem(EBDP).Theobjectiveofthisproblemistoremoveasubsetofedgesofminimumcardinalityfromagraphsuchthattheminimumweightofalldominatingsetsintheremaininggraphisboundedbelowbyagivenintegerr.WebeginthischapterbyprovingthatthedecisionversionofEBDPisNP-hardforanyxedintegerr1.Then,wecontinueformulatingtheEBDPasamixed-integerprogram,anddevisetheprojectionofsuchformulationontothespaceoftheedge-deletionvariables.Moreover,weidentifyfacet-inducinginequalitiesfortheconvexhulloffeasiblesolutionsofthisproblem,andsolvetheproposedformulationviabranchandcut.wenalizethechapterprovidingthecomputationalresultsobtainedaftersolvingatest-bedofrandomlygeneratedinstancesandreal-lifepower-lawgraphs.InChapter 4 ,weaddresstheproblemofdetectingcriticalcliquesofagraphwhosedeletionoptimallydeterioratestheconnectivityofthegivengraph.Webeginthechapterbyintroducingamathematicalformulationasamixedintegerlinearprogram.Then,wepresentaheuristictwo-stagedecompositionstrategythatrst,identiesacandidate 15

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cliquepartitionandthen,usesthispartitiontotranslateandsolvetheproblemasageneralizedcriticalvertexdetectionproblem.Moreover,wereformulatethisproblemasamixed-integerlinearprogramwithalargenumberofvariablesandconstraintsandsolveitviabranch,price,andcut.Wealsointroduceapreprocessingstrategyforthecolumngenerationstageofthealgorithmthatusesthedualinformationofthemasterproblemtoprunethesubproblemgraphinordertoproducenewcolumnseciently.Finally,inChapter 5 ,wepresentareformulationofthemultidimensionalassignmentproblemwithdecomposablecostsforsolvingdataassociationproblemswhereeachoftheassignmentsisassumedtobeastar.Weproposeacontinuousnonlinearformulationfortheproblemanditslinearizationintoamixedintegerprogram.Inadditiontothat,weproposeaseriesofvalidinequalitiestostrengthenthegivenformulation.Then,weimplementabranch-and-priceframeworktosolveasetpartitioningreformulationoftheproblem.Wecomparealltheapproachesandshowthecomputationalresults. 16

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CHAPTER2CRITICALELEMENTDETECTIONPROBLEMSIngeneral,criticalelementdetectionproblemsdealwiththeoptimalremovalofacollectionofelementsfromagivengraphinordertomaximizethedeteriorationofastructuralpropertyofsuchgraph.Dependingonthecontext,anelementremovalmaybecausedbymanydierentissues,suchasadversarialattacks,randomfailures,ornaturaldisasters.Forthisreason,theseproblemsareoftenusedfordesigningoptimalattacksthatinicttheworstpossibledamagetothegraphorfordevisingdefensivestrategiesthatecientlymitigatetheeectsofanypossibledisruption.Mostcriticalelementdetectionproblemslieintheboundariesofdierentresearchareas,includinggraphinterdiction( IsraeliandWood , 2002 ; Wollmer , 1964 ),robustnetworkdesign( Grotscheletal. , 1995 ),graphclusteringandpartitioning( GrotschelandWakabayashi , 1990 ; Shmoys , 1997 ),amongothers.Thus,itisoftencommontoobserveseveralsimilaritiesbetweensomeoftheseproblemsandproblemsinotherareasaswell.Inthischapterwepresentageneraloverviewofcriticalelementdetection,includingsomeoftheobjectivesandstructuralpropertiesthatarecommonlyused.Furthermore,wesurveysomeoftherecentadvancesandsolutiontechniques,suchasheuristicalgorithms,mathematicalprogrammingapproaches,approximatedalgorithms,anddynamicprogrammingschemes.Toavoiddiscrepancieswiththesources,wheneverwepresentamathematicalformulation,wetrytokeeptheoriginalnotationasfoundineachpaper.Therearefewcasesthough,whereweareforcedtointroduceslightmodicationstopreventambiguityandensureconsistency. 2.1ProblemDenition:TheBigPictureGivenasimplegraphG=(V;E),whereVisthesetofverticesandEisthesetofedges.LetTbethesetofelementsinGfromwhichthesetofcriticalelementswillbeselected.ThekindofelementsinTmayvarydependingthecontext,andwithit,thenameoftheparticularproblemthatisbeingsolved.Forinstance,ifTisdenedastheset 17

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ofverticesorthesetofedges,wehaveinstancesofthecriticalvertexorthecriticaledgedetectionproblems,respectively.Ontheotherhand,ifTiscomprisedofmorecomplexsubstructures,suchaspaths,cliques,orclusters,wehaveinstancesofthecriticalpathdetectionproblem,thecriticalcliquedetectionproblem,etc.Furthermore,thetopologyofthegraphmayvarywiththecontextaswell;itcanbedenedasatree,aplanargraph,aseries-parallelgraph,aforest,oramorecomplexgraph;itcanbedirectedorundirected;andthedegreeoftheverticesmayfollowdierentdistributions(e.g.,uniformandpower-law).Letctbeacostassociatedwiththedeletionofelementt2Tandletbbeadeletionbudget.Inthecriticalvertex(edge)context,wehavethatT=V(T=E)andthecostsctrepresentthevertex(edge)deletioncostscifori2V(cefore2E).Formorecomplexcases,suchasfordetectingcriticalcliques,TcontainsthesetofallcliquesinGandthedeletioncostctcanbedenedasthesumofthedeletioncostofalltheverticesandedgesofeachcliquet2T.Furthermore,insomesituations,thedeletioncostctisoftenassumedtobeoneforalltheelements.Therefore,insuchcasesbrepresentsanupperboundofthenumberofelementsthatcanbedeleted.Thislaterversionisfrequentlynamedthecardinalityversionoftheproblem.Ingeneral,mostcriticalelementdetectionproblemsaimtoidentifyasubsetofelementsinT(i.e.,thecriticalelements)whoseremovalmaximizesthedegradationofastructuralpropertyofG,whichismeasuredbyafunctionf:G!R,whilesatisfyingthebudgetconstraint.Alternatively,givenatargeteddeteriorationlevelrofthestructuralproperty(i.e.,dependingonthepropertyanincreaseordecreaseinf(G)),onemayasktominimizethetotaldisruptioncostrequiredtogeneratethedesireddeteriorationofthegraph.Therearetwocrucialfactorsthatmustbeconsideredbesidestheshapeofthegraphandthetypeofthecriticalelements.Thosearethestructuralpropertyofthegraphthatisbeingdeterioratedandthefunctionthatmeasuresthedeteriorationlevel.Severalpropertiesandmeasureshavebeenusedbydierentauthorsand,dependingontheonethatischosen,thecomplexityoftheproblem,thesolutionapproach,andtheoptimal 18

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resultsmayvarydramatically.Eventhoughtheprincipalobjectiveistodamagethegraph,dierentpropertiesmayyielddierentoptimalsolutions.Aswewillpointoutinsection 2.3 ,asimplemodicationintheobjectivefunctioncanleadtoasignicantlymoredicultproblem.Moreover,theselectionoftheappropriatedsolutiontechniquealsodependsonthesefactors.Someobjectivesareeasiertosolvewithsometechniquesthanwithothers.Wenowsummarizetheseproperties. 2.2GraphStructuralPropertiesandCommonDeteriorationMeasuresSeveralstructuralpropertiesandtheircorrespondingmeasurestoassessthedeteriorationoftheresidualgraphhavebeenusedbymanyauthorsovertheyears.Ingeneral,wecategorizethesemeasuresintothreedierentclassesdependingonthecontextoftheproblemthatisbeingsolved.ThemeasuresfromtherstclassaremainlyassociatedwithowproblemsoverG;inparticulartheshortestpath,themaximumow,ortheminimumcostowproblems.Thelogicbehindthemisthatagraphbecomesdeterioratedwhenitstartslosingitsabilitytosendowbetweenapredenedsetofvertexpairsorsimply,whentraversingthegraphbecomestooexpensive( CorleyandSha , 1982 ; GrubesicandMurray , 2006 ; Wollmer , 1964 ).Forthesecases,thecriticalelementsarethosewhosedeletionresultsinthemaximumincreaseoftheshortestpath,orconsequently,themaximumdecreaseoftheowcapacitybetweenthepredenedvertexpair.Thiskindofmeasuresarecommonlyusedinthecontextofnetworkinterdiction( Churchetal. , 2004 ; IsraeliandWood , 2002 ; LimandSmith , 2007 ; MatisziwandMurray , 2009 ; Wood , 1993 )whenidentifyingcriticaledges.Thesecondclassiscomprisedofmeasuresthatquantifyhowwell-connectedtheresidualgraphis.Amongthisclass,themostcommonmeasuresarethetotalnumberofconnectedvertexpairs( Addisetal. , 2013 ; Arulselvanetal. , 2009 , 2010 ; DiSummaetal. , 2011 ; DiSummaetal. , 2012 ; MyungandjoonKim , 2004 ; Shenetal. , 2013 ; Ventresca , 2012 ; VentrescaandAleman , 2014 ; Veremyevetal. , 2014a , b ),thetotalweightoftheconnectionsbetweenthevertexpairs( Arulselvanetal. , 2010 ; DiSummaetal. , 19

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2011 ; Veremyevetal. , 2014b ),thesizeofthelargestmaximally-connectedcomponent( Bodlaenderetal. , 2008 ; Oostenetal. , 2007 ; ShenandSmith , 2012 ; Shenetal. , 2012 ; Veremyevetal. , 2014b ),thetotalnumberofmaximally-connectedcomponents( Albertetal. , 2000 ; ShenandSmith , 2012 ; Shenetal. , 2012 , 2013 ; vanderZwaanetal. , 2011 ),andthegraphinformationentropy( Borgatti , 2006 ; Hewett , 2011 ; Ortiz-ArroyoandHussain , 2008 ; Veremyevetal. , 2014b ).Therstmeasureofthisclassismainlyanexplicitevaluationofhowreachabletheverticesofthegraphareintheabsenceofthecriticalelements.Itisgenerallyusedwhenthereisnoneedtoaccountforcapacitiesorcostsassociatedwiththepathsbetweenthevertices.Forexample,whenanalyzingthevulnerabilityofacity'sinfrastructureduringanevacuationeventtoidentifytheareasofthecitythatcanbetotallydisconnectedfrompredenedsafezones.Thesecondmeasurecanbeseenasaweightedversionoftherstonewheretheweightofeveryconnectedvertex-pairiseitheranarbitrarynumber,liketheimportanceoftheconnection,oramorecomplexvaluelikethecostoftheshortestpathbetweenthevertexpairintheresidualgraph.Thesizeofthelargestcomponentpropertycanbeusedtoachievearelativelymorehomogeneousdisconnection.Thatis,whenminimizingthesizeofthelargestcomponent,despiteapossiblesacriceinthetotalnumberofpairwiseconnections,wecanavoidhavinglargeconcentrationsofconnectionsintheresidualgraph.Apossibleapplicationcouldbetoidentifythekeymembersofancriminalorganizationthathavetobecapturedtomaximizethesegregationoftheremainingmembers.Finally,themeasuresofthethirdclasscorrespondtosizesortherelativeweightsofsometopologicalsubsetsofverticesoredgesinthegraph.Forinstance,theweightoftheminimumspanningtree( FredericksonandSolis-Oba , 1999 ),theweighteddistancetoacentralvertex,(i.e.,eitherthe1-medianorthe1-centervertex( Bazganetal. , 2010 ),thelargestweightofamatching( Zenklusen , 2010 ; Zenklusenetal. , 2009 ),themaximumweightofanindependentset( Bazganetal. , 2011 ),andtheminimumweightofavertexcover( Bazganetal. , 2011 ).Besidesofbeingreferredtoascriticalelement 20

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detectionproblems,itisoftencommontondstudiesthatnametheproblemsthatusethismeasuresasminimumvertex/edgeblocker(disruptor)problems.Someotheradditionalpropertiesthatcanbeincorporatedinthecriticalelementdetectioncontextmayinvolvethediameteroftheresidualcomponents(i.e.,theshortestpathbetweenthetwomostremoteverticesinthegraph),thedegreeoftheremainingvertices,thenumberofpathsbetweeneverypairofvertices,andthenumberofcommonneighborsbetweeneverypairoftheremainingvertices,amongothers.Foradditionalpropertiesthatareusedtomeasurevulnerabilityandconnectivitysee BorgattiandEverett ( 2006 )and Grubesicetal. ( 2008 ).Ingeneral,theselectionoftheadequatepropertyisfundamentalformakingthecorrectanalysis.Despitethefactthatallofthesevariationsaccountforadeteriorationlevelofthegivengraph,usingoneovertheothermayleadtoacompletelydierentsetofcriticalelements.Figure 2-1 providesanexampleofthedierentoptimalsolutionsthatarefounddependingonthepropertythatischosen.Inthisexample,weassumethatthegoalistoidentifythemostcriticalvertexamongthegraph.Notethatifweselectasthestructuralpropertytheshortestpathbetweenverticesoneandve,thecriticalvertexisvertexthree.Ontheotherhand,ifchosethesizeofthelargestcomponent,thecriticalvertexisvertexfour.Finally,ifweusethetotalnumberofcomponents,thecriticalvertexisvertex6.Amongallofthecriticalelementdetectionproblems,theonesofdetectingcriticalverticesandcriticaledgeshaveattractedsignicantlymoreattentionthantheothers,especiallywhendeterioratingconnectivitymeasures.Inthefollowingsectionswewilldescribesomeversionsoftheseproblemsaswellassomeofthetechniquesthatarecommonlyusedtosolvethem. 2.3DetectingCriticalVerticesBecauseofthefactthatdierentversionsoftheseproblemscanbefoundinseveralcontexts,thereappearstobenoagreementonageneralbasename.Mostvariations 21

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AOriginalgraph BMaximizingtheshortestpathbetweenvertices1and5 CMinimizingthesizeofthelargestcomponent DMaximizingthenumberofcomponents Figure2-1. Optimalsolutionsfordierentstructuralproperties areoftencalledcriticalvertexproblems,vertexinterdictionproblems,vertexdeletionproblems,most-vitalvertexproblems,vertexblockerproblems,vertexdisruptorproblems,amongothers.Moreover,thetermsnodesandverticesareoftenusedinterchangeably.Forthesakeofconsistency,wewillremainusingthewordvertexinsteadofnode. 2.3.1ComputationalComplexityFromthecomplexitypointofview,thedecisionversionofthisproblemhasbeenproventobeNP-completeforseveralmeasuresongeneralgraphsby Addisetal. ( 2013 ); Arulselvanetal. ( 2009 ); DiSummaetal. ( 2011 ); DiSummaetal. ( 2012 ); Dinhetal. ( 2011 ); Oostenetal. ( 2007 ); ShenandSmith ( 2012 ); Shenetal. ( 2012 ),and Veremyevetal. ( 2014b ).Furthermore,thecomplexityanalysisoftheproblemforparticulargraphtopologies,suchastrees,series-parallelgraphs,andbipartitegraphscanbefoundin Addisetal. ( 2013 ); DiSummaetal. ( 2011 )and ShenandSmith ( 2012 ).Inparticular, Addisetal. ( 2013 )and DiSummaetal. ( 2011 )provedthatthisproblemisalsoNP-completeontreesandbipartitegraphswhenthetotalweightofthevertexpairwiseconnectionsisminimized.Theyalsoshowedthatwhentheconnectionweightsaresetasone(i.e., 22

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whentheconnectivitymeasureisreplacedwiththesumofthetotalnumberofconnectedvertexpairs),theproblemcanbesolvedinpolynomialtimeusingadynamicprogrammingapproach(section 2.3.5 ).Moreover, ShenandSmith ( 2012 )provedthattheproblemispolynomiallysolvableintreesandseries-parallelgraphsforthecaseswhenthedeletioncostsoftheverticesaresettobeoneandtheobjectiveiseitherminimizingthesizeofthelargestcomponentormaximizingthenumberofresidualcomponents.Theyalsoproposeadynamicprogrammingschemeforsolvingthesecases(section 2.3.5 ).Inadditiontotheserecentndings,oneofthemostrecognizedresultsisdueto LewisandYannakakis ( 1980 ).TheyprovedthatanyproblemthatattemptstominimizethenumberofverticesthatarerequiredtobedeletedsothattheresidualsubgraphsatisesagivenstructuralpropertyisNP-completeaslongasthegivenpropertyishereditaryundervertexsubgraphs.Agraphpropertyishereditaryundervertexsubgraphsif,giventhatagraphGthatsatisestheproperty,anyvertexinducedsubgraphofGsatisesthepropertyaswell.ThisremarkableresultnotonlyrenderstheproblemNP-completeforseveraloftheabovementionedproperties,butmanyothersaswell.Wenowdiscusstheliteratureregardingsomeoftheexistingmethodologiesfordetectingcriticalvertices.Theseincludemathematicalprogramming,approximatedalgorithms,heuristics,anddynamicprogrammingapproaches.Werstintroducesomeofthethemathematicalformulationsandthenwegiveashortdescriptionofotherapproaches. 2.3.2Mixed-integerFormulationsWhensolvingcombinatorialproblems,usingamathematicalformulationisingeneralanaturalstartingpoint.Despitetheinherentdicultyoftheseproblems,techniquessuchasbranchandboundandbranchandcut,areproventobeveryecientapproachestoobtainsolutionsforinstancesofmanageablesize.Recentendeavorsusingmathematicalprogrammingtechniquescanbefoundin Arulselvanetal. ( 2009 ); Churchetal. ( 2004 ); 23

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DiSummaetal. ( 2012 ); GrubesicandMurray ( 2006 ); Oostenetal. ( 2007 ); Shenetal. ( 2012 ); Veremyevetal. ( 2014a , b ).Wenowprovidethreemixed-integerformulationsthatlaterwouldserveasasteppingstoneforsolvingothercriticalelementdetectionproblems.Allthreeformulationsattempttosolvetheproblemofdeterioratingconnectivityproperties.First,webeginwiththemathematicalformulationintroducedby Arulselvanetal. ( 2009 ),whichisdesignedtotackletheproblemwherethetotalnumberofpairwiseconnectionsisminimized.Forthisformulation,theauthorsdeneabinaryvariableyijforeverypairofverticesi;j2Vthattakesthevalueofoneifiandjbelongtothesamecomponent,andzerootherwise.Inaddition,theyintroduceabinaryvariablexiforeveryvertexi2Vthattakesthevalueofoneifvertexiisdeletedintheoptimalsolution,andzerootherwise.Themathematicalformulationfollows:minXi;j2Vyij (2{1)s.t.yij+xi+xj1;8(i;j)2E (2{2)yij+yjl)]TJ /F3 11.955 Tf 11.95 0 Td[(yik1;8i;j;k2V (2{3)yij)]TJ /F3 11.955 Tf 11.95 0 Td[(yjk+yik1;8i;j;k2V (2{4))]TJ /F3 11.955 Tf 11.95 0 Td[(yij+yjk+yik1;8i;j;k2V (2{5)Xi2Vxib (2{6)xi2f0;1g;8i2V (2{7)yij2f0;1g;8i;j2V; (2{8)wheretheobjectivefunction( 2{1 )minimizesthesumofpairwiseconnections.Constraint( 2{2 )ensurethatifthereisanedgebetweentwoverticesandnoneofthemisdeleted,thenbothverticeshavetobeinthesamecomponent.Constraints( 2{3 ){( 2{5 )areknownastriangleinequalitiesandensurethetransitiverelationshipbetweentheconnectionsin 24

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thegraph(i.e.,ifvertexiisconnectedtovertexjandvertexjisconnectedtovertexk,thenverteximustalsobeconnectedwithvertexk).Constraint( 2{6 )setsthebudgetofcriticalvertices,andconstraints( 2{7 ){( 2{8 )denethedomainofthevariables.Wealsoconsiderthefollowingsimplicationoftheformulationproposedby Veremyevetal. ( 2014a ),whichrepresentsanalternativetotheoneby Arulselvanetal. ( 2009 ).minXi;j2Vyij (2{9)s.t.( 2{2 )yij1 nXfk2V:(i;k)2Egykj)]TJ /F3 11.955 Tf 11.96 0 Td[(xi;8i;j;k2V (2{10)Xi2Vxib (2{11)xi2f0;1g;8i2V (2{12)yij2f0;1g;8i;j2V; (2{13)wheretheonlydierencewiththepreviousformulationisconstraint( 2{10 ),whichguaranteesthatifverticesiandjareinthesamecomponent,thenyij=1.Finally,thethirdformulationisby Oostenetal. ( 2007 ).Inthiswork,theauthorssolvethevariationoftheproblemwheretheobjectiveistominimizethenumberofverticestobedeleted,whileensuringthatthesizeresidualcomponentsislessorequalthanagivenparameterr.Themathematicalformulationfollows:minXi2Vxi (2{14)s.t.( 2{2 ){( 2{5 )Xi2Vnfigxijr)]TJ /F1 11.955 Tf 11.96 0 Td[(1;8i2V (2{15)yi2f0;1g;8i2V (2{16)xij2f0;1g;8i;j2V; (2{17) 25

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Figure2-2. Suboptimalsolutionfoundbygreedyapproachesvstheoptimalsolution wheretheobjectivefunction( 2{14 )minimizesthenumberofverticesthataredeleted.Intheirpaper,theauthorsalsoprovideadetailedpolyhedralanalysisofthisformulationanddiscusssomevalidinequalitiesthatareinheritedfromsomecliquepartitioningproblems.Wenowdiscusstheheuristicapproaches. 2.3.3HeuristicsApproachesAsimpleheuristicapproachwasproposedby Albertetal. ( 2000 ).Thisworkisaimedtostudythetoleranceofcomplexgraphswithrespecttostrategicvertexdeletions,ratherthandetectingwhicharethesetofcriticalverticesofthegraph.Theauthorsseektoanalyzetheresultingconsequencesoverthegraphwhenverticeswitharelativehighimportanceareremoved.Inthiscase,theauthorsusedthedegreeofthevertices(i.e.,thenumberofedgesthatareincidenttothevertex)asameasureofimportancetotestthegraphcohesionafterdeletingtheverticeswiththelargestdegree.However,eventhoughitseemsnaturalthatremovingavertexwithalargerdegreemaycausealargedisconnectioninthegraph,itiseasytoshowthatthisisnotnecessarilythecase.Forexample,observethegraphdescribedinFigure 2-2 .Notethatremovingthevertexwiththelargestdegree(vertex4)doesnotaecttheconnectivityoftheresidualgraphatall,whereasremovingavertexwithalesserdegree(vertex6)dividethegraphintwocomponents.Analternativeapproachistouseasimplegreedyheuristic.Notethatndingasinglecriticalvertex(i.e.,solvingthetheproblemwhenthevertex-deletionbudgetisone)canbedoneinpolynomialformanystructuralpropertiesbyindividuallyremovingeachvertexfromthegraph,andthensolvingtherequiredalgorithmtoevaluatethedegradationofthegivenproperty(assumingthatsuchalgorithmrunsinpolynomialtime).Bydoingso,itis 26

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possibletoidentifythevertexdeletionthatgivesabetterresult.Hence,asimplegreedyalgorithmistosequentiallyeliminatethevertexthatgeneratesthelargestdisruptionateachiteration.Unfortunately,greedyalgorithmslikethisareknowntoperformpoorlyinpractice.Toimprovethesuboptimalsolutionsthatareobtainedwhenusinggreedytechniques, Borgatti ( 2006 )introducesalocalsearchheuristic.Inthisalgorithm,aninitialsetofcriticalverticesSofsizebisselectedeitherrandomlyorwithagreedyalgorithm.Then,thelocalsearchperformsaswapbetweeneachpairofverticesiandjsuchthati2Sandj2VnS.Iftheswapleadstoanimprovement,theswapisacceptedandSisupdated,otherwisethealgorithmcontinueswiththeexploration.Thealgorithmstopswhennofurtherimprovementsarefound.Agreedyheuristicthatisbasedonidentifyingmaximalindependentsetswasproposedin Arulselvanetal. ( 2009 )tominimizethetotalnumberofconnectedvertexpairs.Theintuitionbehindthisapproachisthatthesubgraphinducedbyamaximalindependentsetisfullydisconnected.Therefore,deletingtheverticesthatarenotinanindependentsetwouldyieldanobjectiveofzero.Notethat,ifthesizeofthemaximalindependentsetthatisfoundbythealgorithmisgreaterthanthenumberofverticesinthegraphminusthecriticalvertexbudget,theoptimalsolutionistoselecttheverticesthatarenotintheindependentset.However,ifthesizeoftheindependentsetissmallerthanthisvalue,onecangreedilykeepaddingverticesthatyieldthebestimprovementuntiltheupperboundisreached.Toimprovethesolutionoftheproposedalgorithm,alocalsearchproceduresimilartotheonedescribedaboveisalsoimplementedandisembeddedintoamulti-startheuristicthatsequentiallyrepeatsthesamealgorithmreturningthebestoverallsolution. 2.3.4ApproximationAlgorithmsFormtheapproximationalgorithmsperspective, Dinhetal. ( 2011 )aimedtominimizethenumberofverticesthatmustberemovedinordertoachieveacertaindegradation(disruption)inthetotalnumberofpairwiseconnectedvertices.Inadditiontoproviding 27

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aproofoftheNP-completenessoftheseversion.Theauthorsalsoprovethatthisproblemcannotbeapproximatedwithinafactorlessthan1.36oftheoptimalsolutionwhenthedegradationlevelissettobe0.Finally,theyproposeaO(lognloglogn)pseudo-approximationscheme. 2.3.5DynamicprogrammingTheuseofdynamicprogramminghasbeenstudiedby DiSummaetal. ( 2011 )and ShenandSmith ( 2012 ).Intheworkof DiSummaetal. ( 2011 ),theauthorsprovethatthegeneralversionoftheproblemovertrees,fortheweightedpairwiseconnectivityproperty,isstillNP-hard.However,forthecaseswheretheweightsareone,theyprovethattheproblemispolynomiallysolvable.TheauthorsproposetwodynamicprogrammingalgorithmswithcomplexitiesO(n7)andO(n3b2),respectively.Intheworkof ShenandSmith ( 2012 ),theauthorsintroduceapolynomial-timedynamicprogrammingschemeforsolvingtheproblemovertreesandseries-parallelgraphsfortwoconnectivityproperties:thenumberofconnectedcomponentsandthesizeofthelargestcomponent.Forthecaseofthetrees,theoveralltimeandspacecomplexitiesoftheproposedalgorithmsarerespectively,O(n3)andO(n2)forthenumberofconnectedcomponentsmetric,andO(n3logn)andO(n2)forthesizeofthelargestcomponent.Forthecaseoftheseries-parallelgraphs,theauthorspresenttwodynamicprogrammingalgorithmsforthetotalnumberofconnectedcomponentsandthesizeofthelargestcomponentproperties,withatimeandspacecomplexityofO(n3logn)andO(n2),andO(n5logn)andO(n3),respectively. 2.4DetectingCriticalEdgesWenowdiscusssomeoftheapproachesfordetectingcriticaledgesingraphsincludingenumeratingapproaches,mathematicalprogrammingformulations,andapproximationalgorithms.Similarlyasforthevertexcase,itisoftencommontonddierentnamesacrosstheliterature.Furthermore,thetermsarcsandedgesareusedinterchangeably 28

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withinthedierentpapersthattackletheseproblems.Wewillcontinueusingedgesratherthanarcs. 2.4.1Path-EnumerationApproachesAnearlyapproachforsolvingthecriticaledgedetectionproblembyenumeratingpathswasproposedby CorleyandSha ( 1982 ).Inthisworktheauthorsattempttomaximizetheshortestpathbetweentwopredenedverticessandt.TheydenethesetofcriticaledgesLEsothattheshortestpathbetweensandtinG(V;EnL)isgreaterthanorequaltotheshortestpathbetweensandtinG(V;EnL),foranyothersetofnedgesLE.Theideaofthealgorithmisasfollows.SupposethatapathPbetweenverticessandtisdenedasaconsecutivesetofedges,withoutcycles,connectingverticessandt.LetPstbethesetofallpossiblepathsbetweensandtandc(P)thecostofpathP2Pst.AssumethatsetPstissortedbasedonthecostofthepathsandletPkstbethesetofallpathsbetweenvertexsandvertextwiththebestkthcost.i.e.,thesetofpathssuchthat,ifanytwopathsPaandPbbelongtoPkst,thenc(Pa)=c(Pb).Furthermore,ifPc2PhstandPd2Plst,forh
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staticStackelberggamebetweenaleaderandafollower( SimaanandCruz , 1973 ),wheretheleaderattemptstointerdictasetofgraphelements,inaneorttooptimallyrestricttheabilityofthefollowertousethegraph.Forexample,theleadermaytrytoincreasethecostthatthefollowerperceiveswhiletraversingthegraphortodecreasethegraphcapacityforshippingcommodities.Incontrast,thefollowerobjectiveiseithertominimizethetotalcostofusingthegraph,ortomaximizetheamountofcommoditiesshipped.Fromthepointofviewofthecriticalelementdetection,thecriticalelementsarethentheelementsthatwereoptimallyinterdictedbytheleader.i.e.,theelementswhosedeletionresultedinthemaximumincreaseoftheshortestpathsusedbythefollower,orinthemaximumdecreaseoftheowcapacityofthegraph.Anexampleofagraphinterdictionformulationwheretheobjectiveistomaximizetheshortestpathbetweentwogivenverticessandtcanbedescribedasfollows.AssumingthatGisadirectedgraph,letcebethecostassociatedwithtraversingedgee2E.LetsetFS(i)andRS(i)betheforward-andreverse-starsofvertexi,respectively.Letyebeabinaryvariablethattakesthevalueofoneifthefollowerusesedgee,andzerootherwise.Letxebeabinaryvariablethattakesthevalueofoneiftheleaderinterdictsedgee,andzerootherwise.Themathematicalformulationfollows:maxminXe2Eceye (2{18)s.t.Xe2FS(i)ye)]TJ /F9 11.955 Tf 18.85 11.36 Td[(Xe2RS(i)ye=8>>>><>>>>:1;i=s0;8i2Vnfs;tg)]TJ /F1 11.955 Tf 9.3 0 Td[(1;i=t (2{19)ye1)]TJ /F3 11.955 Tf 11.96 0 Td[(xe;8e2E (2{20)Xe2Exeb (2{21)ye2f0;1g;8e2E (2{22)xe2f0;1g;8e2E; (2{23) 30

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wheretheobjectivefunction( 2{18 )maximizestheshortestpathusedbythefollower,constraints( 2{19 )enforcetheowbalanceconditions,constraints( 2{20 )ensuresthatthefollowerdoesnotuseanedgethathasbeeninterdicted,constraint( 2{21 )denestheinterdictionbudget,andconstraints( 2{22 ){( 2{23 )denethedomainofthevariables.Oneoftheapproachesthatiscommonlyusedtosolvethisproblemsistoreformulatethemodelbyreplacingtheinnerproblem(i.e.,thefollowersproblem)byitsdualversion.Theresultisabilinearmaximizationproblemthatcanbesolvedviastandardlinearizationtechniques.Apath-basedmathematicalformulationforthecriticaledgedetectionproblemwasintroducedby MyungandjoonKim ( 2004 ).Inthispaper,theauthorsusedastheconnectivitymeasuretotalweightofthevertexpairwiseconnections.Forthisformulation,aparametercijisdenedastheweightoftheconnectionbetweenanypairofverticesiandjinV.Pijisdenedasthesetofallpossiblepathsconnectingthepairofvertices(i;j).Furthermore,yijisabinaryvariablethattakesthevalueofoneifverticesiandj,arenotconnectedandzerootherwise,andvariablexeisabinaryvariablethattakesthevalueofoneifedgeeisnotdeletedintheoptimalsolution,andzerootherwise.Themathematicalformulationfollows:maxXi2Vcijyij (2{24)s.t.Xe2Exeb;8i2V (2{25)Xe2Pxeyij;8i;j2V;P2Pij (2{26)xe2f0;1g;8e2E (2{27)yij2f0;1g;8i;j2V; (2{28)wheretheobjectivefunction( 2{24 )maximizestheweightedsumofpairwiseconnectionsthataredisrupted,constraint( 2{25 )ensuresthatnomorethanbedgesaredeleted, 31

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constraints( 2{26 )ensurethatifthereisapathbetweenverticesiandj,variableyij=0.Finallyconstraints( 2{27 ){( 2{28 )denethedomainofthevariables.Analternativeformulationregardingthecriticaledgedetectionprobleminthiscontextcanbefoundin MatisziwandMurray ( 2009 ). 2.4.3ApproximationAlgorithms Dinhetal. ( 2011 ),presentedanapproximationalgorithmfordetectingcriticaledges.Theauthorsproposetominimizethenumberofedgesthatmustberemovedinordertoachieveadisruptionlevelinthenumberoftheresidualconnectedvertex-pairs.TheauthorsprovideaproofoftheNP-completenessofthisproblembyareductionfromthebalancedcutproblem Gareyetal. ( 1976 ).Finally,theyproposeaO(log1:5n)pseudo-approximationscheme. 32

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CHAPTER3IDENTIFYINGCRITICALEDGESFORBLOCKINGDOMINATINGSETSInthischapterwepresenttheproblemofidentifyingasubsetofcriticaledgesofminimumcardinalityinaweightedundirectedgraphsothattheweightofanydominatingsetintheremaininggraphisboundedbelowbyagivenintegerr.Adominatingsetisasubsetofverticessuchthatanyvertexinthegrapheitherbelongstothissubsetorisadjacenttoavertexinthissubset.Consequently,aminimumweighteddominatingsetisasubsetofverticesofminimumweight.Dominatingsetsareamongtheearliestconceptsstudiedingraphtheoryandareusedinawidevarietyofgraph-basedapplicationssuchascommunication,locationallocationandcodingtheory.Themotivationfortheproblemisthepracticalneedofevaluatingrobustnessandvulnerabilityofanygivengraphintermsofpreservingthesizeofthesmallestdominatingsetwhileitsedgesarebeingeliminated.AsmentionedinSection 2.2 ,intherelatedliterature,thiskindofproblemsareoftenreferredtoasminimumedgeblockerproblems.Forthisreasonwewillnamethisproblemtheminimumedgeblockerdominatingsetproblem(EBDP). 3.1ProblemDenitionGivenasimplenonemptygraphG=(V;E)withweightsonitsvertices,letV=f1;2;:::;ng,jEj=m,andvectorw=[wi](i=1;:::;n)representtheweightsofallverticesi2V.Foravertexi2V,letNG(i)denotethesetofverticesadjacenttoiinG(neighborhoodofiinG)andG(i)representthesetofedgesincidenttoiinG.GivenasetDVandavertexi2VnD,letG(D;i)=G(i)\(Sj2DG(j))denotethesetofedgesconnectingvertexitoanyvertexofsetD.Furthermore,foragivenasetDV,letW(D)=Pi2DwiandEG(D)=E\(D(VnD)).NotethatW(;)=0.AdominatingsetinGisasetDVsuchthatforanyvertexi2VnD,NG(i)\D6=;.LetGdenotethesetcontainingalldominatingsetsingraphG.AminimumweighteddominatingsetinGwithrespecttotheweightvectorwisadominatingsetDinGsuchthatW(D)isminimum.TheweightofaminimumweighteddominatingsetinGwithrespecttowis 33

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denotedbywG.IftheweightofeachvertexinVisone(i.e.,ifGisanunweightedgraph),thesuperscriptwinwGisomittedforsimplicity.Inthiscase,GisalsoreferredtoasthedominationnumberofG.GivenasetSE,letGhSidenotethesubgraphwithvertexsetVandedgesetS(equivalently,GhSi=(V;S)).Theminimumedgeblockerdominatingsetproblem(EBDP)isdenedasfollows:Givenanintegerr,ndSEsuchthatwGhEnSirandjSjisminimum.Notethatverticeswithnegativeweightsarealwaysincludedinanyminimumweighteddominatingset.Similarly,verticeswithzeroweightsareeitherinaminimumweighteddominatingsetorcanbeaddedtoonesuchsetandresultinanalternateminimumweighteddominatingset.Additionally,ifthereisanedge(i;j)2Esuchthatminfwi;wjg+W(Vnfi;jg)0foralli2V,minfwi;wjg+W(Vnfi;jg)rforall(i;j)2E,andW(V)r.Notethatforr0,S=;isoptimalandthisproblemisagaintrivial. 3.1.1RelatedProblemsInthecontextofunweightedgraphs,acloselyrelatedconcepttotheEBDP,andarguablyoneofthemostwell-studiedextensionsofthedominatingsetproblem,isknownasthebondagenumberofagraph.ThebondagenumberofanunweightedgraphGisthesmallestnumberofedgeswhoseremovalfromGresultsinanincreaseoftheoriginaldominationnumberofG.Theconceptofbondageingraphtheorywasoriginallyintroducedby Baueretal. ( 1983 ),whereitwasinitiallypresentedasthedominationline-stabilityproblem.Thecurrentnameappearedrstintheworkof Finketal. ( 1990 ),whereitwasdenedasameasureforevaluatingrobustnessorvulnerabilityofgraphsunderedgedisruptions.Latter,intheworkof HuandXu ( 2012 ),theauthorsshowedthattheproblemofdeterminingthebondagenumberofagraphisNP-hard.Foradditionalinformationaboutthebondagenumber,wereferthereadertothesurveyprovidedby Xu ( 2013 ). 34

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DespiteapparentsimilaritiesbetweenthebondagenumberandtheEBDPonunweightedgraphs,thesetwoconceptshavemajordierences.First,theEBDPonunweightedgraphsnotonlydealswithincreasingthedominationnumber,butalsoincreasingituptoagiventhresholdr.Moreover,itshouldbenotedthattheresultofusingthesumofrecursivelycalculatedbondagenumbersinordertoincreasethedominationnumberuptormayyieldsuboptimalsolutionsfortheEBDP.Weshowthiswiththefollowingexample.ConsiderthegraphG=(V;E)illustratedinFigure 3-1 A,whereV=f1;2;3;4;5gandE=f(1;2);(2;3);(2;4);(2;5);(3;5);(4;5)g,andassumer=3.Inthisgure,theverticesoftheminimumdominatingsetofGaredepictedinsolidblack.NotethatG=1,thebondagenumberofGisalso1,and(2;4)isanedgewhoseremovalincreasesthedominationnumberto2(Figure 3-1 B).ThebondagenumberofGhEnf(2;4)giis2andonecanremoveedges(1;2)and(4;5)toincreasethedominationnumberto3(Figure 3-1 C).Ontheotherhand,theoptimalsolutiontoEBDPonGwithr=3is2(S=f(1;2);(2;5)g)(Figure 3-1 D),whichisstrictlysmallerthanthecumulativesumofthetwopreviouslyfoundbondagenumbers.Itisalsoimportanttonotethat,unlikethebondagenumber,theEBDPisdenedintheframeworkofweightedgraphs,whereincreasingtheweightofaminimumweighteddominatingsettosomespeciclevelisdesired. 3.1.2ApplicationsinWirelessNetworksandRelatedAreasSimilarlytoothercriticalelementdetectionproblems,oneofthemostimportantapplicationsoftheEBDPisinthecontextofnetworkvulnerabilityassessmentfromboththeattack-anddefense-perspectives.Particularly,theweightoftheminimumweighteddominatingsetisanimportantcharacteristicofreal-worldnetworksindierentscenarios,suchastheanalysisofwireless( Chengetal. , 2003 ; Dingetal. , 2011 ; Lietal. , 2005 )andsocial( Zhuetal. , 2010 )networks.Forinstance,whenmaintainingandoperatingawireless-communicationnetwork,assumingthattheweightsoftheverticesrepresentthecostofaccessingagiventerminal(i.e.,avertex),aminimumweighteddominating 35

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AOriginalgraphG BGraphGaftertherstiteration CGraphGaftertheseconditeration DActualoptimalsolution Figure3-1. ComparisonbetweentheoptimalsolutionoftheEBDPandthesolutionobtainedbyconsecutivecalculationsofthebondagenumberinagraph setrepresentacost-ecientwayofreachingandcommunicatingwithalltheverticesofthenetwork.Fromanattackerperspective,theobjectiveistodelete(i.e.,disable)asetofedges(i.e.,connections)inordertoincreasethecoststheoperatorwouldincurwhenconnectingwiththeterminalsoftheresidualnetwork.Consequently,fromthedefender'sperspective,onewouldbeinterestedinndingwhichofsuchconnectionsmustbereinforcedorprotected.Inthecontextofsocialnetworkanalysis,assumethatweareinterestedindevelopingamarketingcampaignoverasocialnetwork.Clearly,iftheweightassignedtoeachmember(i.e.,vertex)inthesocialnetworkrepresentsthecostofhiringsuchamembertoworkforthecampaign,hiringmembersofaminimumweighteddominatingsetcansignicantlyreducethecostofimplementingsuchcampaign.SolvingtheEBDPcanhelptoeitherimproveordisruptthenalreachandqualityofthecampaignoverthesocialnetwork. 36

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3.2ComputationalComplexityIntuitively,assumingthatthedesiredthresholdrisgivenaspartoftheinstance,itiseasytoseethatthedecisionversionoftheofthecorrespondingunweightedEBDPisNP-hard.SupposethatweareaskedtondifthereexistsasetofcedgessuchthatwhenremovedthedominationnumberoftheinputgraphGbecomeslargerthanr.Notethatforc=0,theresultingproblemcorrespondstothedecisionversionoftheminimumdominatingset,whichisknowntobeNP-complete( GareyandJohnson , 1990 ).ThisresultissomehowexpectedbecauseofthesimilaritybetweentheunweightedversionoftheEBDPandthebondageproblem.Interestinglyenough,thereisastrongerresultaboutthecomplexityofthegeneralversionoftheEBDP.Formally,foranyxedintegerr1,thedecisionversionofEBDP(denotedby)isgivenbyasimplenonemptygraphG=(V;E)withpositiveweightswonitsverticesandanintegerc0,andweaskifthereexistsasetSEsuchthatjSjcandwGhEnSir. Theorem3.1. ThedecisionversionofEBDPisNP-hardforanyxedintegerr1. Proof. TheintractabilityofEBDPisprovedbyapolynomial-timereductionfromtheminimumdominatingsetproblem.GivenasimplenonemptygraphG=(V;E)andanintegerq>0,thedecisionversionoftheminimumdominatingsetproblem(denotedby)isdenedasfollows.IsthereadominatingsetofsizeatmostqinG?Foranyasuchminimumdominatingsetdecisionproblem,wereplicategraphGandassignapositiveweightofr=(q+1)toeachvertexofthisgraph.Letusdenotethisnewgraphby^G=(^V;^E)andthevectorcontainingallweightsonitsverticesby^w=[^wi],where^wi=r=(q+1)foralli2f1;:::;ng.Thisprocedureconstructsinpolynomialtimeaninstance<^G;^w;0>ofEBDPsuchthatthereexistsadominatingsetofsizeatmostqinG,ifandonlyif,theredoesnotexistasetS^EwithjSj0suchthat^w^GhEnSir.NotethattheonlysetS^EwithjSj0istheemptyset(S=;).Toprovenecessityintheabovestatement,supposethereisadominatingsetDofsizeatmostq 37

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inG.SincejDjq,thenjDjr=(q+1)qr=(q+1)<>:1,ife2S0,otherwiseandzDi=8><>:1,iffori2VnD,NGhEnSi(i)\D=;0,otherwise.Inotherwords,x=[x1;:::;xm]2f0;1gmrepresentstheincidencevectorassociatedwithsetS,andvectorzDrepresentstheincidencevectoroftheverticesinVnDthatwillbecomedisconnectedfromDifsetSisremovedfromG.Proposition 3.1 presentsaninterestingresultwhichcanbeusedtodevelopourrstvalidformulation. 38

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Proposition3.1. GivenasetSEandanintegerr1,SisafeasiblesolutionfortheEBDP,ifandonlyif,foranydominatingsetD2G,thefollowingconditionsaresatised:jDj+Xi2VnDzDir (3{1)zDixe;8i2VnD;e2G(D;i) (3{2)zDi1)]TJ /F9 11.955 Tf 24.24 11.35 Td[(Xe2G(D;i)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xe);8i2VnD: (3{3) Proof. Toprovesuciency,assumethattheaboveconditionsaresatisedforalldominatingsetsinGandSisnotafeasiblesolutionoftheEBDP.SinceSisnotfeasible,theremustexistsadominatingsetD2GwithjDj0verticesinVnDwhicharedisconnectedfromD.WecanusethisinformationtondthevaluesofzDasfollows.IfavertexisdisconnectedfromDinGhEnSi,wegivetherespectivevariablezDithevalueofone,otherwisewegivethevariablethevalueofzero.Itiseasytoseethatthissolutionsatisesconditions( 3{1 )-( 3{3 ),acontradiction. 39

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Proposition 3.1 givesavalidcharacterizationofthefeasiblesolutionsfortheEBDP.Wecanusethisresulttopresentthefollowingformulation.minXe2Exe (3{4)s.t.Xi2VnDzDir)-222(jDj;8D2G (3{5)zDixe;8D2G;i2VnD;e2G(D;i) (3{6)zDi1)]TJ /F9 11.955 Tf 24.24 11.36 Td[(Xe2G(D;i)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xe);8i2VnD (3{7)xe2f0;1g;8e2E (3{8)zDi2f0;1g;8D2G;i2VnD: (3{9)Notethatconstraints( 3{7 )canbeomittedwithoutaectingtheoptimalobjectiveoftheproblem.Thisisbecausetheobjectiveconsistofminimizingthesumofthexvariables.Moreover,itiseasytoshowthatiftheconstraintsthatforcevariableszDitobebinaryarereplacedbyzDi0,forallD2G;i2VnD(Constraints( 3{9 )),itisalwayspossibletotransformanyfractionalsolutionintoanintegersolutionhavingthesameobjectivevalue.ThesefactsareshowninPropositions 3.2 and 3.3 . Proposition3.2. WhensolvingtheEBDPforanygivengraphG=(V;E)andanyinteger1
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Proof. Supposewehaveafractionalsolution(x;z).Similarlyasbefore,letzDi=mine2G(D;i)fxeg,forallD2Gandi2VnD.Clearly,solution(x;z)isintegralandsatisesallconstraints( 3{5 )-( 3{9 )andsincexisthesame,ithasthesameobjectivevalue. BasedontheresultsfromPropositions 3.2 and 3.3 ,thefollowingisavalidmixed-integerformulation.minXe2Exe (3{10)s.t.Xi2VnDzDir)-222(jDj;8D2G (3{11)zDixe;8i2VnD;e2G(D;i) (3{12)xe2f0;1g;8e2E (3{13)zDi0;8D2G;i2VnD: (3{14)Oneimportantissueaboutformulation( 3{10 )-( 3{14 )isthefactthat,foreachdominatingsetD2G,wearerequiredtointroduceavariablezDiforeachi2VnD(i.e.,atotalofO(njGj)additionalvariables).SincethetotalnumberofdominatingsetsonagraphcangrowexponentiallyonthesizeofG,providingacompletedescriptionsuchamodelmaybecomputationallyprohibitive.Ratherthansolvingthisformulation,itispossibletoworkwithitsprojectionontothespaceofthexvariables.Inthefollowingsection,weprovideafullcharacterizationofthisprojection. 3.3.2ProjectedFormulationBeforewecanpresenttheproposedprojectionwerequiretointroducethefollowingdenitions. 41

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Denition1. GivenadominatingsetD2G,letPD(G;r)bethesetofsolutionsthatsatisfythefollowingsystem:Xi2VnDzDir)-221(jDj (3{15)zDixe;8i2VnD;e2G(D;i) (3{16)0xe1;8e2E (3{17)zDi0;8i2VnD: (3{18) Denition2. GivenadominatingsetDV,asetHEiscalledawhipofDingraphG,ifandonlyif,HEG(D)andjH\G(D;i)j=1,foralli2VnD.ThesetcontainingallwhipsofDingraphGisdenotedbyDG.Notethatadominatingsetmighthavemorethanonewhipandawhipmightbeassociatedwithmorethanonedominatingset.Figure 3-2 providesanexampleoftwodierentwhipsforthesamedominatingset. AG(V;E) BG(V;H1) CG(V;H2) Figure3-2. ExampleofdierentwhipsfordominatingsetD=f1;2g Theorem3.2. GivenagraphG=(V;E)andaninteger1
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InordertoproveTheorem 3.2 weneedthefollowingresultsrst.SupposewehaveasubsetofedgesSErepresentedbyitsincidencevectorx.FromFormulation( 3{10 )-( 3{14 )andDenition 1 ,itiseasytoseethatSisnotafeasiblesolutionfortheEBDP,ifandonlyif,thereexistsadominatingsetD2Gforwhichthefollowingsystemhasnosolution.Consequently,ifthereisnosuchadominatingset,SisafeasiblesolutionfortheEBDP.Xi2VnDzDir)-222(jDj (3{21)zDimine2G(D;i)fxeg;8i2VnD (3{22)zDi0;8i2VnD: (3{23)BytheFarkas'Lemma,thereexistsasolutionforsystem( 3{23 )-( 3{21 ),ifandonlyif,thereisnovector[]>2RjVnDj+1,where=[i],foralli2VnDthatsatisesthefollowingsystem.(r)-222(jDj))]TJ /F9 11.955 Tf 16.7 11.36 Td[(Xi2VnDmine2G(D;i)fxegi>0 (3{24))]TJ /F3 11.955 Tf 11.96 0 Td[(i0;8i2VnD (3{25)0 (3{26)i0;8i2VnD (3{27)Wenowprovidetherequiredconditionsforwhichsystem( 3{24 )-( 3{25 )isinfeasible.LetQD(G)bethepolyhedralconedescribedby( 3{25 ){( 3{27 )andRbethesetofitsextremerays.Clearly,ifthefollowingconditionissatisedforall[]>2R,( 3{24 ){( 3{27 )isinfeasibleandthus,( 3{21 ){( 3{23 )hasasolution.(r)-222(jDj))]TJ /F9 11.955 Tf 16.7 11.36 Td[(Xi2VnDmine2G(D;i)fxegi0 (3{28) 43

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Let1bethevectorofsizejVnDjwithallcomponentsequaltooneand1ibethei-thunitvectorwhosei-thcomponentisone. Lemma1. ThesetofextremeraysRofconeQD(G)isgivenby:R=f[01i]>:i2VnDg[f[11]>g (3{29) Proof. ItiseasytoseethatthesetofextremeraysRcanbecharacterizedbythesetofextremepointsofthefollowingpolytope,referredtoasQD(G).+Xi2VnDi=1 (3{30))]TJ /F3 11.955 Tf 11.95 0 Td[(i0;8i2VnD (3{31)0 (3{32)i0;8i2VnD: (3{33)Essentially,sinceQD(G)RjVnDj+1+,equation( 3{30 )actsasanormalizationconstraint.Thus,anypositivemultipleofafeasiblesolutionofQD(G)correspondstoafeasiblerayofQD(G).Toprovetheproposedlemma,itsucestoshowthateverysolution[]>2QD(G)canbeproducedasaconvexcombinationofthevectorsetf[01i]>:i2VnDg[f1=(jVnDj+1)[11]>g.Forthisproof,wedivideallfeasiblesolutionsofQD(G)intwosets.First,considerthesetofsolutionsoftheform[0]>(i.e.,solutionsforwhich=0).Clearly,forasolutionofthiskindtobeinQD(G),from( 3{30 ),itisrequiredthatPi2VnDi=1.Hence,suchasolutioncanbewrittenas[0]>=Pi2VnDi[01i]>.Forthesecondsetofsolutions(>0),form( 3{30 )and( 3{31 ),anyfeasiblesolutionrequiresiforalli2VnDand+Pi2VnDi=1.Leti=i)]TJ /F3 11.955 Tf 12.18 0 Td[(foralli2VnD.From( 3{30 )wehavethat+Pi2VnDi=(jVnDj+1)+Pi2VnDi=1.Thus,itiseasytoseethatanysolution[]>canbewrittenas[]>=[11]>+Pi2VnDi[01i]>. UsingtheresultfromLemma 1 ,wearenowinpositiontoproveTheorem 3.2 . 44

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ProofofTheorem 3.2 . AfterintroducingtheextremeraysofconeQD(G)inthefeasibilityconditiongivenby( 3{28 )weobtain:mine2G(D;i)fxeg0;8i2VnD (3{34)Xi2VnDmine2G(D;i)fxegr)-222(jDj: (3{35)Clearly,inequalities( 3{34 )areimpliedbythetrivialboundsofx.Now,notethatsinceinequalities( 3{35 )aresatisedbymine2G(D;i)fxeg,theyaresatisedbyallxefore2G(D;i)andi2VnD.Furthermore,foreveryvertexi2VnD,thesummationin( 3{35 )containsexactlyonevariablexeassociatedwiththeedgesinG(D;i).Therefore,expression( 3{35 )canbedescribedintermsofallthewhipsinDG.Inotherwords,( 3{35 )isequivalenttoXe2Hxer)-222(jDj;8H2DG; (3{36)provingtheresult. Corollary1. GivenagraphG=(V;E)andaninteger1
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3.4.1ProjectedFormulationfortheWeightedcaseSimilarlyasbefore,itiseasytoseethatthefollowingisavalidmixed-integerformulationfortheEBDP:minXe2Exe (3{40)s.t.Xi2VnDwizDir)]TJ /F3 11.955 Tf 11.95 0 Td[(W(D);8D2G (3{41)zDixe;8i2VnD;e2G(D;i) (3{42)xe2f0;1g;8e2E (3{43)zDi0;8D2G;i2VnD; (3{44)whoseprojectedformulationisminXe2Exe (3{45)Xe2Hw(e)xer)]TJ /F3 11.955 Tf 11.96 0 Td[(W(D);8D2G;H2DG (3{46)xe2f0;1g;8e2E; (3{47)andwherew(e)=wiistheweightofthevertexithatistheendpointofeinVnD.Inspiteof( 3{45 )-( 3{47 )beingavalidformulation,wewillshowthatitispossibletoobtainadierentandrathertighterformulationthatalsogeneralizesformulation( 3{37 )-( 3{39 ). 3.4.2AGeneralizationoftheProjectedFormulationBeforeintroducingthisgeneralformulation,wearerequiredtodenenotation(D;r)foranysetDVandintegerr. Denition3. ForanysetDVandintegerr,theinteger(D;r)istheminimumcardinalityofasetD0(VnD)suchthatW(D[D0)r.Algorithm 1 presentsagreedyproceduretocalculate(D;r)foranygivensetDVandintegerr.ThisalgorithmaddsverticesofsetVnDtosetDindecreasing 46

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orderoftheirweightsuntilthesumoftheweightsoverallverticesinDgetslargerthanr.Atterminationofthisprocedure,thenumberofverticesaddedtoDis(D;r).Aworst-caserunningtimeforthisalgorithmandtheproofofitscorrectnessarepresentedinProposition 3.4 . Algorithm1findKappa(G,w,D,r) 1: ifW(D)rthen 2: return(D;r)=0. 3: endif 4: SortverticesinVnDindecreasingorderoftheirweights. 5: k 1 6: D0 ftkg.fwheretkdenotethek-thvertexg. 7: whileW(D)+W(D0)
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DenesetD2=fv(j):1jjD1jg.SinceW(D[D2)W(D2).DuetothefactthatjD1j=jD2jandW(D1)>W(D2),thereexistsavertexv2(D1nD2)suchthatwv>minfwi:i2D2g.Thismeanswv>wv(jD1j),whichcontradictswithv(jD1j)beingthevertexaddedtosetD0duringthejD1j-thiterationofAlgorithm 1 .Therefore,(D;r)=k. Usingnotation(;r),Proposition 3.5 presentsananalyticallowerboundforthecardinalityofanoptimalsolutiontoEBDP. Proposition3.5. LetDbeadominatingsetingraphGandzdenotethecardinalityofanyoptimalsolutiontoEBDPingraphG.Thenz(D;r). Proof. Supposez<(D;r)forsomedominatingsetDVandletSdenoteanoptimalsolutiontoEBDP.LetD0=fi2VnD:NGhEnSi(i)\D=;g.SinceSisafeasiblesolutiontoEBDP,thenD06=;.Ontheotherhand,sincez<(D;r),thenjD0j<(D;r).Now,D[D0isadominatingsetinGhEnSi.Additionally,sincejD0j<(D;r),thenW(D[D0)
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(D;r)1,soPe2Hxe<(D;r).SinceD2GandH2DG,thiscontradictswithPe2Hxe(D;r)forallD2GandH2DG. Remark1. GivenagraphG=(V;E)andvectorw=[wi](i=1;:::;n)suchthatwi=1foralli2V,(D;r)=r)-221(jDj.Fromproposition 3.6 ,wecanobtainthefollowingvalidformulationfortheEBDS:minXe2Exe (3{49)s.t.Xe2Hxe(D;r);8D2G;H2DG (3{50)xe2f0;1g;8e2E; (3{51)which,asmentionedbefore,generalizesformulation( 3{37 )-( 3{39 ). Remark2. Agiveninequality( 3{48 )associatedwithadominatingsetD2GandawhipH2DGisreferredtoas\whipinequality"ofdominatingsetDanditswhipH'andisdenotedby\(D;H)-inequality".ItturnsoutthattheresultofProposition 3.6 canbefurthersimpliedbyremovingasetofredundantwhipinequalitiesfromthischaracterization.Tospecifythissetofredundantwhipinequalities,weneedtoprovidethefollowingdenitions. Denition4. Givenanintegerr,letG(r)denotethesetcontainingalldominatingsetsDingraphGsuchthat(D;r)>(D[fdg;r),wheredisthevertexwithminimumweightinVnD.NotethatV62G(r). Denition5. GivenadominatingsetDVandanintegerr,letDG(r)denotethesetcontainingallwhipsHofDingraphGthatsatisfythefollowingconditions: (i) Foranyvertexi2Dn([j2VnDNGhHi(j)),either(Dnfig)62Gor(Dnfig;r)=(D;r). (ii) Foranyedgee2EG(D)nH,eitherjNGhHi(ae)j+jNGhHi(ce)j2or((Dnfae;ceg)[fbeg;r)(D;r),whereaeistheendpointofeinD,beistheendpointofeinVnD,andceistheuniqueneighborofbeinGhHi. 49

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UsingDenitions 4 and 5 ,thefollowingtheoremidentiesasetofredundantwhipinequalitiesinthecharacterizationpresentedinProposition 3.6 . Theorem3.3. Givenawhip~H2([D2GDG)n([D2G(r)DG(r))andadominatingset~D2Gsuchthat~H2~DG,the(~D;~H)-inequalityusedinProposition 3.6 tocharacterizethesetoffeasiblesolutionstoEBDPisredundantinpresenceofall(D;H)-inequalitiesinwhichD2G(r)andH2DG(r),andmayberemovedfromthischaracterization.Toprovethistheorem,werstneedtodeviseaprocedurethattransformsadominatingset~D2Gandawhip~H2~DGtoadominatingsetD2G(r)andawhipH2DG(r).OnesuchprocedureispresentedbyAlgorithm 2 (whichwillalsobereferredtoasProc(G(V;E);w;r;~D;~H)).ThisalgorithmstartsbylettingD=~DandH=~H,andthenrecursivelyaddingverticeswithminimumweightinsetVnDtosetDwhileremovingtheedgesincidenttothemfromwhipHuntiltheconditionofDenition 4 holdsforD.Then,allverticesinDn([j2VnDNGhHi(j))forwhichCondition(i)ofDenition 5 isviolatedarerecursivelyremovedfromDwhilearbitraryedgesinEG(D)thatareincidenttothem(oneedgeforeachvertexbeingremoved)areaddedtowhipH.Finally,eachedgeeinEG(D)nHthatviolatesCondition(ii)ofDenition 5 isaddedtowhipHwhileaeandceareremovedfromDandbeisaddedtothisset(ae,beandcearetheverticesdenedinDenition 5 ).ThiswholeprocessisrepeateduntilD2G(r)andH2DG(r).Proposition 3.7 showsthatAlgorithm 2 terminatesinpolynomialtime. Proposition3.7. TheworstcaserunningtimeforAlgorithm 2 isO(n3+mn2). Proof. LetusrefertowhileloopsinLines 3 30 , 4 14 , 15 21 and 22 29 asmainloop,loop1,loop2andloop3,respectively.BeforeexecutingLine 2 ,wealsostoreDandHastwobooleanarraysofsizesnandm,respectively.Byusingsucharrays,checkingmembershipofavertexinDoranedgeinHcanbedoneinconstanttime.ConstructingthesearraystakesO(n)+O(m).Moreover,wesorttheelementsofVnDindecreasingorderoftheirweightsandcalculate(D;r),whichcanbedoneinO(nlogn).Duringtheexecutionof 50

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Line 2 ,wegenerateD0andGhHibytraversingH.SincejHjisO(n),thenthesecanalsobedoneinO(n).WestoreD0asabooleanarrayaswell.Thefollowingshowsaworst-caserunningtimeforeachiterationofthemainloop.WhileexecutingLine 3 ofAlgorithm 2 ,conditionofDenition 4 canbecheckedinO(1),Condition(i)ofDenition 5 canbeveriedinO(n(n+1))=O(n2)(foreachi2D0,ndingif(Dnfig)2GtakesO(n)bycheckingifihasaneighborinD;placingiinVnDwhilekeepingthissetsortedindecreasingorderofitselementsweightscanbedoneinO(n);andverifyingif(Dnfig;r)=(D;r)takesO(1)),andCondition(ii)ofDenition 5 canbecheckedinO(m(n+1))=O(mn)(foreachedgee,ndingife2EG(D)nHcanbedoneinO(1);gettingae;be,andcetakesO(1);placingaeandceinVnDwhilekeepingthissetsortedindecreasingorderofitselementsweightscanbedoneinO(n);andverifyingif((Dnfae;ceg)[fbeg;r)=(D;r)takesO(1)).Loop1terminatesinO(n),Loop2willterminateinO(n2),andaworst-caserunningtimeforloop3isO(mn).So,aworst-caserunningtimeforeachiterationofthemainloopisO(n2+mn).Ifinaniterationofthemainloop,noneofloops2and3areexecuted,thenwerefertothatiterationastypeA;otherwise,thatiterationisreferredtoastypeB.NotethatafteraniterationoftypeA,Algorithm 2 terminates.Hence,Algorithm 2 terminateseitherafteraseriesofiterationsoftypeB,orafteraseriesofiterationsoftypeBandoneiterationoftypeA(thiscasecontainsthescenarioinwhichthisalgorithmterminatesafteroneiterationthatisoftypeA).LetnBdenotethetotalnumberofiterationsoftypeBofthemainloopatthebeginningofAlgorithm 2 .NotethatsetDobtainedafteraniterationoftypeBhaslarger(;r)comparedtothesetDusedatthebeginningofthatiteration.Since(:;r)isO(n),thennBisO(n).Therefore,thetotalnumberofiterationsofthemainloopisO(n+1)=O(n).Finally,aworstcaserunningtimeforAlgorithm 2 isO(n(n2+mn))=O(n3+mn2). 51

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Algorithm2UpdateCut(G(V;E);w;r;~D;~H) 1: D ~DandH ~H. 2: D0 Dn([j2VnDNGhHi(j)). 3: whileD62G(r)orH62DG(r)do 4: whileD62G(r)(CheckingtheconditionofDenition 4 )do 5: LetddenoteavertexwithminimumweightinVnD. 6: D (D[fdg). 7: Let(d0;d)denotetheuniqueedgeinHthatisincidenttod. 8: H (Hnf(d0;d)g). 9: ifNGhHi(d0)=;then 10: D0 (D0[fd0;dg). 11: else 12: D0 (D0[fdg). 13: endif 14: endwhile 15: while9i2D0:Condition(i)inDenition 5 isviolatedforido 16: D (Dnfig),D0 (D0nfig). 17: LeteidenoteanarbitraryedgeinEG(D)thatisincidenttoi. 18: H (H[feig). 19: LetjdenotethevertexinDthatisincidenttoei. 20: D0 (D0nfjg) 21: endwhile 22: while9e2EG(D)nH:Condition(ii)inDenition 5 isviolatedforedo 23: LetaedenotetheendpointofeinD. 24: LetbedenotetheendpointofeinVnD. 25: LetcedenotetheuniqueneighborofbeinGhHi. 26: D (Dnfae;ceg)[fbeg. 27: D0 (D0nfaeg) 28: H (H[feg). 29: endwhile 30: endwhile 31: returnDandH. ProofofTheorem 3.3 . Givenawhip~H2([D2GDG)n([D2G(r)DG(r))andadominatingset~D2Gsuchthat~H2~DG,the(D;H)-inequalityassociatedwiththedominatingsetDandwhipHreturnedatterminationofAlgorithm 2 dominates(~D;~H)-inequality.Thiscanbeshownasfollows.Aftereachiterationofloop1,theleft-handsideofthewhipinequalityassociatedwithdominatingsetDandwhipHhasonelessvariablecomparedtotheleft-handsideofthewhipinequalityassociatedwiththesesetsbeforethatiterationwhiletheright-handsideofbothinequalitiesarethesame.Thus,the 52

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whipinequalityassociatedwithdominatingsetDandwhipHobtainedaftereachiterationofloop1dominatesthewhipinequalityassociatedwiththesesetsbeforethatiteration.Similarly,aftereachiterationofloop2(orloop3),theleft-handsideofthewhipinequalityassociatedwithdominatingsetDandwhipHhasonemorevariablecomparedtotheleft-handsideofthewhipinequalityassociatedwiththesesetsbeforethatiterationwhileitsright-handsideisalsolargerbyoneunit.ThisagainmeansthatthewhipinequalityassociatedwithdominatingsetDandwhipHobtainedaftereachiterationofloop2(orloop3)dominatesthewhipinequalityassociatedwiththesesetsbeforethatiteration.Thisshowsthatthe(D;H)-inequalityassociatedwiththedominatingsetDandwhipHreturnedatterminationofAlgorithm 2 dominates(~D;~H)-inequality.So,(~D;~H)-inequalitymayberemovedfromcharacterizationofProposition 3.6 . AsaresultofTheorem 3.3 ,Corollary 2 presentsamoresimpliedcharacterizationofthesetoffeasiblesolutionstoEBDP,andtherefore,asimpliedformulation. Corollary2. AsetSEisafeasiblesolutiontoEBDPifandonlyifXe2Hxe(D;r);8D2G(r);H2DG(r): (3{52)Furthermore,thefollowingrepresentsavalidformulationfortheEBDP.minXe2Exe (3{53)s.t.Xe2Hxe(D;r);8D2G(r);H2DG(r) (3{54)xe2f0;1g;8e2E: (3{55) Remark3. Whipinequalitiesoftype( 3{52 )arereferredtoas\critical-whipinequalities"intheremainderofthisarticle. 3.5PolyhedralAnalysisoftheProposedFormulationInthissection,westudytheconvexhulloftheincidencevectorsofallfeasiblesolutionstoEBDPinordertodiscoverimportantpropertiesofthispolytope.These 53

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propertiesmaybebenecialinthedevelopmentofintegerprogrammingtechniquestosolveEBDP.LetPG(r)denotetheconvexhulloftheincidencevectorsofallfeasiblesolutionstoEBDPonGforagivenintegerr1.Here,weshowthatPG(r)isafull-dimensionalpolytope,theupper-boundconstraintoneachdecisionvariable(xe1,foreachedgee2E)inducesafacetofPG(r),andiftheweightofeachvertexinVistheone(unweightedgraphs),theneachcritical-whipinequalityinducesafacetofPG(r).Theseresultsarepresentedinthefollowingtheorem. Theorem3.4. GivenasimplenonemptygraphG=(V;E)withweightswonitsverticesandanintegerr,thefollowingstatementsaretrue. (a) PG(r)isafull-dimensionalpolytope. (b) Givenanedgee2E,inequalityxe1inducesafacetofPG(r). (c) Fortheunweightedcase(w=1),the(D;H)-inequalityassociatedwithadominatingsetD2G(r)andawhipH2DG(r)inducesafacetofPG(r),ifandonlyif, (c.1) EnH=;,or (c.2) jVjr+2. (d) Forthecaseofunweightedgraphs(w=1),inequalityxe0foranedgee2EinducesafacetofPG(r),ifandonlyif, (d.1) Enfeg=;,or (d.2) jVjr+2. Proof. (a) ToshowthatPG(r)isafull-dimensionalpolytope,itisenoughtoshowthattherearem+1anelyindependentpointsinPG(r).Sinceminfwi;wjg+W(Vnfi;jg)rforall(i;j)2EandW(V)r,thensetsEnfegforalle2EalongwithsetEarefeasiblesolutionstothecorrespondingEBDP.Itcanbeeasilyveriedthattheincidencevectorsofthesefeasiblesolutions,whichbelongtoPG(r),areanelyindependent.Hence,PG(r)isafull-dimensionalpolytope. 54

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(b) Givenanedgee2E,itisenoughtoshowthatthedimensionofthefaceofPG(r)inducedbyxe1ism)]TJ /F1 11.955 Tf 11.96 0 Td[(1.ThefaceofPG(r)inducedbyxe1isPeG(r)=fx2PG(r):xe=1g:NotethatthedimensionofPeG(r)isatmostm)]TJ /F1 11.955 Tf 12.23 0 Td[(1.Hence,itisenoughtoshowthattherearemanelyindependentpointsinPeG(r).Sinceminfwi;wjg+W(Vnfi;jg)rforall(i;j)2EandW(V)r,thensetsEnfe0gforalle02EnfegalongwithsetEarefeasiblesolutionstothecorrespondingEBDP.TheincidencevectorsofthesesetsareanelyindependentandbelongtoPeG(r).Thus,thedimensionofPeG(r)ism)]TJ /F1 11.955 Tf 11.96 0 Td[(1. (c) Toshownecessity,supposethe(D;H)-inequalityassociatedwithadominatingsetD2G(r)andawhipH2DG(r)inanunweightedgraphinducesafacetofPG(r)butEnH6=;andjVj
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beeasilyveriedthatPe2EnHx0e=jEnHj.Therefore,thenumberofincidencevectorsoffeasiblesolutionstoEBDPthatbelongtoPD;HG(r)isjHj.SincejHj
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setS"(Hnf(c";d);(b";c")g)withjS"j=(D;r).Notethatsincecondition(c.2)isvalid,suchasetS"existsinbothcases.Finally,if"2EnHandbothendpointsof"(denotedbya"andb")belongtoVnD,thenlete1ande2denotetheedgesinHthatareincidenttoa"andb",respectively.SelectanarbitrarysetS"(Hnfe1;e2g)suchthatjS"j=(D;r)(notethatsincecondition(c.2)isvalid,suchasetS"exists).AfterselectingsetS"usingtheapproachmentionedforanyofthesethreecases,letI"=S"[(En(H[f"g)):Here,weshowthatforanyedge"2E,setI"isafeasiblesolutiontoEBDP.Tothisaim,let^DdenotethesetcontainingallisolatedverticesinGhEnI"ithatbelongtosetVnD.If"2H,theninGhEnI"i,theredoesnotexistanypairofverticesinDthatbelongtothesameconnectedcomponent.Additionally,wehavej^Dj=(D;r).Hence,GhEnI"ijDj+(D;r)=randsetI"isafeasiblesolutiontoEBDP.If"2EnHandatleastoneofitsendpointsbelongstoD,theninGhEnI"i,thereisexactlyonepairofverticesinDthatbelongtothesameconnectedcomponent(denotedby~G).Itcanbeeasilyveriedthat~G2.Sincej^Dj=(D;r),thenGhEnI"i(jDj)]TJ /F1 11.955 Tf 19.07 0 Td[(2)+2+(D;r)=randsetI"isafeasiblesolutiontoEBDP.If"2EnHandbothendpointsof"belongtoVnD,theneithertheredoesnotexistanypairofverticesinDthatbelongtothesameconnectedcomponentorthereisexactlyonepairofverticesinDthatbelongtothesameconnectedcomponent(denotedby~G).Forbothcases,j^Dj=(D;r).Fortherstcase,GhEnI"ijDj+(D;r)=r.Inthesecondcase,itcanbeveriedthat~G2andhence,GhEnI"i(jDj)]TJ /F1 11.955 Tf 18.21 0 Td[(2)+2+(D;r)=r.Therefore,setI"isagainafeasiblesolutiontoEBDP.Thismeansforeach"2E,theincidencevectorofI"(denotedbyx")belongstoPG(r).Additionally,Pe2Hx"e=(D;r)andhence,x"2PD;HG(r).Itcanbeeasilyveriedthatvectorsx"forall"2Eareanelyindependent.So,thedimensionofPD;HG(r)ism)]TJ /F1 11.955 Tf 11.96 0 Td[(1. 57

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(d)Toshownecessity,supposeinequalityxe0foranedgee2EinanunweightedgraphinducesafacetofPG(r)butEnfeg6=;andjVj
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theincidencevectorofI"(denotedbyx")belongstoPG(r).Additionally,foreach"2E,wehavex"e=0andhence,x"2PeG(r).Itcanbeeasilyveriedthatvectorsx"forall"2Eareanelyindependent.So,thedimensionofPeG(r)ism)]TJ /F1 11.955 Tf 11.96 0 Td[(1. 3.6RowGenerationBasedExactAlgorithmThenumberofcritical-whipinequalitiesinagraphcanbeverylarge.Asaresult,Formulation( 3{53 )-( 3{55 )mayinvolvealargenumberofconstraints,anddealingwiththisformulationmayrequireextensivecomputationaleort.Inthissection,wedeveloptherstexactalgorithmforsolvingEBDP,whichsolvesFormulation( 3{53 )-( 3{55 )byabranch-and-cutapproachwithlazyimplementationofcritical-whipinequalities(constraintsoftype( 3{54 )).Beforepresentingthedetailsofthisbranch-and-cutalgorithm,letusintroducethefollowingnotations.LetFTdenotetheformulationassociatedwithnodeTofthesearchtreeandxTdenoteanoptimalsolutiontothisformulation.Additionally,foranintegralxT,letSdenotethesetwhoseincidencevectorisxT.Atthebeginningofthebranch-and-cutalgorithm,theformulationassociatedwiththerootnodeofthesearchtreeisthelinearprogrammingrelaxationofFormulation( 3{53 )-( 3{55 ),fromwhichallcritical-whipinequalitieshavebeenremoved.TheincumbentsolutionisalsoinitializedtosetE(sinceW(V)r,thenEisafeasiblesolutiontoEBDP).WhennodeTofthesearchtreeisselectedforprocessing,werstsolveformulationFT.IfFTisinfeasible,thennodeTisfathomedbyinfeasibility.IfFTisfeasible,thenPe2ExTeisavalidlowerboundforthecardinalityofthebestsolutionfoundatthesubtreerootedatnodeT.IfPe2ExTeisnotsmallerthanthecardinalityoftheincumbentsolution,thennodeTisfathomedbybound.Otherwise,ifxTisnotintegral,thenwebranchbyvariabledichotomyandtwochildrennodesareaddedtothesetofunprocessedtreenodes.IfxTisintegral,thenfeasibilityofStoEBDPischeckedbysolvingtheminimumweighteddominatingsetproblemonGhEnS(xT)i.IfSisfeasibletoEBDP(wGhEnSir),thenthe 59

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presenttreenodeisfathomedbyfeasibilityandtheincumbentsolutionisupdatedtoS.Otherwise,aviolatedcritical-whipinequalityisdetectedandaddedtoformulationFTasalazyconstraint,andnodeTisprocessedagain.ThisalgorithmterminatesafterprocessingallunprocessednodesinthesearchtreeandthecardinalityoftheincumbentsolutionwillbetheoptimalobjectivetoEBDPongraphG.ForanintegralxT,theminimumweighteddominatingsetproblemonGhEnS(xT)iusedinourexactalgorithmtoverifythefeasibilityofsetS(xT)isformulatedasminXi2Vwiyi (3{56)s.t.yi+Xj2NG0(i)yj1;8i2V (3{57)yi2f0;1g;8i2V; (3{58)whereG0=GhEnSi.IncasesetSisinfeasibletoEBDP(wGhEnSi
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inperformanceofourproposedexactalgorithm.Inthismodiedversion,foreachintegralxTwithaninfeasibleS(xT),theviolatedwhipinequalityassociatedwithaminimumweighteddominatingsetinGhEnS(xT)iandoneofitswhipsinthisgraphisdirectlyaddedtoformulationFTwithoutbeingimproved(transformedtoacritical-whipinequality)byAlgorithm 2 .Intheotherwords,themodiedalgorithmemploysinequalitiesoftype( 3{48 )tocharacterizethesetoffeasiblesolutionstoEBDPinsteadofusinginequalitiesoftype( 3{52 ).Employingviolatedinequalitiesoftype( 3{48 )requireslesscomputationaleortbuttheseinequalitiesareweakercomparedtoinequalitiesoftype( 3{52 ).Oneoftheobjectivesofournumericalexperimentsinthissectionistostudytheeectofthistrade-oinperformanceoftheexactalgorithmforsolvingEBDP.ThecomputationalexperimentswereperformedonaserverwithtwoAMDOpteronTM6128Eight-CoreCPUsand12gigabytesofRAM,runningLinuxx86 64,CentOS5.9.AllalgorithmswereimplementedinC++andGurobiROptimizer5.6.2wasusedtosolvethelinear0{1programmingformulationsinalazyfashion.Thetest-bedofinstancesconsistedofuniformrandomgraphswith25verticesandagroupofreal-lifepower-lawnetworks.FortheURGinstances,theprobabilityforexistenceofanedgebetweenanypairofvertices(expectededgedensity)waschosentobep2f0:1;0:2;0:5;0:7;0:9g.WegeneratedbothunweightedandweightedURGinstances.Fortheweightedinstances,theweightassignedtoeachvertexisauniformlydistributedintegerbetween1and10.Foreachexpectededgedensityp,wegenerated10unweightedand10weightedgraphinstances,resultinginatotalof100instances.Thevalueofparameterrwaschosenfromf8;12;16;20gforunweightedinstances,andfromf40;60;80;100gforweightedones.Thereal-lifenetworkinstancesusedinourexperimentswerecollectedfromtheUniversityofFloridaSparseMatrixCollectiondatabase( DavisandHu , 2011 ).Thegraphsconsideredarekarate 34(asocialnetworkfromakarateclubwith34membersataU.S.university( Zachary , 1977 ),chesapeake 39 61

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(a39-vertexgraphfromtheseasonaldynamicsoftheChesapeakeBayecosystem( BairdandUlanowicz , 1989 ),dolphins 62(asocialnetworkoffrequentassociationsbetween62dolphinsinacommunitylivingoDoubtfulSound,NewZealand( Lusseauetal. , 2003 ),andterror kreb 62(a62-memberterroristnetworkfromthehijackersresponsibleforthe9/11attacks,compiledby Krebs ( 2002 ).Wealsodevelopedtheweightedversionofeachoneoftheseinstancesbyassigningauniformlydistributedintegerbetween1and10toeachvertexofthatinstanceasitsweight.Thevalueofparameterrforreal-lifenetworkinstanceswaschosenfromf10,15,20,25,30,40,45,50gforunweightedinstances,andfromf60,90,120,150,180,240,270,300gforweightedones.TherunningtimelimitforeachinstanceofEBDPwassetat3600seconds. 3.7.1ResultsfortheUniformRandomGraph(URG)InstancesTable 3-1 showspercentageofinstancessolvedtooptimality(Solved(%)),averagerunningtime(Time(s)),averagebestsolutionfound(BSF),andaverageoptimalitygap(Gap(%))obtainedbyusingtheproposedexactalgorithmwhenincludingwhipinequalitiesonthetest-bedofURGinstanceswithin3600secondstimelimit.Table 3-2 showsthevaluesoftheaforementionedattributesacquiredbyemployingthemodiedversionoftheproposedexactalgorithmwhenincludingcritical-whipinequalitiesonthesametest-bedwithinthesametimelimit.AccordingtoTables 3-1 and 3-2 ,foragivenedgedensityp,asrincreases,theoptimalitygapincreases,reachesamaximum,andthendecreases.Thismeansforagivenedgedensityp,theprobleminstancescorrespondingtoverysmallorverylargevaluesofraresimplercomparedtotheinstancesassociatedwithmoderatevaluesofthisparameter.ThisisduetothefactthatthenumberofbinaryvectorssatisfyingtheconstraintassociatedwithagivendominatingsetDandoneofitswhipsH(constraintoftype( 3{52 )forTable 3-1 andconstraintoftype( 3{48 )forTable 3-2 )isequalto)]TJ /F5 7.97 Tf 14.22 -4.38 Td[(H(D;r).ForagivenDandH,asrincreases,(D;r)increasesandthequantity)]TJ /F5 7.97 Tf 14.23 -4.38 Td[(H(D;r)increases,reachesamaximumandthendecreases.Thismeansthesizeofthefeasibleregionforthe 62

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Table3-1. StatisticsfortheURGinstances. Densityr(Unweighted)r(Weighted)(p)8121620406080100 0.11001001001001001001001000.2100904020100801040Solved(%)0.5000000000.7000000000.9000000000.10.00.22.34.20.02.85.19.00.27.0428.22509.63049.341.51071.43242.62567.8Time(s)0.53600.03600.03600.03600.03600.03600.03600.03600.00.73600.03600.03600.03600.03600.03600.03600.03600.00.93600.03600.03600.03600.03600.03600.03600.03600.00.10.24.711.720.51.25.09.816.20.24.617.031.445.05.213.524.733.4BSF0.559.1103.60130.6140.454.979.0102.3135.40.7106.6155.5185.5199.285.8127.9156.9179.90.9151.7212.0246.6258.7108.0152.8193.1223.80.1000000000.2014502157Gap(%)0.541403318534538350.750423219555040270.95244341954484132 correspondingEBDPalsoincreases,reachesamaximumandthendecreases.Thisexplainsthebehavioroftheoptimalitygapwhenparameterrisincreasingwhileedgedensityiskeptunchanged.Itisinterestingtonotethatastheedgedensityincreases,thelargestoptimalitygapoverallvaluesofparameterrhappensatasmallervalueofr.SimilarbehaviorisobservedforSolved(%)andTime(s)attributesinthesetables.WithregardtotheBSFattributeinTables 3-1 and 3-2 ,foragivenedgedensityp,asrincreases,theaveragesizeofthebestsolutionfoundalsoincreases.Similarly,foragivenr,asedgedensitypincreases,theBSFattributealsoincreases.Thisbehaviorisveryintuitivebecauseasparameterr(oredgedensityp)increases,thenumberofconstraintsoftype( 3{52 )forTable 3-1 andconstraintsoftype( 3{48 )forTable 3-2 increases.This 63

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Table3-2. StatisticsfortheURGinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . Densityr(Unweighted)r(Weighted)(p)8121620406080100 0.1100100100901001001001000.210060001007000Solved(%)0.5000000000.7000000000.9000000000.10.00.412.9392.70.13.516.9302.10.24.41481.63600.03600.059.51248.23600.03600.0Time(s)0.53600.03600.03600.03600.03600.03600.03600.03600.00.73600.03600.03600.03600.03600.03600.03600.03600.00.93600.03600.03600.03600.03600.03600.03600.03600.00.10.24.711.720.51.25.09.816.20.24.617.140.147.55.217.733.347.0BSF0.575.7126.0136.4140.974.0112.0150.5145.10.7106.6155.5185.7200.786.5127.9156.9178.70.9151.7212.0246.6259.0108.0152.8193.1224.60.1000100000.2032515093237Gap(%)0.547503619636060420.750423320615443280.95344341958524435 meansthesizeofthefeasibleregionforthecorrespondingEBDPdecreasesandtheoptimalobjectivevalueincreases.ComparingtheresultsshowninTable 3-1 withtheonespresentedinTable 3-2 ,itcanbeconcludedthattheproposedexactalgorithmoutperformsthemodiedversionofthisalgorithmonURGinstancesinallcases.Thismeansimprovingtheviolatedwhipinequalitiesbytransformingthemintoviolatedcritical-whipinequalitiesbyAlgorithm 2 signicantlyimprovestheperformanceofthebranch-and-cutalgorithmontheseinstances. 3.7.2ResultsforthePower-LawInstancesTables 3-3 and 3-4 showtherunningtime(Time(s)),bestsolutionfound(BSF),andoptimalitygap(Gap(%))obtainedbyusingtheproposedexactalgorithmanditsmodiedversiononthetest-bedofunweightedpower-lawinstanceswithin3600secondstimelimit, 64

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respectively.Tables 3-5 and 3-6 showthevaluesoftheaforementionedattributesacquiredbyemployingtheproposedexactalgorithmanditsmodiedversiononthetest-bedofweightedpower-lawinstanceswithinthesametimelimit,respectively. Table3-3. Statisticsfortheunweightedpower-lawinstances. r(Unweighted)Instance1015202530404550 karate 341.25.535.5866.076.8---Time(s)chesapeake 39420.83600.03600.03600.03600.0---dolphins 620.00.01.91219.23600.03600.03600.03600.0terror kreb 620.00.00.1116.0371.23600.03600.03600.0karate 34717314966---BSFchesapeake 39224494114134---dolphins 62017205978124117terror kreb 62051020336482106karate 3400000---Gap(%)chesapeake 3909311911---dolphins 6200004212297terror kreb 6200000546 Accordingtothesetables,similartotheobservationmadefortheURGinstances(withsomeexceptions),foragiveninstance,asrincreases,theoptimalitygapincreases,reachesamaximum,andthendecreases.Therefore,theprobleminstancescorrespondingtomoderatevaluesofraremorechallengingcomparedtotheonesassociatedwithverysmallerorverylargervaluesofthisparameter.ThisbehavioroftheoptimalitygapforagiveninstancecanbeexplainedbythesamelogicusedfortheURGinstances.Thevalueofparameterratwhichtheoptimalitygapreachesitsmaximumdependsontheinstanceandvariesfromoneinstancetoanother.SimilarbehaviorisobservedforTime(s)attributeinthesetables.ConsideringBSFattribute,foreachinstance,asparameterrincreases,thesizeofthebestsolutionfoundalsoincreases.ThisobservationisagainsimilartotheonemadeforURGinstancesandcanbejustiedusingasimilardiscussion.Accordingtothesetables,similartotheobservationmadefortheURGinstances(withsomeexceptions),foragiveninstance,asrincreases,theoptimalitygapincreases, 65

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Table3-4. Statisticsfortheunweightedpower-lawinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . r(Unweighted)Instance1015202530404550 karate 3416.95.13600.03600.03600.0---Time(s)chesapeake 393600.03600.03600.03600.03600.0---dolphins 620.00.04.13600.03600.03600.03600.03600.0terror kreb 620.00.00.13600.03600.03600.03600.03600.0karate 34717324966---BSFchesapeake 39234494114134---dolphins 620172259105124135terror kreb 62051020426583115karate 3400365---Gap(%)chesapeake 3999332113---dolphins 620001847383223terror kreb 620005299816 Table3-5. Statisticsfortheweightedpower-lawinstances. r(Weighted)Instance6090120150180240270300 karate 34185.13600.03600.03600.03600.0---Time(s)chesapeake 393600.03600.03600.03600.03600.0---dolphins 620.01.73600.03600.03600.03600.03600.03600.0terror kreb 620.046.53600.03600.03600.03600.03600.03600.0karate 34925465867---BSFchesapeake 39416291119143---dolphins 620411304983110125terror kreb 62162234427892111karate 3403641286---Gap(%)chesapeake 395447443726---dolphins 62025185049353425terror kreb 6200504736322720 66

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Table3-6. Statisticsfortheweightedpower-lawinstanceswhentheinequalitiesarenottransformedintocritical-whipinequalitiesbyexecutingAlgorithm 2 . r(Weighted)Instance6090120150180240270300 karate 34317.23600.03600.03600.03600.0---Time(s)chesapeake 393600.03600.03600.03600.03600.0---dolphins 620.00.53600.03600.03600.03600.03600.03600.0terror kreb 620.052.63600.03600.03600.03600.03600.03600.0karate 34931395871---BSFchesapeake 39416291119143---dolphins 620311304983110125terror kreb 62162234427892111karate 34048333121---Gap(%)chesapeake 395648454230---dolphins 6200185049393628terror kreb 6200504736352824 reachesamaximum,andthendecreases.Therefore,theprobleminstancescorrespondingtomoderatevaluesofraremorechallengingcomparedtotheonesassociatedwithverysmallerorverylargervaluesofthisparameter.ThisbehavioroftheoptimalitygapforagiveninstancecanbeexplainedbythesamelogicusedfortheURGinstances.Thevalueofparameterratwhichtheoptimalitygapreachesitsmaximumdependsontheinstanceandvariesfromoneinstancetoanother.SimilarbehaviorisobservedforTime(s)attributeinthesetables.ConsideringBSFattribute,foreachinstance,asparameterrincreases,thesizeofthebestsolutionfoundalsoincreases.ThisobservationisagainsimilartotheonemadeforURGinstancesandcanbejustiedusingasimilardiscussion.ComparingtheresultsshowninTable 3-3 withtheonespresentedbyTable 3-4 (andtheresultsshowninTable 3-5 withtheonespresentedbyTable 3-6 ),itisobservedthattheproposedexactalgorithmperformsbetterthanthemodiedversionofthisalgorithmonpower-lawinstancesaswell.ThisagainindicatesthatimprovingtheviolatedwhipinequalitiesbyAlgorithm 2 andtransformingthemintoviolatedcritical-whipinequalitiesimprovestheperformanceoftheproposedbranch-and-cutapproach. 67

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CHAPTER4IDENTIFYINGCRITICALCLIQUESINSPARSEGRAPHSInthischapter,wepresenttheproblemofidentifyingasubsetofatmostbdisjointcliquesinanundirectedgraphsothatafterdeletingtheverticesandedgesofthecliquesthegraphgetsmaximallydisconnected.Acliqueisasubsetofverticessuchthatanytwoverticesinthecliqueareadjacent.Alongwiththedominatingset,theconceptofacliqueisoneofthemostwell-studiedingraphtheoryandisusedinseveralgraph-basedapplications,suchassocialnetworkanalysis.Themotivationfortheproblemisthepracticalneedofevaluatingrobustnessandvulnerabilityofanygivengraphintermsofpreservingconnectivitywhilestructuredvertexsubsetsareremovedfromthegraph. 4.1ProblemDenitionGivenaconnectednonemptygraphG=(V;E)whereV=f1;2;:::;ngandE=f1;2;:::;mgarethesetofverticesandedges,respectively,andanonnegativeintegerb,thecriticalcliquedetectionproblem(CCP)involvesndingasetofbdisjointcliquessuchthatitsdeletionresultsinthemaximumgraphdisconnection.Additionalconstraintsregardingthestructureofthecliquescanalsobeimposed,forinstance,upperorlowerboundsonthesizeofthecriticalcliques.NoticethatCCPcanbeseenasageneralizationofthecriticalvertexdetectionproblem,wheretheobjectiveistondcliquesinsteadvertices.Thecriticalvertexdetectionproblemisthenthecasewherethesizeofthecliquesislimitedtobeone.Figure 4-1 presentsanexampleofCCPovera9-vertexgraph,whereb=2.Figure 4-1 Adisplaystheoriginalgraph,andFigure 4-1 Btheoptimalsolutionwherethecliquesselectedarecoloredinblackandgray.AmongthedierentconnectivitymeasuresdescribedinChapter 1 ,wediscusstwo:thesizeofthelargestmaximally-connectedcomponentandthetotalnumberofconnectedvertexpairs.Adescriptionofthisobjectivesfollows:Beforepresentingtheobjectives,weneedtointroducethefollowingnotation.ForanysubsetNV,letEG(N)=E\(NN)bethesetofedgessothat,foreachedge 68

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AOriginalgraph BOptimalsolution Figure4-1. Examplefora9-nodegraph e2EG(N),bothendpointsofebelongtoN.WealsodeneGhNiasthesubgraphwithvertexsetNandedgesetEG(N).Weassumethattwoverticesi;j2NareconnectedoverGifthereexistatleastonepaththatconnectsiwithjinG.LetGbethesetofallmaximally-connectedcomponentsofG.Wedeneamaximally-connectedcomponentQ2GasasubsetQVsothateverypairofverticesi;j2QisconnectedoverGhQiandeveryvertexinVnQisdisconnectedfromallverticesinQ.Fromnow,onwewillrefertothemaximally-connectedcomponentsonlyascomponentsunlessadditionalclaricationisrequired.WedenethenumberofconnectedvertexpairsofacomponentQ2Qas)]TJ /F10 7.97 Tf 5.48 -4.38 Td[(jQj2.AcliqueinGisasetKVsuchthat(KK)E.LetTbethesetofallcliquesinGandbGTbeasetofsizeatmostbofcriticalcliquesinG.Furthermore,K=SK2bGKisthesubsetofverticesthatcompriseallthecliquesinbG.Finally,letGhVnKibetheresultinggraphafterthedeletionofthecriticalcliquesinbGandGhVnKithecorrespondingsetofremainingcomponents.Thedenitionofthetwoobjectivesfollows: 69

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Minimizethesizeofthelargestcomponent(LC):GivenanetworkG=(V;E)andanintegerb,wetrytondasetofatmostbcliquessuchthatafterremovingthesecliquesthesizeofthelargestcomponentisminimized:minmaxQ2GhVnKifjQjg (4{1)Minimizethetotalconnectedvertexpairs(VP):GivenagraphG=(V;E)andanintegerb,wetrytondasetofatmostbcliquessuchthatafterremovingthesecliquesthetotalconnectedvertexpairsleftisminimized:minXQ2GhVnKijQj(jQj)]TJ /F1 11.955 Tf 17.94 0 Td[(1)=2 (4{2) 4.2ComputationalComplexitySinceCCPcanbeseenasanextensionofthecriticalvertexdetectionproblem,whichisNP-hardforbothoftheobjectivesdescribedabove( Arulselvanetal. , 2009 ; Shenetal. , 2012 ),theintuitiondictatesthatCCPisalsoNP-hard,especially,takingintoaccountthatothercliquerelatedproblemsareNP-hardaswell( GareyandJohnson , 1990 ).Formally,thedecisionversionofCCP(denotedby)isgivenbyasimplenonemptygraphG=(V;E)andtwononnegativeintegerscandb.WeaskifthereexistsasetbGofatmostbdisjointcliquessothatagivenconnectivitymeasure(e.g.,thenumberofconnectedpairsorthesizeofthelargestcomponent)getsboundedbycifthecliquesinbGareremovedfromG.Ifweuseastheconnectivitymeasurethesizeofthelargestcomponent,wenametheresultingproblemCCP-LC,whereasweuseCCP-VPforthemeasurethataccountsforthetotalnumberofconnectedvertexpairs. Theorem4.1. ThedecisionversionofCCP-LCisNP-complete. Proof. Clearly,foranygivengraphG=(V;E),thedecisionversionofCCP-LCbelongstotheclassNP.Notethat,givenacollectionofcliquesbGTinG,identifyingthesizeof 70

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eachcomponentofGhVnKicanbedoneinpolynomialtimeusingabreadthrstalgorithm( HopcroftandTarjan , 1973 ).Further,theintractabilityofCCP-LCisprovedbyapolynomial-timereductionfromthepartitionintocliquesproblem,knowntobeNP-complete( GareyandJohnson , 1990 ).GivenasimplenonemptygraphG=(V;E)andanintegerq>0,thedecisionversionofthepartitionintocliques(denotedby)isdenedasfollows.IsitpossibletopartitionsetVintoatmostqdisjointcliques?ItcanbeeasilyarguedthatthepartitionintocliquesproblemisaparticularcaseofCCP-LC.Notethat,thereisacollectionbGofatmostb=qcliquessothateverycomponentleftinthenetworkhassizec=0(i.e.,GhVnKi=;),ifandonlyif,everyvertexi2VbelongstoK=SK2bGK.Inotherwords,graphGcanbepartitionedintocliques,ifandonlyif,thereisasetofcriticalcliquesthatwhenremovedthesizeofthelargestcomponentiszero.Thereby,thedecisionversionofCCP-LCisNP-complete. Theorem4.2. ThedecisionversionofCCP-VPisNP-complete. Proof. Similarly,foranygivengraphG=(V;E),thedecisionversionofCCP-VPbelongstotheclassNP.Notethat,givenacollectionofcliquesbGTinG,identifyingthenumberofconnectedvertexpairscanbedoneinpolynomialtimebysequentiallyexecutingabreadthrstsearchforeachvertexinVnK.Now,theintractabilityofCCP-VPisprovedbyapolynomial-timereductionfromthegraphk-colorabilityproblem,knowntobeNP-complete( GareyandJohnson , 1990 ).GivenasimplenonemptygraphG=(V;E)andanintegerk>0,thedecisionversionofthek-colorability(denotedby)isdenedasfollows.IsitpossibletocolortheverticesGwithatmostkcolorssothatanytwoadjacentverticesarecoloredwithadierentcolor?Theproposedtransformationisasfollows.Givenak-colorabilityinstancedenedoveragraphG=(V;E),wecanconstructthefollowinggraphH.Foreachvertexi2V,createacopydenotedbyi0.LetV0bethesetcomprisedofsuchvertexcopies.Furthermore,letFbethesetofedgesconnectingeachvertexi2Vwithitsrespective 71

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AOriginalgraphGfromthek-coloring BTransformedgraphHfortheCCP-VP Figure4-2. Exampleoftheproposedtransformation copyinV0andH=(V[V0;((VV)nE)[F).Figure 4-2 providesanexampleofa4-vertexgraphGanditscorrespondinggraphH.WearguethatgraphGisk-colorable,ifandonlyif,thereisacollectionbHofatmostb=kcliquessothatthetotalnumberofconnectedpairsinHhVnKiisc=0.Toprovenecessity,assumethereexistasetofkcliquesbHsuchthatthetotalnumberofconnectedvertexpairsHhVnKiiszero.ThecliquesofsetbHcanbeclassiedintothreegroups(twoofthosepossiblyempty).First,thecliquesthatarecomprisedofverticesonlyfromV,second,thecliquesthataregivenbyavertexinVanditscorrespondingcopyinV0,andthird,thecliquesthataresingletonverticesofV0.Itiseasytoseethat,ifthereisacriticalcliqueK=fi0gfori02V0(i.e.,acliqueofthethirdgroup)inthecurrentoptimalsolution,thereisanalternativeoptimalsolutionwhere,insteadofselectingthecliquegivenbyi0weusethesingletoncliquegivenbyi.Then,withtheexceptionoftheverticesinV0,wewillgivetheverticesofeachcliqueinbHthesamecolor,foratotalofkcolors.Wenowprovethatthegivencoloringisvalid.Ifavertexbelongstoacliqueofthesecondgroup,therewillnotbeothervertexinGwiththesamecolor.Moreover,notethatthecliquesfromtherstgroupformindependentsetsinG 72

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(i.e.,thoseverticesarenonadjacentinG).Thus,coloringtheverticesofeachcliqueinbHwiththesamecolorwillyieldavalidk-coloring.Toprovesuciency,assumethatgraphGisk-colorable.TheverticescoloredwitheachofthekcolorsformanindependentsetinGandeachindependentsetinGinducesacliqueinH.LetbHbethesetofthosekcliques.Notethat,whenthecliquesofbHareremovedfromH,theonlyverticesleftaretheonesofsetV0.SincebydenitiontheverticesofV0arenonadjacentinH,thetotalnumberofconnectedvertexpairsleftinHhVnKiiszero.ThisprovesthatthedecisionversionofCCP-VPisNP-complete. 4.3AggregatedFormulationsWerstintroducetwoaggregatedformulationsforsolvingboththeCCP-LCandtheCCP-VP.TheideabehindtheseformulationsistoconstructthecliquesofsetbG.Wedenevariablesx,y,andzasfollows:xKi=8><>:1,ifvertexi2VisincriticalcliqueK2bG0,otherwise,zi=8><>:1,ifvertexi2Vbelongstoanycriticalclique(i.e.,i2K)0,otherwise,andyij=8><>:1,ifverticesiandj2Vareconnected0,otherwise.Letv(e)bethesetofendpointsofedgee2EandbGbethesetofcriticalcliques(recallthatjbGj=b).AvalidmathematicalformulationfortheCCP-VPobjectivefollows:minXi;j2Vyij (4{3)s.t.xKi+xKj1;8e2(VVnE);i;j2v(e);K2bG (4{4) 73

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XK2bHxKi=zi;8i2V (4{5)yij+zi+zj1;8e2E;i;j2v(e) (4{6)yij+yjl)]TJ /F3 11.955 Tf 11.95 0 Td[(yil1;8i;j;l2V (4{7)yij)]TJ /F3 11.955 Tf 11.95 0 Td[(yjl+yil1;8i;j;l2V (4{8))]TJ /F3 11.955 Tf 11.95 0 Td[(yij+yjl+yil1;8i;j;l2V (4{9)xKi2f0;1g;8i2V;K2bG (4{10)zi2f0;1g;8i2V (4{11)yij2f0;1g;8i;j2V; (4{12)wheretheobjectivefunction( 4{3 )minimizesthesumofconnectedvertexpairs.Notethatyijisequalto1ifnodesiandjbelongtothesamecomponent.Thus,Pi;j2VyijisequivalenttoPQ2GhVnKijQj(jQj)]TJ /F1 11.955 Tf 18.37 0 Td[(1)=2.Constraint( 4{4 )ensuresthatifthereisnoedgee2Ebetweenverticesiandj(i.e.,e2(VVnE)),bothverticescannotbeassignedtothesameclique;constraint( 4{5 )ensuresthatifvertexibelongstoacriticalclique,itscorrespondingvariablezimustbeequaltoone;constraints( 4{6 )denetherelationshipbetweentheyvariablesandzvariables;constraints( 4{7 )-( 4{9 )denethetriansitiverelationshipofyvariables;andnally,constraints( 4{10 )-( 4{12 )denethedomainofthevariablesused.ForsolvingtheCCP-LC,wecanadaptformulation( 4{3 )-( 4{12 )byintroducinganewvariabledenedasthesizeofthelargestcomponent.Then,theproblemcanbeformulatedasfollows:min (4{13)s.t.( 4{4 )-( 4{12 )Xi2Vyij+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zi);8i2V; (4{14) 74

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whereobjectivefunction( 4{13 )combinedwithconstraints( 4{14 )enforcestheminimizationofthesizeofthelargestcomponent. Remark4. TheupperboundforthenumberofcliquesisenforcedbythesizeofsetbG.Notethat,thesetofconstraintsthatforcevariablesziandyijtobebinarycanberelaxed.ThisfactisshowninProposition 4.1 . Proposition4.1. WhensolvingCCP-VPorCCP-LCforanygivengraphG=(V;E)andanyintegerbusingformulations( 4{3 )-( 4{12 )and( 4{13 ){( 4{14 ),respectively,constraintszi2f0;1gandyij2f0;1gcanbereplacedreplacedbyzi0andyij0foralli;j2Vwithoutaectingtheoptimalsolutionoftheproblem. Proof. Clearly,sincevariablesxarebinary,constraints( 4{5 )forceztobebinaryaswell.Forthecaseoftheyvariables,werstconsiderthecaseoftheformulationforCCP-VP.Sincetheobjectiveofthisformulationminimizesthesumoftheyvariables,foranygivenpairofverticesi;j2V,variableyijwilltrytotakethesmallestpossiblevalue.Thus,ifzi+zj=0,thelargestvaluethatyijwilltakeisonebecauseofconstraints( 4{6 )-( 4{9 ).Now,forthecaseoftheCCP-LC,usingasimilarargument,itiseasytoseethattheyvariablesassociatedwiththeverticesofthelargestcomponentwillalwaysbeeitherzeroor1. Thereareseveralshortcomingsofformulations( 4{3 )-( 4{12 )and( 4{13 )-( 4{14 ).First,theseformulationsarehighlysymmetric.Notethat,anypermutationofanyfeasiblesetofcriticalcliquescanbeencodedasanalternativefeasiblesolutionhavingthesameobjective.Thisissueisparticularlyproblematicbecauseitcansignicantlyincreasethesizeoftheresultingbranch-and-boundtree,hinderingtheeectivenessofthesolutionalgorithm.Itistruethatitispossibletointroducesymmetry-breakingconstraints.However,basedonourexperience,theimprovementthatthoseadditionalconstraintsyieldisminor.Second,thelowerboundthatisobtainedwiththeseformulationsisveryloose.Thisisnotsurprisingasseveralotherformulationsthatuseindependent-setconstraintsforidentifyingcliques(cf.constraints( 4{4 ))areknowntobehavepoorly.Third,these 75

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formulationsarediculttoadaptforidentifyingothercriticalvertexsubsetsdierentthancliques.Asanalternativesolutionapproach,werstprovideadecompositionstrategyforsolvingtheCCP.Laterinthechapter,weprovidetwodierentformulationsthatproducebetterresults. 4.4DecompositionApproachforSolvingtheCCPInthissection,wepresentadecompositionapproachforsolvingtheCCP.TheproposedapproachisbasedonareductionfromtheCCPtoageneralizedcriticalvertexproblem(GCVP)bymeansofsolvingacliquepartitioningproblemrst.Wealsopresenttwoalgorithmsthatcanbeusedtoobtaincandidatecliquepartitions,aswellasaformulationfortheGCVPthatisusedtosolvetheresultingproblem.Thedecompositionstrategyproposedinthissectionisbasedonthefollowingtheorem: Theorem4.3. Thesetofcriticalcliquesofanyfeasiblesolution(x;y;z)belongstoatleastonecliquepartitionoftheoriginalgraphG. Proof. LetbHbethesetofdisjointcriticalcliquesrepresentedbysolution(x;y;z)andKbethesetofverticescomprisingsuchcliques.LetRbeanycliquepartitionoftheresidualgraphGhVnKi.NotethatR=bH[RisacliquepartitionofGasbHandRaretwodisjointsetsofcliquesthatcoveralltheverticesinG. Sinceeverysetofcriticalcliquescanbeassociatedwithacliquepartition,weproposetosolvetheCCPby:rst,generatingacliquepartition;second,collapsingeachcliqueofthegivenpartitionintoasinglevertexforminggraphG0;andthird,usinganexactorheuristicmethodforsolvingtheresultingGCVPoverG0(Algorithm 3 ). Algorithm3CCPCollapseAlgorithm(G) R generateacliquepartition G0 collapse(R) bH SolveGeneralizedCNP(G0) returnbH 76

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4.4.1ConstructingCliquePartitionsThemaincomponentofthisapproachisthewayinwhichthecliquepartitionisgenerated.Thisisbecause,inordertoobtainagoodsolution,wewouldliketogenerateacliquepartitioncontainingtheoptimalsetofcriticalcliques,oratleastagoodproxy.Weproposetoheuristicallygeneratecandidatecliquepartitions.Theideabehindourapproachisthat,ifwewanttogreedilyreducethenumberofpairwiseconnections,wecaneithereliminatealargeclique,oracliquewithalargedegree(i.e.,acliquewithmanyedgesemanatingfromit),weusetwodierentalgorithmsforpartitioningthenetworkfollowingthisanalysis.Therstapproachistouseasacliquepartitionthesolutionofamaximumedgecliquepartitionproblem(Max-ECP).TheMax-ECPproblemlooksforacliquepartitionthatmaximizesthenumberofedgeswithinthecliques.EventhoughtheMax-ECPisproventobeNP-hard,thereareseveralapproximationalgorithmstosolvethisproblem.Wedecidedtousethe2-approximationalgorithmproposedby Dessmarketal. ( 2006 )thatwecalledMaxECP(Algorithm 4 ).SinceAlgorithm 4 requiressolvingsequentiallyamaximumcliqueproblem,weusedthealgorithmproposedin Ostergard ( 2001 ).Notethatitisalsopossibletogetthecliquepartitioningbysolvingthecorrespondingmathematicalproblemsorbymeansofanyothertechnique,exactorheuristic. Algorithm4MaxECP(G) Dessmarketal. ( 2006 ) 1: R ; 2: G G 3: repeat 4: SelectthemaximumcliqueKinGhVnKi. 5: R K[R 6: G GhVnKi 7: untilG=; 8: returnR Forthesecondapproach,weproposetouseacliquepartitionbasedonthedegreeofthecliques.WeuseaheuristicthatgreedilyndsacliquewithalargedegreeinG 77

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(Algorithm 5 ).Oncewendthisclique,weremoveitfromthenetworkandcontinuefollowingthesameprocessuntilallthenodesareeliminated(Algorithm 6 ). Algorithm5GreedyGetClique(G) 1: K ; 2: G G 3: repeat 4: SelectvertexiwiththelargestdegreeinG. 5: K K[fig 6: G G\NG(i) 7: untilG=; 8: returnK Algorithm6MaxDegree(G) 1: R ; 2: V V 3: whileV6=;do 4: K GreedyGetClique(V) 5: R R[K 6: V VnK 7: endwhile 8: returnR 4.4.2CliqueCollapsingFirst,assumethatwehaveacliquepartitionR=fK1;K2;:::;Ktg.WecancollapseeachofthecliquesinRintoasinglevertex.LetV0bethesetofverticesrepresentingthecliquesandE0betheedgesconnectingtheverticesinV0.Thereexistsanedge(Ki;Kj)inE0ifthereexistsatleastoneedgeinEconnectingavertexinKiwithavertexinKj.LetG0=(V0;E0)bethegraphinducedbythecollapsedcliques.Figure 4-3 providesanexampleofthecliquecollapsing,givenacliquepartition.Inthisgure,thepartitioniscomprisedoffourcliquesdepictedinwhite,lightgray,darkgray,andblack.TheresultinggraphG0hasatotalof4vertices,oneforeachoftheoriginalcliques. 4.4.3CNPGeneralizationforSolvingtheCCPAssumingthatwehaveapartitionR,oncewehavethecollapsedgraphG0wecanobtainthesolutionoftheCCPbysolvingageneralizedcriticalvertexproblemproblem. 78

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ACliquePartition BCliqueCollapse Figure4-3. Exampleoftheclique-collapsealgorithm WewilldiscussonlythereformulationfortheCCP-VPcase,although,thisresultcanbetriviallyextendedfortheCCP-LC.Noticethat,ifwewanttocountthetotalnumberofconnectedvertexpairsinG0,weneedtotakeintoaccounttheconnectionsattheinteriorofeachvertexinV0(recallthatatthispoint,eachcliqueisnowrepresentedbyavertex),aswellastheconnectionassociatedwitheachedgeinE0.Forthesakeofclarity,weabusethenotationinthisformulationusingiandjwhenreferringtothecollapsedverticesinV0andbydeningxiasabinaryvariablethattakesthevalueofoneifcollapsedvertexiisremovedandzerootherwise.WithineachcliqueKi2R,thetotalnumberofconnectionsisgivenbypi=)]TJ /F10 7.97 Tf 5.48 -4.38 Td[(jKj2.Moreover,notethatifverticesiandjareconnectedinG0,thenumberofconnectedvertexpairsrepresentedbyedge(i;j)2E0isgivennowbytij=jKijjKjj.Thus,theformulationfortheGCVPfollows:minXi2V0pi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi)+Xi;j2V0tijyij (4{15)s.t.yij+xi+xj1;8i;j2E0 (4{16) 79

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yij+yjl)]TJ /F3 11.955 Tf 11.96 0 Td[(yli;18i;j;l2V0 (4{17)yij)]TJ /F3 11.955 Tf 11.96 0 Td[(yjl+yli1;8i;j;l2V0 (4{18))]TJ /F3 11.955 Tf 11.96 0 Td[(yij+yjl+yli1;8i;j;l2V0 (4{19)Xi2V0xib (4{20)xi2f0;1g;8i2V0 (4{21)yij2f0;1g;8i;j2V0 (4{22)whereobjective( 4{15 )accountsfortheminimizationofthetotalnumberofconnectedpairstakingintoaccounttheconnectionsattheinteriorofthecliques.Constraints( 4{16 )-( 4{22 )aredenedexactlyasin Arulselvanetal. ( 2009 ).Similarlyasforthecriticalvertexdetectionproblem,eventhoughthesegreedyapproachescanyieldgoodsolutions,itisalwayspossibletoprovideexamplesforwhichsuchalgorithmswouldfail.Forthisreason,weprovidetwoimprovedformulationsthatwesolveviabranch,price,andcut. 4.5DisaggregatedFormulationsThemaindierencewiththeaggregatedformulationspresentedinSection 4.3 werethecriticalcliquesareconstructedfromscratch,isthatthefollowingformulationswillratherselectthebcriticalcliquesfromthecompletesetofcliquesTinG.WewillfocusontheformulationfortheCCP-VPasitcanbeeasilyadaptedtosolvetheCCP-LCvariation.AllofthetechniquesdescribedbelowworkfortheCCP-LCaswell.Thesetupofthisformulationisasfollows.ForeachcliqueK2T,letaKibeaparameterthattakesthevalueofoneifvertexiisincliqueKandzero,otherwise.LetPstbethesetofallpathsbetweenverticessandtinG.Forthisformulation,letvariablexKbeabinaryvariablethattakesthevalueofoneifcliqueKisselectedtobecriticalandzero, 80

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otherwise.Then,theresultingformulationisasfollows:minXi;j2Vyij (4{23)s.t.XK2TxKb (4{24)XK2TaKixK=zi;8i2V (4{25)Xi2Pzi+yst1;8s;t2V;P2Pst (4{26)xKi2f0;1g;8i2V;K2T (4{27)yij0;8i;j2V (4{28)zi0;8i2V: (4{29)ContrarytotheaggregatedformulationswheretherestrictiononthemaximumnumberofcriticalcliqueswasgivenbythesizeofsetbG,inthedisaggregatedformulationsweenforcethisboundwithconstraint( 4{24 ).Constraints( 4{25 )ensurethatifvertexibelongstoacriticalclique,itscorrespondingvariablezimustbeequaltoone.Constraints( 4{26 )guaranteethatifthereexistapathbetweenverticessandtinwhichnointermediatevertexibelongstoacriticalclique,thecorrespondingvariableysttakesthevalueofone(werefertothisconstraintsaspathconstraints).Finally,constraints( 4{27 )-( 4{29 )denethedomainofthevariables.Inprinciple,wearenotrequiredtousevariablesz,asthosecanbereplacedinconstraints( 4{26 )bytheexpressionPK2TaKixKforeachi2V.However,aswewillpointoutinSection 4.5.1 ,thesevariableswillhelptoimprovetheprocessofsolvingthisproblem.Theadvantageofformulation( 4{23 )-( 4{29 )overformulation( 4{3 )-( 4{12 )isthefactthatthecliqueconguration,requiredbythefeasiblesolutions,isimplicitlyguaranteedbythedenitionofthecliquesinT.SincealltheelementsinTarecliques,wecandropthesomehowinecientcliqueconstraints( 4{4 ).Nonetheless,tofullydescribethisformulation 81

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weneedtogenerateT,whichcangrowverylargeasthesizeofGincreases.Moreover,constraints( 4{26 )aredenedoverthesetofpathsthatexistbetweeneverypairofverticesinGandthesesetscanalsogrowexponentiallylarge.BecauseofthepotentialsizeofsetsTandPstforalls;t2V,solvingthisformulationisratherchallenging.Todealwiththisdiculty,wesolvetheproblemviabranch,price,andcut( DesrosiersandLubbecke , 2010 ).Thatis,ateachnodeofthebranch-and-boundtree,wesolveamodiedversionoftheproblemdenedoveramanageablesubsetofcliquesT0TandpathsPst0foralls;t2V.Werefertothisformulationasthemasterproblem(MP).Inthebranch-price-and-cutalgorithmwegenerateadditionalcliquesviacolumngenerationifthecurrentsetofcliquesT0isnotsucienttodeclareoptimalityforMP.Then,ifthereexistsanypathconstraint( 4{26 )thatisbeingviolatedbythecurrentsolutionofMP,weseparatesuchconstraintandintroduceitbacktoMP.Finally,iftheoptimalsolutionofMPisnotintegral,webranchtoreducethesolutionspace,eliminatingundesiredfractionalsolutions.TheMPisasfollows:(MP):minXi;j2Vyij (4{30)s.t.XK2T0xKb (4{31)XK2T0aKixK=zi;8i2V (4{32)Xi2Pzi+yst1;8t;s2V;P2Pst0 (4{33)0xKi1;8i2V;K2T0 (4{34)yij0;8i;j2V (4{35)zi0;8i2V: (4{36)Inthemathematicalprogrammingparlance,wheneverasubsetofvariablesisinitiallydropped,likeinthebranch-and-pricecontext(Chapter 5 ),theresultingproblemisoftencalledtherestrictedmasterproblem.Thisisbecausetheabsenceofthesevariables 82

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reducesthesetofpossiblesolutions.Ontheotherhand,whenasubsetofconstraintsisdropped,theresultingproblemisnamedtherelaxedmasterproblem,asthenumberofsolutionsincrease.Note,however,thatinthecaseofaboveformulation,sincewearedroppingbothvariablesandconstraints,theresultingMPisnotarestrictionnorarelaxationoftheoriginalformulation.Beforestartingthecolumngenerationphase,werequiretoconstructaninitialsetofcliquesT0,forthisformulation,ndinganinitialfeasiblesolutionisrathersimple:anycollectionofbcliquesissucient.Forthisreason,wearenotrequiredtouseacomplexalgorithmtoinitializeT0,wecansimplybeginwiththesetofcliquesinducedbytheedgesinE.Thatis,thecliquesgivenbythepairsofendpointsofeachedge.Tointroducelargercliques,wealsoincludeacollectionofcliquesthatarefoundusingavariationofAlgorithm 5 wheretheinitialvertexisselectrandomly. 4.5.1NewCandidateCliquesGenerationAsmentionedbefore,whilesolvingMP,itpossiblethatwerequiretoupdatesetT0withnewcandidatecliquesinordertodeclareoptimality.ThisisnotaparticularissueoftheCCPasithappensoftenwithmostproblemsthataresolvedusingthistechnique( Barnhartetal. , 1998 ).TondtheoptimalsolutionofMP,weuseacolumngenerationapproachthatintroducesnewcliquestoT0,incasetheyareneededtodeclareoptimality.Hence,ateachiteration,wearerequiredtondwhetherthereexistsanewcliqueK2TnT0thatimprovesthecurrentsolution.Inotherwords,weaimtondavariablexKsuchthatitsreducedcostisnegative,or,fromthedualperspective,avariablesuchthatitscorrespondingdualconstraintisviolated.Atthisstageofthealgorithm,itistruethatinMPmanyofthepathconstraints( 4{26 )arerelaxed.Forthisreason,itispossiblethatthesolutionthatisproducedafterthenewcliquesareintroducedisnotfeasibleforthecompletesetofpathconstraints.Whatisgenerallydoneinthiskindofalgorithms,duringthecolumngenerationstage, 83

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istomomentarilydisregardthepossibleviolationsthatmayexistduetothepathconstraintsthatarecurrentlynotpresentinMP.Inotherwords,anewsetofcliqueswillbeintroducedtoT0untilthesolutionthatisobtainedisoptimalforthecurrentsubsetofpathconstraints( 4{33 )givenbyPst0foralls;t2V.Then,oncethereisnocandidatecliquethatcanimprovethecurrentsolution,wemovetothestagewherepossibleviolatedconstrainsareidentiedandincludedinMP.WewilldescribehowsuchconstraintscanbeidentiedinSection 4.5.3 ,butrstwedescribethecolumngenerationstage.Notethat,oneofthereasonswhythiscanbedonewhensolvingMPwithoutaectingthecolumngenerationstageisbecausethexvariablesarenotpresentinthepathconstraints.Therefore,thedualvariablesassociatedwiththepathconstraintsarenotrequiredtobeconsideredtogeneratethenewcliques.Thisisoneofthereasonswhythezvariablesandconstrains( 4{25 )areoriginallyintroducedinthisformulation.Thedualvariablesofsuchconstraints,alongwiththedualvariableofconstraint( 4{24 )willcarrytherequiredinformationtogeneratethenewcolumns.ConsiderthedualconstraintsthatarecomplementarytothexvariablesnotpresentinMP.Xi2VaKii!;8K2TnT; (4{37)whereiisthedualvariableassociatedwitheachconstraint( 4{25 )foreachi2Vand!isthedualvariableassociatedwith( 4{24 ).Notethat,ifthereexistacliqueK2TnT0forwhich( 4{37 )isviolatedbythecurrentsolution,i.e.,acliqueKforwhich!)]TJ /F9 11.955 Tf 11.96 8.96 Td[(Pi2Vakii<0,thenwecanincludesuchcliqueinT0.Thepricingsubproblemthatidentiesnewcliquesfollows:!)]TJ /F1 11.955 Tf 11.29 0 Td[(maxXi2Viai (4{38)s.t.ai+aj1;8e2(VVnE);i;j2v(e) (4{39)ai2f0;1g;8i2V (4{40) 84

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wheretheobjectivefunction( 4{38 )minimizesthereducedcostoftheclique,constraints( 4{39 )ensurethecliquestructureofthenewcandidates,andconstraints( 5{56 )denethedomainofthevariables.Formulation( 4{38 )-( 4{40 )correspondstoaformulationforsolvingmaximum-weightedcliqueproblem.Here,theweightofeachvertexi2Visdualvariablei.AssumethatWistheweightofthemaximum-weightedclique.Thus,ifW>!,thecorrespondingcliquecanbeincludedinT0.AsmentionedinSection 4.3 ,formulationsthatincludecliqueconstraints,suchas( 4{39 ),areofteninecientforsolvingcliquerelatedproblemsbecausetheyproducealooselowerbound.Incontrast,thereexistcombinatorialbranch-and-boundalgorithmsthatgenerallyoutperformmathematicalprogrammingformulationstondoptimalcliques.Toobtainnewcliquecandidatesweusethealgorithmproposedby Ostergard ( 2001 ).Solvingamaximum-weightedcliqueproblemisingeneraladiculttask.Thisisindeedawell-knowNP-hardproblemandobtainingoptimalsolutionscanbecomputationallychallenging.TakingintoaccountthatseveralinstancesofthisproblemmustbesolvedtodeclareoptimalityfortheMP,itisimportanttoidentifyecientwaystoreducethesolutiontimesrequiredtosolvethisproblem.Inthefollowingsection,wepresentanalgorithmthatusesthedualinformationofthemasterproblemtoprunethesubproblemgraphandthusreducethecomputationaleortofndingnewcliques. 4.5.2UsingtheDualInformationtoImprovetheSearchforNewColumnsAsmentionedintheprevioussection,thedualvariable!representsalowerboundontheweightsofthecandidatecliques.Forthisreason,itisnotrequiredtoanalyzeanycliquewithasmallerweightinG.Havingalowerboundontheweightsofthecliquesisaremarkablyusefulinformationthatcanbeusedtopruneunnecessaryvertices,thusreducingthesearchtime.Bythedenition,alltheverticesofacliquemustbeadjacent;hence,anupperboundonthepotentialweightofacliquethatcontainsvertexi2Visthesumofthevertexweightplustheweightofallofitsneighbors.Ifsuchpotentialweight 85

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issmallerthan!,thenvertexicanbeprunedfromG.Moreover,notethatwheneveravertexispruned,theweightofthepotentialcliquesofitsneighborsdecreases.Thus,wecanrepeatthesameprocessiterativelyuntilnoadditionalvertexifpruned.Algorithm 7 presentsaproceduretopruneunnecessaryverticesinGwhilesearchingfornewcandidatecliques.Thisalgorithmiterativelyeliminatesavertexifthesumofitsweightandtheweightofitsneighborsissmallerthan!.ThecomplexityofthisalgorithmispresentedinProposition 4.2 . Algorithm7VertexPruning(G;;!) 1: V V 2: G G 3: stop false 4: while:stopdo 5: stop true 6: SelectvertexiwiththelargestdegreeinG. 7: foralli2Vdo 8: ifi+Pj2NG(i)j!then 9: V Vnfig 10: stop false 11: UpdateG 12: endif 13: endfor 14: endwhile 15: returnG Proposition4.2. GivenagraphGandthevaluesofdualvariables!andi,foralli2V,theworst-caserunningtimeforAlgorithm 7 isO(n2). Proof. Theinnerforloopofthealgorithm(steps 7 to 13 )isexecutedinO(n)and,sincethealgorithmprunesintheworstcaseallofthevertices,theoverallrunningtimeofthealgorithmisO(n2). Notethatthisprocedurecanbeexecutedateachiterationofthecolumngenerationstage.Wefoundthisalgorithmtobeveryecientatreducingthesizeofthegraph,especiallyinlatterstagesofthealgorithm. 86

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4.5.3UpdatingtheSetofPathConstraintsAsmentionedbefore,thenumberofpathsbetweenanytwoverticesinagraphcanbeverylarge.Asaresult,solvingthisformulationmayrequireextensivecomputationaleortbecauseofthenumberofconstraintsthatthosepathsrepresentinMP.Inthissection,weshowhowtoseparateandaddthesetoconstraintstoMPinalazyfashion.ThelazygenerationofthepathconstraintsforCCPdiersfromtheonepresentedinChapter 3 inthat,whensolvingEBDP,weonlyintroducewhipinequalitieswheneveraninfeasibleintegersolutionisfound,whereashere,weaddpathconstraintsevenwhenfractionalsolutionsarefoundtobeinfeasible.Theprocedureisasfollows.Oncethecolumngenerationstageisover,i.e.,whenwehaveanoptimalsolutionfortheMPwiththecurrentsetofconstraintsgivenbyPst0,foralls;t2V,weproceedtocheckifthereexistsapathconstraintforanyP2PstnPst0,foralls;t2V,thatisviolated.Toidentifysuchviolatedconstraintswecansolvethefollowingsubproblemforeverypairofverticess;t2V.Let(x;y;z)beanoptimalsolutionforthecurrentMP.Wecreateadirectedgraph^G=(V;^E)whereset^Eiscomprisedoftwoarcs(i;j)and(j;i)withi;j2v(e)foreachedgee2E.Theproposedsubproblemfollows.(SP(s;t)):yst+zt+minX(i;j)2^Eziwij (4{41)s.t.Xfj:(i;j)2^Egwij)]TJ /F9 11.955 Tf 24.44 11.35 Td[(Xfj:(j;i)2^Egwji=8>>>><>>>>:1,i=s0,8i2Vnfs;tg-1,i=t (4{42)wi;j2f0;1g;8(i;j)2^E: (4{43)Iftheoptimalsolutionof( 4{41 )-( 4{43 )islessthanone,theconstraintassociatedwiththecorrespondingoptimalpathisbeingviolatedbythecurrentsolutionofMP.Hence,wecanaddsuchaconstrainttoMP.Clearly,theproposedseparationproblem 87

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( 4{41 )-( 4{43 )isashortest-pathproblem.Sincethisproblemmustbesolvedforeachpairs;t2V,inordertoimprovethecomputationaltimerequiredtosolveallO(n2)subproblems,wecaninsteadusethewell-knownFloyd-Warshallalgorithmdesignedforsolvingtheall-pairshortestpathproblem( Ahujaetal. , 1993 ).WekeeprunningthisalgorithmuntilnoadditionalpathconstraintsarerequiredtobeaddedtoMP. 4.5.4BranchingRuleOneofthemajordicultiesthatarisewhensolvingaproblemviabranchandpriceisdeningthebranchingrule.Therearemanyreasonswhystandardbranching(i.e.,xingafractionalvariabletoeitherzeroorone)isundesirableinthiscontext.Firstofall,whenxingavariabletoone,theverticesofthecliquecorrespondingtosuchvariableareselectedtobedeletedfromGand,sincethecriticalcliquesmustbedisjoint,alltheothercliquesinTcontainingatleastoneofthoseverticesarediscarded(i.e.,thecorrespondingvariablesarexedtozero).Hence,thenumberofvariablestobeconsideredinthatbranchissignicantlyreduced.Ontheotherhand,whenavariableisxedtozero,onlyoneofexponentiallymanyvariablesisxed.Thisparticularbehaviorresultsinahighlyunbalanced,andthusinecient,branchingtree.Furthermore,whenxingavariabletozero,wemustensurethatthepricingsubproblemdoesnotproducethesamevariableagain.Thisisgenerallydonebyeitherndingthenextbestsolution(possiblythenthbestafternbranches),orbyincludingadditionalconstraintsinthesubproblem.Inbothcases,solvingthepricingproblembecomesremarkablyharder.ForsolvingCCPweproposethefollowingalternative.ConsiderafractionalsolutionoftheMP.Fromconstraints( 4{25 ),itiseasytoseethatifasolutionisfractional,thereexistsavertexthatpartiallybelongstoatleasttwocliques.LetxK1andxK2betwofractionalvariables(i.e.,xK1;xK22(0;1))suchthatthecorrespondingcliquesshareavertexi.Clearly,sincecliquesK1andK2aredierent,thereexistsothervertexj6=ithatiseithercoveredbyK1andnotbyK2,orviceversa.Thus,wecangenerateabranchinwhichweforcebothverticesiandjtobeinthesamecriticalclique,andasecondone 88

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thatensuresthattheverticescannotbeinthesamecriticalclique.Itiseasytoseethatthecurrentfractionalsolutionisnolongervalid,whileanyfeasibleintegersolutionbelongstooneofthetwobranches.Thisbranchingprocedureisverysimilartothewell-knownRyan-Fosterbranchingrule( Barnhartetal. , 1998 ; RyanandFoster , 1981 ).However,themaindierenceisthattheRyan-Fosterisusedforproblemswithsetpartitioningconstraints(Chapter 5 ).Inthecontextofsetpartitioningallthevertexmustbecoveredbyexactlyonepartition,whereashere,notalltheverticesarerequiredtobepartofacriticalclique.Oncewegeneratethenewbranches,weneedtoenforcetherespectivebranchingdecisionsinthesubsequentiterations.InsteadofsequentiallyaddingsuchconstraintstotheMP,wecanalternativelypropagatethemoversetT0.Forthebranchthatrequiresbothverticestonotbeinthesamecriticalclique,wexto0allthevariablesofthecliquesthatcontainbothverticesFurthermore,inthegraphofthesubproblemforgeneratingnewcandidatecliques,wedeletetheedgeconnectingbothvertices.Conversely,forthebranchthatrequiresbothverticestobeinthesameclique,wexto0allthevariablesofthecliquesthatcontainonlyoneoftheseverticesandxthecorrespondingzvariablesofbothverticestoone. 4.6ComputationalExperimentsThissectionpresentsthecomputationalresultsobtainedaftersolvingCCPonatest-bedofinstancesthatincludesUniformRandomGraphs(URG),randompower-lawgraphsgeneratedusingtheBarabasi-Albert(BA)model( BarabasiandAlbert , 1999 ),andrandomsmall-worldgraphsgeneratedusingtheWatts-Strogatz(WS)model( WattsandStrogatz , 1998 ).ThecomputationalexperimentswereperformedonaserverwithtwoAMDOpteronTM6128Eight-CoreCPUsand12gigabytesofRAM,runningLinuxx86 64,CentOS5.9.AllalgorithmswereimplementedinCandSCIPOptimizer3.0.0( Bertholdetal. , 2012 )wasusedtosolvetheproposedformulation.Intheseexperiments,wefocused 89

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oureortsonsolvingtheCCP-VPwiththedisaggregatedformulationusingtheproposedbranch-price-and-cutscheme,asitwasthetechniquethatyieldedbetterresults.Thetest-bedofinstancesconsistedofuniformrandomgraphs,Barabasi-Albertgraphs,andWatts-Strogatzgraphswithf30;40;60;70;100gvertices.Forthegraphswith30vertices,wegeneratedinstanceswithf60;90;170gedges;forthegraphswith40vertices,theinstanceshadf80;120;240gedges;forthegraphswith60vertices,theinstanceshadf120;180;360gedges;forthegraphswith70vertices,theinstanceshadf140;210;420gedges;andforthegraphswith100vertices,theinstanceshadf200;300;600gedges.Forthe30-and40-vertexinstancesweselectedthecliquebudgetbfromf1;3;5gandfromf3;5;10gforthe60-,70,and100-vertexinstances.Foreachofthesevertex,edge,andbcongurations,wegenerated3dierentrandominstancesforatotalof405(i.e.,135instancesforeachoftheURG,BA,andWSversions).Wesetatimelimitof7200seconds. Table4-1. Computationaltimesandoptimalitygapsforthe30-and40-vertexinstances Time(s)Path-cutstime(s)Pricertime(s)Gap(%)b135135135135m 600.450.250.120.130.070.020.010.010.010.000.000.00BA-30900.470.350.170.070.090.040.010.000.000.000.000.001800.344.690.170.070.630.020.000.280.000.000.000.00601.269.223.210.231.161.120.030.060.030.000.000.00WS-30900.7813.975.330.121.451.260.010.090.030.000.000.001800.5459.120.820.054.140.120.010.330.010.000.000.00601.102.780.750.160.340.210.030.030.010.000.000.00URG-30900.7510.5413.780.101.062.540.020.050.060.000.000.001800.4044.7631.160.032.755.630.020.240.200.000.000.00800.921.450.480.230.380.210.020.010.000.000.000.00BA-401201.0912.414.160.271.721.220.030.030.030.000.000.002400.9485.2893.580.167.4415.490.010.940.770.000.000.00804.0520.41146.050.652.5325.290.040.090.520.000.000.00WS-401203.0428.22476.340.452.4948.060.040.091.010.000.000.002401.56171.62305.930.1213.5142.510.020.511.160.000.000.00801.1133.9450.870.173.9010.220.010.130.230.000.000.00URG-401201.7086.35833.540.297.2176.380.030.251.800.000.000.002401.56165.197200.000.1410.62463.590.040.5420.290.000.0058.37 90

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Table4-2. Computationaltimesandoptimalitygapsforthe60,70-and100-vertexinstances Time(s)Path-cutstime(s)Pricertime(s)Gap(%)b3510351035103510m 1207.0612.871.231.663.450.610.050.030.030.000.000.00BA-6018039.12260.534.863.9629.342.170.110.360.040.000.000.00360678.277200.0010.0834.36647.603.623.1026.360.080.0091.290.00120562.794606.324731.8053.73470.961913.401.217.0116.100.001.5614.78WS-60180754.195420.427200.0061.52403.402393.521.125.5521.750.007.1288.703601514.837200.003601.53106.43449.37220.531.978.2011.180.0055.7714.50120370.881852.48350.1835.07204.09156.830.842.931.180.000.000.00URG-601801621.826639.035605.56134.16444.731509.012.196.7713.600.009.7444.643602538.907200.007200.00205.28383.791268.333.387.7010.290.0023.871325.5214055.1277.952.729.5018.701.500.110.240.030.000.000.00BA-70210296.721539.7510.1231.35135.425.350.571.980.070.000.000.00420333.097200.007200.0025.05417.64551.451.0210.3820.470.0095.46100.00140814.077140.607200.0065.73723.512628.771.258.5722.480.0011.9951.03WS-702101339.677200.007200.0099.83448.911575.791.496.7012.660.0012.47264.624202984.887200.007200.00192.14395.691926.633.385.6314.600.0035.131767.951402806.856604.505024.14303.91554.211745.324.477.5712.850.0014.3821.91URG-702101440.037200.007200.00109.32419.391064.501.604.947.340.0014.83326.314204231.267200.007200.00369.87317.84709.325.434.135.460.0019.11829.202004788.07518.9454.04665.12310.0037.444.131.220.284.450.000.00BA-1003003761.631466.003757.99397.16528.91303.202.872.222.769.400.005.806007200.007200.007200.00341.99800.76766.134.466.206.1944.23727.28100.002007200.007200.007200.00431.901131.813740.773.276.7416.0927.30360.67100.10WS-1003007200.007200.007200.00298.79682.182853.152.264.5812.9021.63387.63381.576007200.007200.007200.00235.23660.202259.812.543.8710.2941.76891.3517735.402007200.007200.007200.00458.72966.043842.962.965.4118.5921.41403.5471.46URG-1003007200.007200.007200.00302.41282.321481.892.922.076.3916.63235.391211.736007200.007200.007200.00262.66165.99861.942.411.584.7814.00242.341964.55 91

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InTables 4-1 and 4-2 wepresenttheaveragetotaltimerequiredforthealgorithmtosolvetheproblem(Time(s)),theaveragetimespentbythealgorithmgeneratingnewpath-inequalities(Paths-cutstime(s)),theaveragetimespentbythealgorithmpricingnewcliquevariables(Pricertime(s)),andtheaverageoptimalitygapattheendoftherun(Gap(%)).AccordingtoTables 4-1 and 4-2 ,asthesizeoftheinstancesandthebudgetbincreases,theoptimalitygapalsoincreases.Thisbehaviorisparticularlyemphasizedwhenthedensityoftheinstanceishigh.ThisisduetothefactthatthesetofpossiblecliquesTissignicantlylarger.Anotherexplanationforthisbehavioristhatthenumberofalternativewaysofconstructingdisjointcriticalcliquesincreasesexponentiallywiththesizeofthegraph.ThissetofalternativesolutionsdecreasethelowerboundsproducedbyMPwhichinturn,increasethetimerequiredbythealgorithmtosolvetheproblem.Forsomeoftheinstances,particularlytheverylargeanddense,theoptimalitygapsissurprisinglyhigh.Thisbehaviorhappensmoreoftenwhensolvinginstanceswithalargecliquebudgetb.Onepossibleexplanationisthatsomeofthefractionalsolutionsconsistofalternativeoverlappingcriticalcliquesthatcoverlargeportionsofthegraph.Inconsequence,thelowerboundstendtobelow,whichincreasestheoverallcomplexityrequiredtoclosethegap.Inourobservations,thishappenswhentheinitiallowerboundproducedbyMPisclosetozero.Withrespecttotheexecutiontimesofthealgorithm,itisimportanttonoticethatthetotaltimespentgeneratingnewcolumnsisrelativelylowcomparedwiththetotalexecutiontime.Thisfactgivesevidencethattheproposedtechnique,whichusesthedualinformationtoreducethesizeofthesubproblemgraph,canhelptosignicantlyreducetheoverallcomputationaltime.Moreover,eventhough,thetimerequiredbythealgorithmtoidentifynewpathconstraintsislargerthantheoneneededforpricingvariables,thistimeisstilllowcomparedwiththetotalrunningtime. 92

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Onepossiblevariationthatweareinterestedtoanalyzeinthefutureistoremovetheauxiliaryvariableszi,foralli2Vbyreplacingthoseinconstraints( 4{26 )bytheexpressionsPK2TaKixK.Basedonourinitialexperiments,thisalternativeformulationproducesabetterlowerbound.Although,asmentionedbeforeinSection 4.5.1 ,thereexistatradeobetweentheimprovementofthelowerboundandthecomputationaltimerequiredtosolvethesubproblem. 93

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CHAPTER5MULTIDIMENSIONALASSIGNMENTFORMULATIONSFORSOLVINGDATAASSOCIATIONPROBLEMSInthischapterwepresentavariantofthemultidimensionalassignmentproblemwithdecomposablecostsforsolvingdataassociationproblemsinwhichtheresultingoptimalassignmentisdescribedasasetofdisjointstars.Themultidimensionalassignmentproblem(MAP),originallyintroducedby Pierskalla ( 1968 ),aimstominimizetheoverallcostofassignmentwhenmatchingelementsfromN=fN1;:::;Nng(n>2)disjointsetsofequalsizem.Itcomesasanaturalgeneralizationofthetwo-dimensionalAssignmentProblem(AP),knowntobepolynomiallysolvable( EdmondsandKarp , 1972 ; Kuhn , 1955 ).AmongallthedierentgeneralizationsoftheMAP,theoneconsideredinthispaperistheaxialMAP(hereafterreferredtoasMAP).InanaxialMAP,eachelementofeverysetmustbeassignedtoexactlyoneofmdisjointn-tuples,andeachn-tuplemustcontainexactlyoneelementofeachset.ContrarytotheAP,theMAPisknowntobeNP-hard( Karp , 2010 )forallvaluesofn>2. 5.1ProblemDenitionTheMAPisusuallypresentedasthefollowinginteger(0-1)programminXi12N1Xi2=N2Xin2Nnci1i2:::inxi1i2:::in (5{1)s:t:Xi22N2Xi32N3Xin2Nnxi1i2:::in=1;8i12N1 (5{2)Xi12N1Xis)]TJ /F12 5.978 Tf 5.76 0 Td[(12Ns)]TJ /F12 5.978 Tf 5.75 0 Td[(1Xis+12Ns+1Xin2Nnxi1i2:::in=1;8is2Ns;s=2;:::;n)]TJ /F1 11.955 Tf 11.95 0 Td[(1 (5{3)Xi12N1Xi22N2Xin)]TJ /F12 5.978 Tf 5.76 0 Td[(12Nn)]TJ /F12 5.978 Tf 5.75 0 Td[(1xi1i2:::in=1;8in2Nn (5{4)xi1i2:::in2f0;1g;8is2Ns;s=1:::n; (5{5)where,foreveryn-tuple(i1;i2;:::;in)2N1N2Nn,variablexi1i2:::intakesthevalueofoneifelementsofthegivenn-tuplebelongtothesameassignment,andzerootherwise.Thetotalassignmentcost( 5{1 )iscomputedasthecostofmatchingelementsfrom 94

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dierentsetstogether.Asanexample,anassignmentwhichselectselements(i1;i2;:::;in)tobegroupedtogetherwouldhaveacostofci1i2:::in. 5.1.1AssignmentCostsDependingonthedenitionoftheassignmentcosts,thereareseveralvariationsoftheMAPthatcanbeconsidered.Thesevariationsaremainlyassociatedwithcaseswheretheassignmentcostofeachn-tuplecanbedecomposedasafunctionofallpossiblepairwiseassignmentcostsbetweenelementsofdierentsets.Thatis,ci1i2:::in=f(ci1i2;:::;cinin)]TJ /F12 5.978 Tf 5.76 0 Td[(1),wheref:N1N2[[Nn)]TJ /F11 7.97 Tf 6.59 0 Td[(1Nn!Randcisitisthecostofassigningtogetherelementsis2Nsandit2Nt,fors6=t.Ingeneral,themainadvantageofhavingdecomposablecostfunctionsisthattheremaybewaysoftacklingtheproblemwithouthavingtocompletelyenumerateallofthedierentassignmentcosts,whichcanbeexponentiallymany.Moreover,mostoftheseMAPvariationscanbeassociatedwithaweightedn-partitegraph,inwhichtheelementsarerepresentedbytheverticesofthegraph,eachoftheedgesdescribesthedecisionofassigningtwoelementswithinthesamen-tuple,andtheweightsontheedgesaccountforthecorrespondingassignmentcosts.WeprovideadetailedexplanationofthisrepresentationinSection 5.2 .Basedontheapplicationsandthecontextoftheproblem,therearedierentdenitionsoftheMAPwithdecomposablecoststhatcanbefoundintheliterature( AnejaandPunnen , 1999 ; Bandeltetal. , 1994 ; Burkardetal. , 1996 ; CramaandSpieksma , 1992 ; KurokiandMatsui , 2009 ; Malhotraetal. , 1985 ).Inthispaperweconsiderthecasewhereeachn-tupleofanyfeasibleassignmentisassumedtoformastar( Bandeltetal. , 1994 ).Nonetheless,sincemostoftheextantliteratureconcentratesonthecasewherethen-tuplesformcliques,wealsoprovideabriefdescriptionofthelattertoemphasizethedierencesandenrichthediscussion.Forthecaseofthecliques,afeasibleassignmentincludesallpossiblepairwiseconnectionswithintheelementsofeachtupleandthus,thecostoftuple(i1;i2;:::;in)2 95

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N1N2Nnisdenedasthesumofallpairwiseassignmentcosts.Thatis,ci1i2:::in=nXs=1nXt=s+1cisit (5{6)Ontheotherhand,forthecaseofthestars,oneelementofeachtupleisassignedtobeacenter(orrepresentative)andtheotherelementsareconsideredtobetheleafs(orlegs)ofthestar.Notethat,contrarytothecaseofthecliques,eachtuplecangeneratemanydierentstarcongurations,dependingonwhichelementisselectedasthecenter.Assumingforexample,thatthecenteriselementis,thecostoftheinducedstaristhesumofthepairwisecostsbetweenisandtheotherelementsofthetuple.Inviewofthesemultiplepossiblecongurations,thecostoftuple(i1;i2;:::;in)2N1N2Nnisdenedastheminimumcostamongthecostsofallthepossiblestarcongurationsofthetuple.Thatis,ci1i2:::in=minis2fi1;i2;:::;ing8<:Xt2f1;2;:::;ngnfsgcisit9=; (5{7)WenametheaforementionedMAPversion,themultidimensionalstarassignmentproblem(MSAP),becauseoftheparticularstructurethateachfeasibleassignmenthas.Despitethefactthatthisvariantisoftenreferredtoasaparticularcaseofthecliqueversion( Bandeltetal. , 1994 ),weconsiderthatitisrelevanttostateitinaseparateform.Webaseourargumentonthefactthatthereexistapplicationsforwhichtheuseofthisvariantcouldbeofbenet.Moreover,thereareformulationsandtechniquesspecicallytailoredtosolvetheMSAP. 5.1.2PreviousWorksSeveralmethodologieshavebeenproposedtosolvedierentvariantsandgeneralizationsoftheMAP,includingexactapproaches,approximationalgorithms,heuristics,andmetaheuristics.Inparticular,giventheinherentNP-hardnessoftheproblem,heuristicapproacheshavegainedpracticalinterestovertheyears.Theseincludegreedyheuristics( BalasandSaltzman , 1991 ),generalizedrandomizedadaptivesearchprocedures(GRASP) 96

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( Murpheyetal. , 1999 ; RobertsonIII , 2001 ),GRASPwithpathrelinking( Aiexetal. , 2005 ),randomizedalgorithms( OliveiraandPardalos , 2004 ),geneticalgorithms( GaofengandLim , 2003 ),memeticalgorithms( KarapetyanandGutin , 2011b ),localsearchheuristics( Bandeltetal. , 2004 ; KarapetyanandGutin , 2011a ),simulatedannealing( Clemonsetal. , 2004 ),decompositionschemes( Vogiatzisetal. , 2014 ),Lagrangianbasedprocedures( BalasandSaltzman , 1991 ; FriezeandYadegar , 1981 ; PooreandRobertsonIII , 1997 ),andbranch-and-boundtechniques( Larsen , 2012 ; Pasiliaoetal. , 2005 ).Fromtheperspectiveofapproximationalgorithms,thereexisttheworksof CramaandSpieksma ( 1992 )and Bandeltetal. ( 1994 ).Furthermore,contributionstothestudyofthepolyhedralstructureoftheMAPformulationandothergeneralizationscanbefoundin Appaetal. ( 2006 ); BalasandSaltzman ( 1989 );and MagosandMourtos ( 2009 ).Finally,studiesrelatedtotheasymptoticbehavioroftheexpectedoptimalvalueoftheMAP,aswellastoolstoperformprobabilisticanalysisofMAPinstancesaregivenin Krokhmaletal. ( 2007 ), Grundeletal. ( 2004 ),and GutinandKarapetyan ( 2009 ).Amongalltheproposedtechniqueslistedabove,wenextfocusourattentiononapproachesthatareeitherproposedtotackletheMSAP,orthataredesignedtosolvegeneralizationsoftheMAP,andthuscanalsobeusedtosolvethisproblem.TosolvetheMSAP, CramaandSpieksma ( 1992 )introducedanapproximationalgorithmdesignedtosolvethethree-dimensionalcase(i.e.,n=3).Theproposedalgorithmconsistsofsequentiallysolvingtwolinearassignmentproblems.First,theelementsofsetN1areassignedtotheonesofsetN2andthen,theresultingpairsareassignedtotheelementsofsetN3.Theauthorsprovedthatifthepairwiseassignmentcostssatisfythetriangleinequality,theproposedalgorithmproducesa1 2approximation.Moreover,notingthatthisalgorithmcanproducethreedierentsolutionsbysimplyvaryingtheassignmentorderofthesets(e.g.,assigningrsttheelementsofN1totheonesofsetN3andthen,assigning 97

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theresultingpairstotheelementsofsetN2), CramaandSpieksma provedthatselectingthebestofthethreesolutionsyieldsa1 3approximation.Inasubsequentstudy, Bandeltetal. ( 1994 )proposedtwotypeofheuristics,namelythehubandtherecursiveheuristics.Theycanbeviewedasgeneralizationsoftheapproachproposedby CramaandSpieksma ( 1992 ),butdesignedtosolvethegeneraln-dimensionalcase.Theauthorsalsoprovideanupperboundontheratiobetweenthecostofthesolutionsproducedbytheseheuristicsandthecostoftheoptimalsolution.Asmentionedbefore,theMSAPisaparticularcaseoftheMAPandtherefore,itcanbesolvedusingformulation( 5{1 ){( 5{5 ).Thepolyhedralstudiesintroducedby BalasandSaltzman ( 1989 ),forthethree-dimensionalcase,andby Appaetal. ( 2006 )and MagosandMourtos ( 2009 ),foramoregeneralversionoftheMAP,canbeusedtoenhance( 5{1 )-( 5{5 )viatheintroductionofcuttingplanes.Usingformulation( 5{1 )-( 5{5 )tosolvetheMSAPhasonemaindrawback.Itrequiresthatallpossiblestarcostsbegeneratedbeforehand.Thiscouldbeproblematicbecausethetotalnumberofpossiblestarsgrowsexponentiallywiththesizeoftheproblem(Section 5.3 ).Tocircumventthisissue,itispossibletoembed( 5{1 )-( 5{5 )withinabranch-and-pricescheme(Section 5.3 ).Therefore,insteadofenumeratingallpossiblestarsfromthebeginning,thosearegeneratedviacolumngeneration,incasetheyareconsideredsuitable.Thedownsideofthisapproach,though,isthatmixingcuttingplanesandcolumngenerationisingeneraladiculttask( Barnhartetal. , 1998 ; Desaulniersetal. , 2011 ; LubbeckeandDesrosiers , 2005 ).ForadditionalinformationabouttheMAPanditsvariations,wereferthereadertothesurveysprovidedby Burkard ( 2002 ); BurkardandCela ( 1999 ); Burkardetal. ( 1998 ); GilbertandHofstra ( 1988 ); PardalosandPitsoulis ( 2000 ); Pentico ( 2007 ),and Spieksma ( 2000 ). 5.1.3DataAssociationApplicationsfortheMSAPThischapterisinspiredbythecontextofmulti-sensormulti-targettrackingproblems,thatinvolvetheassignmentofaseriesofsensorobservationsintoasetofdierenttargets. 98

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TherelationshipbetweentheseproblemsandtheMAP,hasbeenstatedandstudiedbymanyauthorsincluding Bandeltetal. ( 2004 ); Chummunetal. ( 2001 ); Debetal. ( 1993 , 1997 ); Moreeld ( 1977 ); Murpheyetal. ( 1999 ); Poore ( 1994 ),and Pusztaszerietal. ( 1996 )amongothers.Thereareseveralcontextsinwhichusingstarcostscanprovebenecialwhensolvingmulti-sensormulti-targettrackingproblems.Inparticular,whentheassignmentcostshavemetricproperties(i.e,nonnegativity,symmetry,andsubadditivity),itisinterestingtoseethatinsomecases,consideringallpairwisecostswithintheassignments(e.g.,thecliquecase)isnotnecessarytoobtainavalidsolution.ConsidertherstcasewedescribedinChapter 1 ,whereweaimtoidentifyasetoflandminesthatareplantedonaeld.Tondthelocationofthemines,adroneissenttoyovertheeldemittingasignal.Oncethesignalreacheseachofthemines,itbouncesbackandisanalyzedbythesensorsofthedrone.Afterthedronehasownovertheeldanumberoftimesanditssensorshavecollectedthesetofdierentsignals(severalofthoseassociatedwitheachofthemines),itispossibletocalculateasetofestimatedlocationswheretheminescouldbelocated.TheideabehindsolvingaMAPistoassociatethelocationsthatareclosetoeachother,whichwouldhelppinpointtheactualpositionsofthemines.Inthiscontext,theassignmentcostsrepresenttheEuclideandistancesbetweentheestimatedlocationsand,sincethecostssatisfythetriangleinequality,notallthecostsneedtobeconsideredtoobtainavalidassociation.ConsiderthesimplecasedepictedinFigure 5-1 A,wheretheverticesrepresentthreeestimatedlocationsthatwhereassignedtogether.Here,thecenterofthestarisclearlyvertex1.Noticethat,thecostofthelegs(i.e.,edges(1;2)and(1;3))alreadycontainsinformationaboutthecostofedge(2;3).Inthisexample,sinceweexpectthatcostc23satisesmaxfc12;c13gc23c12+c13,someinformationaboutc23isalreadyconsideredwithinthevalueofc12+c13.Insomesituations,forinstanceinthepresenceofafaulty 99

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sensor,addingc23couldcontributetoobtainawrongassignment,especiallywhenthisvalueisclosetoitsupperbound.Moreover,theexpectedpositionofthemineisoftenconsideredtobeaconcurrencypointassociatedwiththepositionsthatareinthesamen-tuple,generallythecentroid.Ontheotherhand,ifstarcostsareused,theregionaroundthecenterofthestarscanbeseenasthezonewheretheminesaremostlikelytobelocated.Underthetriangleinequalityassumption,itiseasytoprovethatmanyofthetriangleconcurrencypoints(e.g.,thecentroid,theincenter,orthecircumcenter)arealwaysclosertothecenterofthestar.Forinstance,observethelocationsofthecentroid(1)andtheincenter(2)ofFigure 5-1 B(thiscanalsobegeneralizedforn>3).Thisimpliesthat,iftheassignments(i.e.,then-tuples)producedbyboththecliqueorthestarversionsarethesame,thentheconclusionregardingthepositionsofthemineswouldbeverysimilar.Followingthisidea,lettingCMSAPandCMAPbetheoptimalassignmentcostsoftheMSAPandthecliqueMAP,respectively.Itcanbeseenthat,forthe3-dimensionalcase,theinequalityCMSAPCMAP2CMSAPholds.Thisiseasilygeneralizedforthen-dimensionalcasetoCMSAPCMAP(n)]TJ /F1 11.955 Tf 12.22 0 Td[(1)CMSAP,whichimpliesthat,incaseasolutionforthecliqueversionirrequired,theMSAPgeneratesbothupperandlowerboundsontheoptimalvalue. AEstimatedlocations BThepositionsofthecentroidandtheincenter Figure5-1. Exampleofthreeestimatedlocations 100

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Figure 5-2 representsacaseinwhichoneoftheobservationsofagivensensor(vertex(1,1))iswronglyassignedtoapositionfarawayfromtheotherobservations.Inthisexample,weassumethattherearetwominesintheeldandsixestimatedpositionscalculatedbythreedierentsensors.First,inFigure 5-2 B,notethat,iftherepresentativelocationsoftheminesareassumedtobethecentroidsofthecliques,thosepositionsarestronglybiasedbythewrongobservationandthen,arepositionedsignicantlyfartherawayfromanyoftheoriginalobservations.Thisparticularissueisdepictedincliquef(1;1);(1;2);(1;3)g.Ontheotherhand,inFigure 5-2 Citcanbeseenthatthiseectisreducedwhenusingastarassignment.Wemaketheremarkthatsimilarexamplescanalsocanbeconstructedtopointoutnegativeeectsofusingthestarcosts.Therefore,inthiscontextitisoftenconsideredsolvingtheproblemwithboththestarandthecliquecoststohavemoreinformationandobtainabetteranalysis.TheexamplesthatweprovidedrepresenttwocasesofmanyothersforwhichtheMSAParisesinthecontextofdataassociation.WewouldliketoemphasizethatthemodelsintroducedinthispaperareintendedtotacklemoregeneralinstancesoftheMSAPand,therefore,canbeusedtosolvenotonlytheseproblems,butamultitudeofothersaswell. 5.2AggregatedFormulationsforSolvingtheMSAPGivenacollectionN1;:::;NnofndisjointvertexsetsofequalsizemandthecorrespondingcollectionofpairwisearcsetsE12;:::En)]TJ /F11 7.97 Tf 6.59 0 Td[(1;n,whereEst=NsNtfors
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AAssignmentcostsofthesensorobservations BOptimalcliqueassignment COptimalstarassignment Figure5-2. Exampleofa3-sensor2-targettrackingproblem anypairofleafs.Then,theMSAPaimsforasetofmdisjointvalidstarsthatcoveralltheverticesinG.Anexampleofavalidstarassignmentofaninstance,wheren=4andm=3isgiveninFigure 5.2 .Thethreevalidstarsarecoloredgray,black,andwhite.TheMSAPisknowntobeNP-hard( CramaandSpieksma , 1992 ). 5.2.1ContinuousNonlinearFormulationLetvariableszandxbedenedasfollows:zsi=8><>:1,ifvertex(i;s)2Nisastarcenter0,otherwise, 102

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Figure5-3. Avalidstarassignmentforagraphwithn=4andm=3 andxstij=8><>:1,ifvertices(i;s)and(j;t)2Nbelongtothesamestar0,otherwise.Observethatxstij=xtsjiforall(i;s;j;t),andxssij=0,since(i;s;j;s)=2E.Theinitialformulationispresentedin( 5{8 )-( 5{15 ),wheretheobjectivefunction( 5{8 )aimstominimizetheoverallassignmentcost.Constraints( 5{9 )ensurethat,ifvertex(i;s)isacenter,itmustbeconnectedto(n)]TJ /F1 11.955 Tf 12.49 0 Td[(1)vertices.Conversely,ifitisnotacenter,thenitshouldbeconnectedtoonevertex.Constraints( 5{10 )-( 5{11 )guaranteethatifvertex(i;s)isacenter,itmustbeconnectedtoexactlyonevertexfromeachsetNt,forallt6=s.Further,constraints( 5{12 )enforcethatthereareexactlymstarsintheoptimalsolution.Nonlinearconstraints( 5{13 )guaranteethateachvertex(i;s)2Niseitherconnectedtoacenter,orisacenteritself,andhence,itcanonlybeconnectedtoleafs.Last,( 5{14 )-( 5{15 )denethedomainofvariableszandx.(INLP):minmXi=1mXj=1nXs=1Xft=1;:::;n:s
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mXi=1nXs=1zsi=m (5{12)zsi+mXj=1nXt=1xstijztj=1;8i=1;:::;ms=1;:::;n (5{13)zsi2f0;1g;8i=1;:::;ms=1;:::;n (5{14)xstij2f0;1g;8i=1;:::;ms=1;:::;n: (5{15)LetRNLPbethecontinuousrelaxationofformulation( 5{8 )-( 5{15 ).Thatis,bothzandxareallowedtotakefractionalvaluesbetweenzeroandone.WenowproceedtoprovethatinanyfeasiblesolutionofRNLP,allzvariablestakeintegervalues.Forthisproof,assumewehaveafeasiblesolution(z;x).Werefertovertex(i;s)aswhiteifzsi=0,blackifzsi=1,orgrayifzsi2(0;1). Lemma2. InanyfeasiblesolutionofRNLP,awhitevertexcanonlybeconnectedtoblackvertices. Proof. Letvertex(i;s)bewhite.From( 5{9 )and( 5{13 ),sincezsi=0,wehavethatmXj=1nXt=1xstij=1and,mXj=1nXt=1xstijztj=1:Thus,mXj=1nXt=1xstij=mXj=1nXt=1xstijztj;whichcanberewrittenasmXj=1Xft=1;:::;n:ztj=0gxstij+mXj=1Xft=1;:::;n:ztj>0gxstij==mXj=1Xft=1;:::;n:ztj=0gxstijztj+mXj=1Xft=1;:::;n:ztj>0gxstijztj=Xfj=1;:::;m:ztj>0gxstijztj: (5{16) 104

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Moreover,sincexstij2[0;1]andztj2[0;1],Xfj=1;:::;m:ztj>0gxstijXfj=1;:::;m:ztj>0gxstijztj:Hence,from( 5{16 )weobtainmXj=1Xft=1;:::;n:ztj=0gxstij=0;and (5{17)mXj=1Xft=1;:::;n:ztj>0gxstij=mXj=1Xft=1;:::;n:ztj>0gxstijztj: (5{18)Equations( 5{17 )and( 5{18 )implythattherearenoconnectionsbetweenwhitevertices,andwhiteandgrayvertices,respectively. Lemma3. InanyfeasiblesolutionofRNLP,ablackvertexcanonlybeconnectedtowhitevertices. Proof. Letvertex(i;s)beblack,i.e.,zsi=1.From( 5{9 ),and( 5{13 )respectively,wehavemXj=1nXt=1xstij=n)]TJ /F1 11.955 Tf 11.95 0 Td[(1;and (5{19)mXj=1nXt=1xstijztj=mXj=1nXft=1;:::;n:xstij=0gxstijztj+mXj=1Xft=1;:::;n:xstij>0gxstijztj=mXj=1Xft=1;:::;n:xstij>0gxstijztj=0 (5{20)Now,noticethatinordertosatisfy( 5{20 ),ifxstij>0,thenztj=0.This,inturn,impliesthatablackvertexcannotbeconnectedtootherblackvertices,orgrayvertices. BasedonLemmata 2 and 3 ,wecandeducethat,ifthereexistgrayvertices,thentheyareonlyconnectedtoothergrayvertices. Lemma4. Ifthereexistgrayverticesinafeasiblesolution,thenthenumberofthoseisamultipleofn. 105

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Proof. Assumethereexistkblackvertices.Then,eachofthoseverticesisconnectedto(n)]TJ /F1 11.955 Tf 12.67 0 Td[(1)whitevertices,overallcoveringknverticesinthegraph.Hence,theremustbeexactly(m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)ngrayverticesremainingtobecovered. Theorem5.1. InanyfeasiblesolutionoftheRNLP,allzvariablesarebinary(i.e.,therecannotexistgrayvertices). Proof. Assumeforacontradictionthatthereexistgrayvertices.First,notethatfromLemma 4 ,ifthereexistsagrayvertexinG,wecanassumewithoutlossofgeneralitythateveryvertexisalsogray.Thisisbecausewecanremoveallblackandwhiteverticesfromthegraph,andtheremaining(gray)verticesstillformann-partitegraph.Sinceallverticesaregray,wehavethatzsi2(0;1),forvertex(i;s).Further,constraint( 5{13 )canberewrittenaszsi+mXj=1nXft=1;:::;n:xstij>0gxstijztj=1: (5{21)Now,considerthefollowingexpressionzsi+mXj=1Xft=1;:::;n:xstij>0gztj: (5{22)Clearly,expression( 5{22 )mustbeeitherlessthan,equalto,orgreaterthan1.Thus,wedividetheproofintothesethreecases.Case1.Letusassumethatitislessthan1.Itiseasytoseethatzsi+mXj=1nXft=1;:::;n:xstij>0gztjzsi+mXj=1Xft=1;:::;n:xstij>0gxstijztj;whichimpliesthatzsi+mXj=1nXft=1;:::;n:xstij>0gxstijztj<1;contradicting( 5{21 ). 106

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Case2.Assume( 5{22 )isequaltoone.Tosatisfyconstraint( 5{21 ),allxstijthatarepositivemustbeequaltoone.Further,fromconstraint( 5{11 ),wegetthatvertex(i;s)isconnectedtoatleastonevertexfromeachsetNt,t6=s.Sinceallconnectionsemanatingfrom(i;s)satisfyxstij=1weobtainmXj=1nXft=1;:::;n:xstij>0gxstijn)]TJ /F1 11.955 Tf 11.95 0 Td[(1:Byassumptionvertex(i;s)isgray.Hence,sincezsi<1,wewouldviolateconstraint( 5{9 ).Case3.Considerthecasewhereexpression( 5{22 )isgreaterthanone.First,notethatzsi>0impliesthatvertex(i;s)ispartiallyassignedtobepartofastarcenteredatitself.Moreover,foreachvertex(j;t)suchthatxstij>0,sinceztj>0,vertex(i;s)isalsopartiallyassignedtobepartofastarcenteredat(j;t).Thus,expression( 5{22 )accountsforthenumberofstarsthatvertex(i;s)isassignedto,includingtheonecenteredatitself.Weprovedabovethatthisexpressioncannotbelessthanorequaltooneforanygrayvertex.Hence,thisimpliesthatallverticesinGareassociatedwithmorethanonestar.Byconstraint( 5{12 )thereareonlymstars.Also,thesumofthexvariablesofeachstarmustbeequalton)]TJ /F1 11.955 Tf 12.2 0 Td[(1.Thus,if( 5{22 )isgreaterthanoneforallverticesinG,wewouldviolateatleastoneoftheconstraintsin( 5{9 ).ThethreecontradictionsdescribedaboveimplythatinanyfeasiblesolutionoftheRNLPtherearenograyvertices. Contrarytothecaseofthezvariables,itiseasytoseethattherecanbefeasiblesolutionswheresomeofthexvariablesarefractional.ConsidertheexamplepresentedinFigure 5-4 .ItiseasytoseethatthefractionalsolutiondepictedinFigure 5-4 C,satisesallconstraintsin( 5{9 )-( 5{13 ).Moreover,suchsolutionisactuallyaconvexcombinationofintegralsolutionsdisplayedinFigures 5-4 Aand 5-4 B.Notethatweareabletoconstructthiscombinationbecausethestarcentersinbothintegralsolutions 5-4 Aand 5-4 Barethesame.BecauseoftheresultpresentedinTheorem 5.1 ,astrictlyconvexcombinationbetweentwointegralsolutionswithdierentstarcenterswouldyieldfractionalzvariables, 107

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andthereforeaninfeasiblesolution.Wenowformallyprovethatifthereexistsanoptimalsolutionwherethexarefractional,thereisalsoanalternativeintegralsolution. ASolution1 BSolution2 CFractionalsolution Figure5-4. Exampleofafractionalsolution Theorem5.2. IfRNLPisfeasible,thentherealwaysexistsanoptimalsolutionofRNLP,whereallxvariablestakeintegervalues. Proof. Theorem 5.1 impliesthat,inanyfeasiblesolution,thereareexactlymblackvertices.Withoutlossofgenerality,assumethatallofthoseverticesareelementsofthesamesetNs(Figure 5-4 ).Assumethereisanoptimalsolution(z;x),wherexisfractional.Observethatifwexz=zin( 5{8 ){( 5{15 ),theresultingformulationcanbedividedinton)]TJ /F1 11.955 Tf 12.54 0 Td[(1independentlinearassignmentproblemsbetweentheelementsinNs(theblackvertices)andtheelementsinNt;t6=s,respectively(thewhiteverticesateachset).Note,thatsincetherealwaysexistsanintegeroptimalsolutioninalinearassignmentproblem,wecandeducethatthereisalwaysanalternativeoptimalsolutionfortheRNLPwherexstij2f0;1g,foralltheedgesinG. AsaresultfromTheorems 5.1 and 5.2 ,constraints( 5{14 )and( 5{15 )canberelaxedinINLP,leavingthecontinuousnonlinearoptimizationproblemRNLP,henceforthreferredtoonlyasNLP.Tosolvethisformulationwecanuseanyavailableoptimizerthathandlesnonlinearprograms.Although,inspiteofhavingacontinuousformulation,ratherthananintegerone,nonlinearconstraints( 5{13 )stillposeadicultchallengebecauseoftheirnon-convexnature.Asanalternativeapproach,insteadofsolvingNLP,weproposeusingastandardlinearizationtechniquethatinvolvesintroducingadditionalvariablestoreplacethebilineartermsinconstraints( 5{13 ).Furthermore,fromthe 108

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resultspresentedinTheorems 5.1 and 5.2 ,wederiveadditionalvalidinequalitiesthatstrengthentheproposedlinearformulation.Adescriptionofthelinearizationandthevalidinequalitiesfollows. 5.2.2LinearizationThebilineartermsofconstraints( 5{13 )representthegreatestdicultyoftheNLPformulation.Thus,weapplyastandardlinearizationtechniquebyreplacingthosetermswithadditionalvariables(w).Unfortunately,byrelaxingthenonlinearconstraints,welosetheintegralitypropertiesdescribedinTheorems 5.1 and 5.2 .Hence,theresultingformulationisamixedintegerlinearprogram(MIP).Ontopofthat,thisreformulationcomeswithanoverheadofO(n2m2)variablesandconstraints.Theproposedreformulationfollows:(MIP):minmXi=1mXj=1nXs=1Xft=1;:::;n:s
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Finally,usingasimilarargumentasinTheorem 5.2 ,itiseasytoseethatintheaboveformulationwecanrelaxtheintegralityconstraintsofvariablesxandw. 5.2.3ValidInequalitiesInthissubsection,weintroducevefamiliesofvalidinequalitiesthatstrengthentheMIPformulation.Sincewenolongerhavethenonlinearsetofconstraints( 5{13 ),itispossible(andlikely)thatthelinearrelaxationofthisformulationproducesfractionalsolutions(i.e.,grayvertices).Tocopewiththisissue,wecanusetheresultsfromLemmata 2 and 3 todenethefollowinginequalities. Proposition5.1. Foranyedge(i;s;j;t)2E,theinequalityxstij2)]TJ /F3 11.955 Tf 11.95 0 Td[(zsi)]TJ /F3 11.955 Tf 11.96 0 Td[(ztjisvalid. Proof. Fromconstraints( 5{13 ),forvertices(i;s)and(j;t),weobtainzsi+xstijztj1andztj+xstijzsi1:Addingbothinequalitiesyields2zsi+ztj+xstij(zsi+ztj): (5{32)Furthermore,sincexstij2[0;1]andzsi2[0;1],wehavethatxstijzsixstij+zsi)]TJ /F1 11.955 Tf 11.96 0 Td[(1andxstijztjxstij+ztj)]TJ /F1 11.955 Tf 11.96 0 Td[(1:Addingupbothinequalities,resultsinxstij(zsi+ztj)2xstij+zsi+ztj)]TJ /F1 11.955 Tf 11.96 0 Td[(2: (5{33) 110

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Finally,from( 5{32 )and( 5{33 ),weobtainxstij2)]TJ /F3 11.955 Tf 11.95 0 Td[(zsi)]TJ /F3 11.955 Tf 11.96 0 Td[(ztj: (5{34) Inequality( 5{34 )comesfromthefactthatinanyfeasiblesolutionoftheMIPtherearenoconnectionsbetweenblackvertices(starcenters).Clearly,ifbothvariableszsiandztjtakethevalueofone,thecorrespondingvariablexstijcannotbepositive. Proposition5.2. Foranyedge(i;s;j;t)2E,theinequalityxstijzsi+ztjisvalid. Proof. FromLemma 2 ,theproposedinequality:xstijzsi+ztj (5{35)istriviallyderived. Similarlyasbefore(cf.Proposition 5.1 ),inanyfeasiblesolutionoftheMIPtherearenoconnectionsbetweenwhitevertices(leafs).Thus,ifbothvariableszsiandztjarezero,thecorrespondingvariablexstijmustalsobezero.Therearetwointerestingfeaturesofinequalities( 5{34 )and( 5{35 ).First,thesizeofbothfamiliesisO(n2m2),whichimpliesthatbothcanbedirectlyincludedintheMIPwithouttheneedofaseparationalgorithm.Second,asprovedinTheorem 5.3 ,ifconstraints( 5{34 )and( 5{35 )areincludedinformulation( 5{23 )-( 5{31 ),itispossibletorelaxconstraints( 5{28 ),andthereforeremovetheextrawvariables.Theresultingformulation,referredtoasMIPa,notonlyhaslessvariablesandconstraintsthanMIP,butalsoproducesabetterdualbound.(MIPa):minmXi=1mXj=1nXs=1Xft=1;:::;n:s
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s.t.( 5{9 )-( 5{12 ) (5{37)xstijzsi+ztj;8i;j=1;:::;ms;t=1;:::;n (5{38)xstij2)]TJ /F3 11.955 Tf 11.96 0 Td[(zsi)]TJ /F3 11.955 Tf 11.95 0 Td[(ztj;8i;j=1;:::;ms;t=1;:::;n (5{39)zsi2f0;1g;8i=1;:::;ms=1;:::;n (5{40)xstij2f0;1g;8i=1;:::;ms=1;:::;n: (5{41) Theorem5.3. MIPaisavalidformulationfortheMSAP. Proof. SinceMIPaincludesconstraints( 5{9 )-( 5{12 ),itsucestoshowthatanyoptimalsolutionofthisformulationsatisesthat,(1)therearenograyvertices(i.e.,zsi2f0;1g;8(i;s)2N)and(2)theonlyconnectionsthatcanexistarebetweenwhiteandblackvertices.Clearly,MIPacontainstheintegralityconstraintsforthezvariables.Thus,nograyverticeswouldexist.Furthermore,observethatconstraints( 5{38 )guaranteethatnoconnectionexistsbetweenwhitevertices.Hence,whiteverticescanonlybeconnectedtoblackones.Similarly,constraints( 5{39 )guaranteethatnoconnectionexistsbetweentwoblackvertices,henceblackverticescanonlybeconnectedtowhiteones. Wenowdescribethreeadditionalfamiliesofvalidinequalities. Proposition5.3. Foranyvertex(i;s)2N,theinequalityzsi+mXj=1nXt=1qstij1; (5{42)whereqstij2fxstij;ztjgisvalid. Proof. Fromnonlinearconstraints( 5{13 ),itiseasytoseethatanyintegralsolutionofMIPasatisesthefollowingequation:zsi+mXj=1nXt=1minfxstij;ztjg=1: 112

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Sincebydenition,qstijminfxstij;ztjg,allintegralfeasiblesolutionssatisfytheproposedinequality. Werefertothisfamilyastheminimumsuminequalities.Notethat,foreachvertex(i;s),thenumberofinequalitiesofthisfamilyisO(2mn).Thus,weproposetheseparationprocedurepresentedinAlgorithm 8 .Givenafractionalsolution(z;x),foreachvertex(i;s),wesearchfortheinequalitythatisviolatedthemost.Forthispurpose,foreach(j;t)suchthat(i;s;j;t)2E,weselectasqstijin( 5{42 )thevariableassociatedwiththeminimumvaluebetweenztjandxstij.Werepeatthisprocedureforall(i;s)2N,addingallviolatedinequalities. Algorithm8minSumSeparation(G;i;s;z;x) 1: sum=zsi 2: cut=zsi 3: forall(j;t)suchthat(i;s;j;t)2Edo 4: ifztj
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Figure5-5. ExampleofafractionalsolutionofthelinearrelaxationofMIPa Proposition5.4. Forany4-cycleinvolvingvertices(i;s);(j;t);(k;u),and(l;v),xstij+xtujk+xuvkl+xsvil2isavalidinequality. Proof. Assumethatthereisanintegralsolutionwherexstij+xtujk+xuvkl+xsvil>2.Hence,thereexistatleastthreearcsinthe4-cycleforwhichtherespectivexvariablesareone.Inturn,thisimpliesthatatleasttwoverticeshavetwoconnectionseach.From( 5{9 ),itcanbededucedthatthesetwoverticesneedtobeblack(sincetheyhavemorethanoneconnection).Hence,allthepossiblevertexcongurationscanbesummarizedwiththetwocases,showninFigure 5-6 .Thecasesthatarenotincludedinthisgurecorrespondtomirroredversionsoftheonesthatarepresented. ACase1 BCase2 Figure5-6. CyclecutexampleforProposition 5.4 Case1canbedismissedbecauseawhitevertexisconnectedtotwoblackvertices,contradicting( 5{9 ).Similarly,case2cannothappenbecauseitcontradictsthefactthattwoblackverticescanneverbeconnected(Proposition2). Werefertothisfamilyasthe4-cycleinequalities.SincethenumberofinequalitiesofthisfamilyisO(m4n4),wecanndallviolationsbytotalenumeration. 114

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Finally,observethetripletformedbyvertices(1;3);(2;1),and(2;2)inFigure 5-5 .Itiseasytoseethatallthreeverticescannotbeblackandsimultaneouslyhaveanyconnectionbetweenthem.WeanalyzethisinequalityinProposition 5.5 . Proposition5.5. Foranytripletofvertices(i;s);(j;t);(k;u),wherevertex(i;s)isthecenterofthetriplet,xstij+xsuik+zsi+ztj+zuk3isavalidinequality. Proof. Sincetherecannotbeanyconnectionbetweentwoblackverticesortwowhitevertices,itiseasytoseethattheonlyfeasibleoptionscanbesummarizedbytheonesdepictedinFigure 5-7 .Asbefore,wedonotincludemirroredcongurations,orthecaseswheretheinequalityistriviallysatised(e.g.,thecasewherealltheverticesarewhiteandtherearenoconnectionsbetweenthem).Notethatinallthiscases,ifwesumupthecorrespondingzvariablesofallthreevertices,andthexofthearcsconnectingthecenterwiththeleafs,theresultingsummationisalwayslessorequalthanthree. ACase1 BCase2 CCase3 Figure5-7. TripletcutexampleforProposition 5.5 Wedescribedtheaboveinequalityforthecasewherevertex(i;s)isthecenterofthetriplet.Although,notethatitispossibletoobtaintwoalternativeinequalitiesbychoosinganyoftheothertwoverticesasthecenter.Werefertothisfamilyasthetripletinequalitiesand,sincethetotalnumberofposibletripletsisO(n3m3),wecanndviolationsbytotalenumeration.Finally,notethatthisfamilyofinequalitiescanbeeasilygeneralizedfork-tuples,wherek>3.However,inpracticetheyarelesseectiveandhardertoevaluate,asthetotalnumberofk-tuplesifO(nkmk). 115

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5.3ASetPartitioningFormulationforSolvingtheMSAPFromformulation( 5{1 )-( 5{5 ),itiseasytoseethattheMSAPisaparticularcaseofthesetpartitioningproblem,andthuscanbeformulatedassuch.Thesetupofthisformulationisasfollows.LetKbethesetofallfeasiblestarsthatcanbeusedtopartitionG.Foreachstark2K,letakisbeaparameterthattakesthevalueofoneifstarkcoversvertex(i;s)andzero,otherwise.Letykbeabinaryvariablethattakesthevalueofoneifstarkisselectedandzero,otherwise.Further,let(k)beafunctionthatreturnsthecenterofstark,andnally,letckbethecostofstark.Thatis,assumingthat(k)=(i;s),thecostofstarkisgivenby,ck=mXj=1Xft=1;:::;n:t6=sgcstijakjt: (5{43)Then,theresultingsetpartitioning(SP)formulationcanbedenedasfollows:(SP):minXk2Kckyk (5{44)s.t.Xk2Kakisyk=1;8i=1;:::;ms=1;:::;n (5{45)yk2f0;1g;8k2K: (5{46)Notethatconstraintset( 5{45 )iscreatedbyaggregatingallconstraintsin( 5{1 )-( 5{5 ).Hence,bothformulationsareequivalent.TheadvantageoftheSPoverMIPaisthefactthatthestarconguration,requiredbyanyfeasiblesolutionoftheMSAP,isimplicitlyguaranteedbythedenitionofsetK.Inotherwords,sincealltheelementsinKarevalidstars,theremainingdecisionisndingtherightpartition.However,thedownsideisthatjKjgrowsexponentiallyasthesizeofGincreases(Proposition 5.6 ).Forthisreason,( 5{44 )-( 5{46 )hasfarmorevariablesthancanbereasonablyhandledandthereforesolvingitexplicitlycanbecomputationallyimpractical. Proposition5.6. Thetotalnumberoffeasiblestarsisnmn. 116

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Proof. Tobeafeasiblestar,itscentermustbeconnectedtoexactlyonevertexfromeachoftheothern)]TJ /F1 11.955 Tf 12.78 0 Td[(1sets.Sincetherearemverticesineachset,thetotalpossiblestarscenteredatanygivenvertexismn)]TJ /F11 7.97 Tf 6.59 0 Td[(1.Furthermore,sincethetotalnumberofverticesisnmandeachofthemcanbethecenterofastar,thetotalnumberofstarsisnmn. Toovercomethisdiculty,wesolvetheproblemviabranchandprice( Barnhartetal. , 1998 ; LubbeckeandDesrosiers , 2005 ).Thatis,ateachvertexofthebranch-and-boundtree,wesolvealinearrelaxationofSP,denedoveramanageablesubsetofstarsK0K.Werefertothisformulationastherestrictedmasterproblem(RMP).IfthecurrentsetofstarsK0isnotsucienttodeclareoptimality,weproceedtogenerateadditionalstarsviacolumngeneration.Furthermore,iftheoptimalsolutionattheendofthecolumngenerationphaseisnotintegral,webranchtoreducethesolutionspace,eliminatingundesiredfractionalsolutions.TheRMPisasfollows:(RMP):minXk2K0ckyk (5{47)s.t.Xk2K0akisyk=1;8i=1;:::;ms=1;:::;n (5{48)0yk1;8k2K0; (5{49) 5.3.1GeneratingtheInitialSetofColumnsBeforestartingthecolumngenerationphase,werequiretoconstructaninitialsetofstarsK0forwhichRMPisfeasible.Therearedierentoptionsforconstructingthisset.Onecan,forexample,useatypicalinitializationmethodsuchasthetwo-phaseorthepenalizationmethods( Bazaraaetal. , 2009 ).Alternatively,wecaninitializeK0withanyfeasiblesolutionproducedheuristically.ThisisingeneralapreferredoptionbecauseagoodinitialsolutionprovidesRMPwithahighqualityupperbound,whichinturnmayhelpprunesomeunnecessarybranchesinthebranch-and-boundtree.TondtheinitialsetK0,weproposeaverysimplegreedyalgorithmthatgeneratesiterativelyasetofvalidstars.Forthispurpose,wedeneafeasiblestarptobeaminimumcoststarofG,if 117

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cpck,forallk2K.AssumewebeginthealgorithmwithanemptysetK0.Then,wesequentiallyaddtoK0aminimumcoststarofgraphGnK0.Werepeatthisprocessuntilalltheverticesarecovered.Afulldescriptionofthisheuristicispresentedinalgorithms 9 11 . Algorithm9minCenteredStar(G;m;n;i;s) 1: S f(i;s)g 2: cost 0 3: forallt=1;:::;n,wheret6=sdo 4: l2argminj=1;:::;mfcstijg 5: S S[f(l;t)g 6: cost cost+cstil 7: endfor 8: return[S;cost] Algorithm10minStar(G;m;n) 1: S ; 2: cost 1 3: foralls=1;:::;ndo 4: foralli=1;:::;mdo 5: [S;cost] minCenteredStar(G;m;n;i;s) 6: ifcost
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Algorithm11findInitialSet(G) 1: K0 ; 2: G0 G 3: cost=0 4: m0=m 5: whileG06=;do 6: [S;cost]=minStar(G0;m0;n) 7: K0 S 8: cost cost+cost 9: G0 G0nS 10: mp=m0)]TJ /F1 11.955 Tf 11.95 0 Td[(1 11: endwhile j2argminj=1;:::;mfcstijg,fromeachoftheremainingn)]TJ /F1 11.955 Tf 12.2 0 Td[(1sets.Furthermore,eachsethasintotalmvertices.Thus,ndingtheminimumcoststarcenteredat(i;s)takesO(nm)(Algorithm 9 ).Now,observethatsincetherearenmverticesinG,wecanrepeatthesameprocessforeachvertex,selectingtheoverallminimumcoststarinO(m2n2)(Algorithm 10 ). Theorem5.4. GivenagraphG,theproposedheuristicndsafeasiblesetK0inO(m3n2). Proof. WeinitializeAlgorithm 11 withagraphG0=G.First,notethatateachiterationofthewhileloop(steps 5 11 ),wegenerateavalidstarcomprisedbyexactlyonevertexfromeachsetN1;:::;Nn.SinceweremovefromG0theverticesofsuchstar,weguaranteethatthoseverticesarenotcoveredbyotherstarsinfurtheriterations.Moreover,ateachiteration,thenumberofverticesinG0decreasesbyn.InviewofG0havinginitiallymnvertices,byiterationmalltheverticesmustbecoveredbyonestar.Finally,ndingtheminimumcoststartakesO(m2n2).Thus,generatingK0takesO(m3n2). Naturally,thereareadditionalelementsthatcanbeusedforimprovingtherunningtimeofAlgorithm 11 andobtainingadditionalstarstoaddinK0.Forexample,aftereachrunofAlgorithm 9 ,wecantemporarilystoretheminimumcoststarcenteredateachvertex(i;s)2G.Then,infurtheriterationswecanwarmstartthealgorithminitializingthesearchwiththosestars.Moreover,notingthateachoftheseminimumcoststarsisa 119

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memberofK,wecanincludesomeofthemintheinitialsetK0.However,sincewewouldliketoavoidhavingalargeinitialsetofcolumns,orworseincludingpoorcandidates,inpracticeweimposealimitonthesizeoftheinitialsetK0.Thus,besidesaddingthesolutionfoundbyAlgorithm 11 ,wekeepanorderedsetofthebestmgeneratedstarsandincludethoseinK0.Wehavetesteddierentvaluesfor,being=f2;3;4gtheonesthatproducedthebestresults.Weonlyusethislimitwhengeneratingtheinitialsetofcolumns.Onceinthecolumngenerationstage,werelaxthislimit. 5.3.2OptimalityConditionsandNewCandidateStarsGenerationWhilesolvingRMP,itispossible(andingeneralcommon)thattheinitialsetofstarsK0doesnotincludeanoptimalpartitionofG.Moreover,itisalsopossiblethatwecannotevenndanoptimalsolutionofthelinearrelaxation,usingthestarsinK0.ThisisnotaparticularissueoftheMSAPasithappensoftenwithmostproblemsthataresolvedusingsetpartitioningformulations( Barnhartetal. , 1998 ).Tondtheoptimalsolutionofthelinearrelaxation,weuseacolumngenerationapproachthatintroducesnewstarstoK0incasetheyareneededtodeclareoptimality.Hence,ateachiteration,wearerequiredtondwhetherthereexistsanewstark2KnK0,thatimprovesthecurrentsolution.Inotherwords,weaimtondavariableyk,fork2KnK0,suchthatitsreducedcostisnegative,or,fromthedualperspective,avariablesuchthatitscorrespondingdualconstraintisviolated.ConsiderthedualoftheRMP,denedastherelaxeddualproblemRDP.(RDP):maxmXi=1nXs=1si (5{50)s.t.mXi=1nXs=1akissick;8k2K0: (5{51)Noticethat( 5{51 )includesonlytheconstraintsassociatedwiththestarsinK0.Thisimpliesthat,ifwendastark2KnK0,suchthatitscorrespondingconstraint( 5{51 )isviolatedbythecurrentsolution,i.e.,astarkforwhichck)]TJ /F9 11.955 Tf 12.81 8.97 Td[(Pmi=1Pns=1akissi<0, 120

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wecanincludesuchstarinK0.Inconsequence,theoptimalityconditionsofthecolumngenerationstagecanbedenedasfollows.ck)]TJ /F5 7.97 Tf 16.84 14.95 Td[(mXj=1nXt=1akjttj0;8k2KnK0; (5{52)or,alternatively,from( 5{43 ),mXj=1Xft=1;:::;n:t6=sgcstijakjt)]TJ /F5 7.97 Tf 16.85 14.95 Td[(mXj=1nXt=1akjttj=mXj=1Xft=1;:::;n:t6=sgcstijakjt)]TJ /F3 11.955 Tf 11.96 0 Td[(si)]TJ /F5 7.97 Tf 16.85 14.95 Td[(mXj=1Xft=1;:::;n:t6=sgakjttj==)]TJ /F3 11.955 Tf 11.96 0 Td[(si+mXj=1Xft=1;:::;n:t6=sg(cstij)]TJ /F3 11.955 Tf 11.96 0 Td[(tj)akjt0;(i;s)=(k);8k2KnK0: (5{53)Toidentifypromisingstars,weproposeasimilarapproachtotheoneusedtoobtaintheinitialsetK0.Werstxthecenterofthestaratvertex(i;s)andthen,searchforstarsthatviolatecondition( 5{53 ).Werepeatthesameprocessforeachvertex(i;s)2G.Thus,givenacentervertex(i;s),theproposedpricingsubproblem(P(i;s))isasfollows:(P(i;s)):)]TJ /F3 11.955 Tf 9.3 0 Td[(si+minmXj=1Xft=1:::;n:t6=sg(cstij)]TJ /F3 11.955 Tf 11.96 0 Td[(tj)ajt (5{54)s.t.mXj=1ajt=1;8t=1;:::;n;t6=s (5{55)ajt2f0;1g;8j=1;:::m;t=1;:::;n;t6=s; (5{56)wheretheobjectivefunction( 5{54 )minimizesthereducedcostofthestar,constraints( 5{55 )guaranteethatthecentervertex(i;s)isconnectedwithavertexineachlayert6=s,andconstraints( 5{56 )denethedomainofthevariables.Then,wesolveaproblemP(i;s)foreachvertex(i;s)2G,ndingnewcandidatestars(possiblymany)ateachiteration.Notethat,ifforallvertices(i;s)2G,)]TJ /F3 11.955 Tf 9.29 0 Td[(si+minPmj=1Pnt=1;t6=s(cstij)]TJ /F3 11.955 Tf 11.95 0 Td[(tj)ajt0,wehaveanoptimalsolutionfortheRMP.Inprinciple,insteadofsolvingasequenceofmnP(i;s)subproblems,wecouldadaptAlgorithm 10 toobtainthesameresultinO(m2n2).Thisapproachwillindeedworkforsolvingtherootvertexofthebranch-and-boundtree.However,ifthesolutionofthelinear 121

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relaxationisnotintegral,wemustproceedwiththebranchingstagetoeliminatefractionalsolutions.Then,thebranchingdecisionsmustbeenforcedinthesubproblemintheformofconstraintstopreventgeneratingstarsthatviolatethem.Becauseoftheseadditionalconstraints,wecannotsolelyapplyAlgorithm 10 .Ontheotherhand,P(i;s)canbeeasilyupdatedtoincludethebranchingconstraints.Nonetheless,solvingsequentiallymnofsuchintegerprograms,afterincludingthebranchingconstraints,couldpotentiallyhinderthecolumngenerationprocedure.TocircumventthisissuewewillshowthatitispossibletoreformulateP(i;s)asashortestpathproblemwithsideconstraints.Beforedescribingtheproposedreformulation,werstintroducethebranchingrule. 5.3.3BranchingRuleAsmentionedinChapter 4 ,whensolvingaproblemviabranchandprice,deningthebranchingruleisoneofthemostimportanttasks.SimilarlyasfortheCCP,therearemanyreasonswhystandardbranching(i.e.,xingafractionalvariabletoeitherzeroorone)isundesirableforsolvingtheMSAP.Firstofall,whenxingavariabletoone,thenverticesofthestarcorrespondingtosuchvariablearecovered.Sincethoseverticescannotbecoveredagain,alltheotherstarsinKcontainingatleastoneofthoseverticesarediscarded(i.e.,thecorrespondingvariablesarexedtozero).Hence,thenumberofvariablestobeconsideredinthatbranchissignicantlyreduced.Ontheotherhand,whenavariableisxedtozero,onlyoneofexponentiallymanyvariablesisxed.Thisparticularbehaviorresultsinahighlyunbalanced,andthusinecient,branchingtree.Furthermore,whenxingavariabletozero,wemustensurethatthepricingsubproblemdoesnotproducethesamevariableagain.Thisisgenerallydonebyeitherndingthenextbestsolution(possiblythenthbestafternbranches),orbyincludingadditionalconstraintsinthesubproblem.Inbothcases,solvingthepricingproblembecomesremarkablyharder.Analternativeoption,thatiswidelyusedwhensolvingsetpartitioningproblems,istheso-calledRyan-Fosterbranchingrule( Barnhartetal. , 1998 ; RyanandFoster , 1981 ). 122

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ConsiderafractionalsolutionoftheRMP.Fromconstraints( 5{48 ),itiseasytoseethatifasolutionisfractional,thereexistsavertexthatispartiallycoveredbyatleasttwostars.Letykandylbetwofractionalvariables(i.e.,yk;yl2(0;1))suchthatthecorrespondingstarssharevertex(i;s).Clearly,sincestarskandlaredierent,thereexistsavertex(j;t)6=(i;s)thatiseithercoveredbykandnotbyl,orviceversa.Thus,wecangenerateabranchinwhichweforcebothvertices(i;s)and(j;t)tobecoveredbythesamestar,andasecondonethatensuresthattheverticesarecoveredbydierentstars.Itiseasytoseethatthecurrentfractionalsolutionisnolongervalid,whileanyfeasibleintegersolutionbelongstooneofthetwobranches.Oncewegeneratethenewbranches,weneedtoenforcetherespectivebranchingdecisionsinthesubsequentiterations.InsteadofsequentiallyaddingsuchconstraintstotheRMP,wecanalternativelypropagatethemoversetK0.Forthebranchthatrequiresbothverticestobeindierentstars,wexto0allthevariablesofthestarsthatcoverbothverticessimultaneously.Conversely,wexto0allthevariablesofthestarsthatcoveronlyoneofthesevertices,forthebranchthatrequiresbothverticestobetogether.Moreover,wemustensurethatthepricingproblemalsosatisesthebranchingconstraints.IfweuseformulationP(i;s)asthepricingproblem,theseconstraintsareeasilyenforcedasfollows.Assumethat(i;s)and(j;t)aretheverticesofthecurrentbranchingdecision.IfthesubproblemwearesolvingisP(i;s),wexajttobeeitheroneorzeroforthecaseswherebothverticesarerequiredtobetogetherorseparated,respectively.ForP(j;t),theprocessissimilarbutxingaisinstead.Ifwearesolvingthesubproblemforotherverticesdierentthan(i;s)or(j;t),weintroduceconstraintsais=ajtorais+ajt1ineachofthebranches,respectively. 5.3.4StarGenerationasaShortestPathProblemwithSideConstraintsIngeneral,thequalityofanycolumngenerationapproachreliesontheabilitytogeneratenewcandidatevariablesinanecientway.Thisisbecausethesubproblem,thateitherdeclaresoptimalityorgeneratesnewvariables,maybesolvedaconsiderablenumber 123

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oftimesbeforeterminating.Despitethefactthatsolvingthesubproblemviaintegerprogrammingissucientfortacklingsmallinstances,inpractice,theuseofspecializedalgorithmsisproventoyieldbetterresults( Feilletetal. , 2004 ).Furthermore,whenpossible,itisoftendesiredtotransformthesubproblemintoashortestpathproblembecauseofthevarietyofdierentsolutiontechniquesthatcanbeusedtoecientlysolvethiskindofproblems( Feilletetal. , 2004 ; IrnichandDesaulniers , 2005 ; ValeriodeCarvalho , 1999 ).Forthisreason,asanalternativetosolvingP(i;s)usingintegerprogrammingtechniques,weproposereformulatingit,asashortestpathproblemoveradirectedacyclicnetworkH(i;s)=(N0;A).Thesetupforthisnetworkisasfollows.First,weintroducetwovertices( i;s)and(i;s )referredtoasthesourceandsinkvertices,respectively.Thereisasetofvertices,N,comprisedofonevertexassociatedwitheachsetN1:::;Ns)]TJ /F11 7.97 Tf 6.59 0 Td[(1;Ns+1;:::;Nn)]TJ /F11 7.97 Tf 6.59 0 Td[(1(i.e.,N=f1;:::;s)]TJ /F1 11.955 Tf 12.53 0 Td[(1;s+1;:::;n)]TJ /F1 11.955 Tf 12.52 0 Td[(1g).ThevertexsetisdenedasN0=f( i;s);(i;s )g[N[NnNs,andthearcsetAisconstructedasfollows.First,ifs=1,thenthereisanarcbetweenthesourcevertex( i;s)andvertex(j;2),forallj=1;:::;m.Otherwise,ifs>1,thereisanarcbetweenthesourcevertex( i;s)andvertex(j;1),forallj=1;:::;m.Second,thereisanarcbetweenvertices(j;t)andt,forallj=1;:::;m,suchthatt2N.Third,thereisanarcbetweenverticestand(j;t+1),suchthat1
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vertices(j;t)and(i;s )iszero,forj=1;:::;mandt=fn)]TJ /F1 11.955 Tf 12.15 0 Td[(1;ng,alsodependingonthevalueofs.AgraphicalrepresentationofH(i;s)ispresentedinFigure 5-8 B. ANetworkH(i;s) BH(2;3)forn=4andm=3 CCandidatestar Figure5-8. Subproblemnetwork Notethatthecostofthearcsdonotdependontheverticestheyemanatefrom.Thisisbecausesucharcsrepresenttheconnectionbetween(i;s)andthevertexattheheadofthecorrespondingarc.Moreover,itiseasytoseethateachpathfromthesourcetothesinkinH(i;s)canbeassociatedwithastarcenteredat(i;s).Forinstance,thepathformedbythegrayverticesinFigure 5-8 BcanbeassociatedwiththevalidstarshowninFigure 5-8 C.Thus,ifwesubtractsifromthecostoftheshortestpathbetweenthesourceandsinkverticesinH(i;s),weobtaintheminimumreducedcostofthevariableassociatedwiththestarthatcorrespondstotheshortestpath.Intheabsenceofbranchingconstraints,sinceH(i;s)isacyclicandthenumberofarcsisO(mn),ndingashortestpathcanbedoneinO(mn)( Ahujaetal. , 1993 ).Thus,similarlyasinAlgorithm 10 ,solvingtheproblemforalltheverticestakesO(m2n2).Ontheotherhand,oncethebranchingstagebegins,ndingthecorrespondingshortestpathbecomesamorediculttask.Theadvantageoftheproposedrepresentationisthatwe 125

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canincludetheadditionalbranchingconstraintsandsolvetheresultingshortestpathwithsideconstraintsviaDynamicProgramming( IrnichandDesaulniers , 2005 ).Inthefollowingsubsectionweprovideadescriptionofthealgorithmsthatweusedtocalculatethecandidatestars.AftersolvingalltheP(i;s)subproblemsforallvertices(i;s)2N,itispossibletoobtainseveralcandidatestarswiththesamesetofvertices,butwithdierentcenters.Forinstance,forthenetworkdepictedinFigure 5-8 C,assumethataftersolvingP(2;3)weobtainthestardescribedinthegure,andwhensolvingP(1;4)weobtainastarwiththesamesetofverticesbutcenteredat(1;4).Clearly,fromthedenitionofthestarcostspresentedin( 5{7 ),amongthosestars,weshouldonlyconsidertheonewiththeminimumcost.Thus,whilesolvingthesubproblems,wecreateahashmapthat,incasemorethanonestarisgeneratedwiththesamesetofvertices,keepsonlytheonewiththeminimumcost. 5.3.5DynamicProgrammingAlgorithmforFindingCandidateStarsTherearedierentalgorithmsavailableintheliteraturetosolvetheshortestpathproblemwithsideconstraints(e.g., IrnichandDesaulniers , 2005 ; LozanoandMedaglia , 2013 ; Santosetal. , 2007 ).Weoptedforadaptingthedynamicprogrammingapproachdescribedby IrnichandDesaulniers ( 2005 ),which,basedonourexperiments,wassucientforourpurposes.Theproposedapproachisasfollows.Letbethesetofbranchingconstraints.AsmentionedinSection 5.3 ,eachbranchingconstraint!2hasapairofvertices(werefertothoseasN(!)=f(i;s);(j;t)g)associatedwithitandrepresentsthebranchinwhichthegivenverticesmustbeeithercoveredbydierentstars(DIFF)orbythesamestar(SAME).Lettype(!)beafunctionthatreturnsthetypeoftheconstraint.Atanystageofthealgorithm,weusePtorefertothesetofintermediatecandidatepaths(i.e.,pathsthathavenotreachedthesinkvertex).AssociatedwitheachcandidatepathP2P,weuseP:cost,P:last,andP:ltorepresentthecurrentcost,thelastvertex,andthevectoroflabelsofpathP,respectively.ThevectoroflabelsP:lcontains 126

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oneentryforeachbranchingconstraint!2andhelpstodetermineifthecandidatepathisfeasible.Dependingonwhichtypebranchingconstraint!isofandtheverticesthatcomprisepathP,labelP:l[!]cantakethefollowingpossiblevalues:P:l[!]=8>>>>>>><>>>>>>>:2,ifbothverticesinN(!)arepartofP(onlyfortype(!)=SAME)1,ifonlyonevertexinN(!)ispartofP0,ifnovertexinN(!)iscurrentlypartofP,-1,ifnovertexinN(!)canbeaddedtoP(onlyfortype(!)=SAME).AllofthepreviousvaluesofP:l[!]areself-explanatorywiththeexceptionofP:l[!]=)]TJ /F1 11.955 Tf 9.3 0 Td[(1.AssumingthatN(!)=f(i;s);(j;t)g,thiscasecorrespondstothesituationwherethereisalreadyavertex(k;u)inP,suchthateitheru=soru=t.Sinceineachstartheremustbeexactlyonevertexfromeachset,thisimpliesthatneither(i;s)nor(j;t)canbeincludedinP.IfafeasiblepathPhasreachedthesinkvertexandforeachconstraint!2,suchthattype(!)=SAME,wehavethatP:l[w]iseither0or2,wesaythatPisacompletepath.NotethatifPhasreachedthesinkandthereisstilloneofsuchconstraintsforwhichP:l[w]=1,thepathPisinfeasibleasitcontainsonlyoneoftheverticesofconstraint!.Finally,wenameQasthesetofcompletepathsthathavebeengenerated.Theproposedapproachispresentedinalgorithms 12 { 14 .Theinitializefunction(Algorithm 12 )createsapaththatbeginsatthesourcevertexwithaninitialcostof)]TJ /F3 11.955 Tf 9.3 0 Td[(si.Italsoupdatestheconstraintlabelsaccordingtothevaluesdescribedabove.Theappendfunction(Algorithm 13 )identiesifappendingavertextotheendofpathPisafeasibleoption.Insuchcase,thefunctionupdatestheconstraintlabelsaswellasthecost.Iftheresultingpathisnotfeasible,itreturnsanemptypath.Notethat,ifthevertexvthatisbeingaddediseitheramemberofsetNorisequaltovertex(i;s )(step 2 ),thealgorithmappendsitattheendofPasdoingsodoesnotaectthefeasibilityofthepath(i.e.,visnotpresentinanyconstraintofset). 127

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Algorithm12initialize(G;;i;s) 1: P f( i;s)g 2: P:cost )]TJ /F3 11.955 Tf 24.57 0 Td[(si 3: forall!2do 4: if(i;s)2N(!)then 5: P:l[!]=1 6: elseiftype(!)=SAMEand9(j;t)2N(!)suchthatt=sthen 7: P:l[!]=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 8: else 9: P:l[!]=0 10: endif 11: endfor 12: returnP ThequalityofAlgorithm 14 ishighlydependenton(1)whichpathisselectedateachiteration(step 5 )and(2)howandwhenthedominancetestisperformed(step 16 ).TheselectioncriteriathatweuseisbasedonthefactthateverycompletepathP2Qwillpassthroughthesamenumberofvertices,i.e.,itwillhavethesamesize(Figure 5-8 foranexample).Thisisbecauseeachcompletepathisassociatedwithafeasiblestarandeachfeasiblestarhasintotalnvertices.Moreover,becauseoftheparticularstructureofnetworkH(i;s),itisalwaysposibletokeepadierenceofatmostonevertexbetweenthesizesofallcandidatepathsinsetP.Thiscanbedonebyalwaysselectinginstep 5 apathPwiththecurrentlysmallestnumberofvertices.Thereasonforthis,isthatperformingthedominancetestiseasierwhenallthecandidatepathsareofthesamesize.Theobjectiveofthedominancetest(step 16 )istodiscardcandidatepathsthataredominatedbyotherpathsinP.Thatis,ifwehavetwopathsP1andP2forwhichP1:last=P2:last,wesaythatP1dominatesP2if: forall!2withtype(!)=SAME,P1:l[!]andP2:l[!]arebothequalto1,ortakethevalueof0or2(e.g.,P1:l[!]=0andP2:l[!]=2isavalidpossibility); forall!2withtype(!)=DIFF,P1:l[!]P2:l[!]; P1costP2:cost. 128

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Algorithm13append(G;v;P) 1: ifv2Norv=(i;s )then 2: P P[fvg 3: returnP 4: endif 5: (j;t) v 6: forall!2do 7: if(j;t)2N(!)then 8: iftype(!)=DIFFthen 9: ifP:l[!]=1then 10: return; 11: else 12: P:l[!] 1 13: endif 14: else 15: ifP:l[!]=1then 16: P:l[!] 2 17: elseifP:l[!]=0then 18: P:l[!] 1 19: else 20: return; 21: endif 22: endif 23: elseiftype(!)=SAMEand9(k;u)2N(!)suchthatu=tthen 24: P:l[!]=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 25: endif 26: endfor 27: P P[v 28: P:cost P:cost+cstij)]TJ /F3 11.955 Tf 11.95 0 Td[(tj 29: returnP Clearly,ifP1dominatesP2,anycompletepathconstructedbyextendingP1willyield,intheworstcase,apathwiththesamecostastheoneconstructedbyextendingP2withthesamesetofvertices.Forthisreason,thereisnogaininconsideringP2asacandidatepathandthus,itcanberemovedfromP.BecausetheexecutiontimeofthedominancetestdependsonthenumberofbranchingconstraintsandthesizeofP,basedonourexperiments,thebestperformanceweobtainedwaswhenthedominancetestwasperformedonlyafterreachingeachvertexv2N. 129

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Algorithm14findCandidateStar(G;m;n;;i;s) 1: P initialize(G;;i;s) 2: P fPg 3: Q ; 4: whileP6=;do 5: selectapathP2PandremoveitfromP 6: forallvertexvintheforwardstarofP:lastdo 7: P0 append(G;v;P) 8: ifP06=;then 9: ifP0iscompletethen 10: Q Q[P0 11: elseifP0hasnotreachedthesinkthen 12: P P[P0 13: endif 14: endif 15: endfor 16: performadominancetesttoltersetP 17: endwhile 18: ifQ=;then 19: return; 20: endif 21: ndP=argminP2QfP:costg 22: reconstructthecandidatestarSfromsolutionP 23: returnS 5.3.6StabilizationTechniquesFinally,itiswellknownthattheSPformulationishighlydegenerate.NoteforexamplethatfortheMSAPcase,SPhasintotalmnconstraints(withoutincludingtheboundsonthevariables),whereasanyfeasibleintegersolutionwouldonlyhavemnonzerovariables.Primaldegeneracyiswellknowntocauseslowconvergence( LubbeckeandDesrosiers , 2005 ).Oneoftheprincipalcausesofthisbehavioristheinstabilityofthedualvariables.Unfortunately,theydonotsmoothlyconvergetotheirrespectiveoptima,astheyconstantlyoscillatewithoutaregularpattern.Sincethecolumnsgenerateddependonthedualvariablevalues,eectsonthequalityofsolutionsandontherunningtimeposeanimportantissuethatmustbeconsidered.Severalmethodshavebeenproposedtoovercomethisissue( Agarwaletal. , 1989 ; Marstenetal. , 1975 ; Rousseauetal. , 2007 ).Besidesapplyingsuchmethods,itisalso 130

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knownthatwhenusinganinteriorpointmethodtosolvetheRMP,thedualsolutionsthatareproducedareoftenmorebalanced( Bixbyetal. , 1992 ),leadingtoabetterperformance.Weareawarethough,thatstabilizationstrategiesgenerallyoutperformthisoption.However,forthepurposeofthispaper,thecombinationofaninteriorpointmethodandthefactthatweoftenincludeseveralstarcandidates(possiblyoneforeveryP(i;s)),provestobesucient. 5.4ComputationalExperimentsToanalyzetheperformanceofthedierentformulations,wecreatedatestbedof37problemsets,rangingnfrom3to20andmfrom4to30.WerefertoeachsetasnDm.Moreover,everysetiscomprisedof20randominstancesgeneratedusingthemethoddescribedin( GrundelandPardalos , 2005 ),havingintotal3720=740.ThecomputationalexperimentswereperformedonaserverwithtwoAMDOpteronTM6128Eight-CoreCPUsand12gigabytesofRAM,runningLinuxx86 64,CentOS5.9.FormulationsMIPandMIPawereimplementedinC++andsolvedusingCPLEX12.3,whilethebranch-and-priceframeworkfortheSPwascodedinCandsolvedusingSCIPOptimizer3.0.0( Bertholdetal. , 2012 ). 131

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Table5-1. Computationaltimes,optimalitygapsandthenumberofsolvedinstances. SetMIPMIPaMIPa-RMIPa-ASPTime(s)Gap(%)SolvedTime(s)Gap(%)SolvedTime(s)Gap(%)SolvedTime(s)Gap(%)SolvedTime(s)Gap(%)Solved 3D50.020.00200.010.00200.000.00200.000.00200.000.00203D100.660.00200.090.00200.080.00200.100.00200.000.00203D151.580.00200.160.00200.150.00200.170.00200.010.00203D2031.610.00201.560.00201.460.00201.610.00200.010.00203D2555.040.00201.830.00201.780.00201.970.00200.020.00203D3067.640.00203.270.00203.270.00203.450.00200.030.00204D50.110.00200.010.00200.000.00200.010.00200.000.00204D105.310.00200.640.00200.600.00200.660.00200.030.00204D15149.650.00207.550.00207.650.002012.950.00200.160.00204D20266.370.002011.830.002011.440.002018.200.00200.320.00204D253,427.100.0020103.900.002099.270.0020188.700.00202.680.00204D307,200.0011.3301,193.000.0020836.200.00201,447.000.00208.760.00205D50.900.00200.200.00200.180.00200.220.00200.010.00205D1055.250.00204.450.00204.390.00206.310.00200.180.00205D15581.670.002031.360.002029.420.002045.130.00205.300.00205D207,200.002.470518.090.0020196.800.0020952.120.0020454.490.00205D257,200.0014.4207,152.000.00204,104.000.00207,200.000.5803,758.900.88116D54.160.00200.620.00200.590.00200.750.00200.080.00206D10110.960.002010.040.00208.170.002017.810.002013.600.00206D157,200.004.830592.290.0020463.480.00201,017.200.00202763.940.05167D55.600.00200.980.00200.790.00201.210.00200.860.00207D101,745.900.002079.610.002069.750.0020143.130.00201738.210.30177D157,200.007.5107,200.001.0206,120.000.11197,200.001.03204,871.150.00208D521.110.00201.960.00201.450.00203.190.00200.500.00208D105,209.300.9117323.730.0020229.080.0020638.230.00204,614.732.0189D526.130.00202.790.00202.310.00203.180.00202.650.00209D107,200.0011.540581.020.0020389.300.0020912.840.00207,200.009.342010D5105.350.002011.340.002010.810.002017.610.002016.340.002010D107,200.0023.230706.980.0020631.120.00202,791.700.00207,200.0014.912011D5152.450.002022.770.002020.690.002031.480.00201,344.100.002011D107,200.0026.1107,200.002.0307,193.800.4617,200.0010.0807,200.0034.5012D5403.090.002036.750.002031.810.002050.660.0020774.420.002012D107,200.0031.9207,200.0021.4607,200.007.5407,200.0022.9207,200.0039.222015D52,137.280.002068.630.002058.850.0020185.110.00207,200.0016.81015D107,200.0043.3307,200.0029.8307,200.0018.0107,200.0032.4607,200.0041.42020D47,200.0017.110248.930.0020239.570.0020439.760.00207,200.0018.34020D57,200.0029.960579.410.0020415.410.00201,016.200.00207,200.0029.6020 132

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InourexperimentswecomparetheperformanceofformulationsSP,MIP,andMIPawithouttheminimumsum,4-cycle,andtripletcuts,aswellastheMIPaincludingthecutsat(1)therootnode(MIPa-R),and(2)allnodesofthebranch-and-boundtree(MIPa-A).Foralltheapproaches,weimposedatimelimitof7,200seconds.Table 5-1 reportstheaveragecomputationaltimeforalltheproblemsets,wherethefastesttimesarepresentedinbold.Additionally,weinclude(1)theaverageoptimalitygapoftheinstancesthatwerenotsolvedbythetimelimitand(2)thenumberofinstancesthatweresolvedtooptimality,ifany.Table 5-2 comparestheaveragenumberofnodesofthebranch-and-boundtree,whilealsopresentingtheaveragenumberofinequalitiesthatwereproducedforeachofthethreefamiliesofcuts.Theaveragenumberofcutsgeneratedattherootnodeisalsogiven.Forthesakeofsimplicity,theselastvalueswereroundeduptothenextinteger.Firstofall,examiningTable 5-1 wecaneasilyobservethatMIPisgenerallyoutperformed.Thisisreasonable,sincetheoverheadofaddingthelinearizationvariablesrendersitveryslow.WecanalsoobservethatSPisveryfastintheinstanceswherenissmall.However,asitincreases,SPbecomessignicantlyslowerthantheMIPa,withandwithoutthecuts.Furthermore,inthesmallerinstances,thecutfamiliesintroducedarenotaseective,whichistobeexpected.Hence,thesolutiontimesofMIPa,MIPa-R,andMIPa-Aarecomparable.Ontheotherhand,inlargerinstances,theeectofthecutsbecomesvisible,renderingMIPa-Raclearwinner.Notethat,becauseofthecomplexityoftheseparationalgorithms,MIPa-RisfasterthanMIPa-A.Nonetheless,asitcanbeseeninTable 5-2 ,thebranch-and-boundtreeforMIPa-AisremarkablysmallerthanMIPa-R,whichinturnissmallerthanMIPa.Finally,wereportthenumberofcuttingplanesintroducedateachproblemset.Observethatallfamiliesofinequalitiesareused,beingtheminimumsumcutstheonesthatappearmoreoften. 133

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Table5-2. Numberofnodesandcutsgenerated. NumberofnodesNumberofcutsSetMIPMIPaMIPa-RMIPa-AMinimumsum4-CycleTripletTotalRootTotalRootTotalRoot 3D500000000003D1000001000003D15141422433700323D209682513818643002533D2510112860442551040039113D30113160816410073760051194D521005310214D1037613118296282482144D15457945336272313282464D20320260815956313847289064D25279828391901411082349113624181104D30121951404726015201549310971193311447245D514141394424203615D102202191086310308015283845D15988132761831387151544001712665D2011861675871846862115346053626261155D251475384002330595326542123964629293166D53300151210106D10851919938121957208241406D15140963746712216829202122309362521947D54543181400007D101085161410594141427772976812317D15172113300617139625192196766644013638D51116149607183108D10914910738964512142812714721616319D5101168942610442451281509D101044190151192757017361241123218102210D51182037945130949368917110D101693320761442182112072115173614130311D51853029567163258440831111D1020013830313082109421102122249110157312D5257506190114227556501652112D103974389001951018272489344527089158215D55015442011472291376071253115D10528339975262082340289911207280644201320D46121486362429216193632148120D527310162776438093092048331971 Finally,Table 5-3 reportsadditionalstatisticsabouttheperformanceofformulationSP,includingtheaveragenumberofnodesofthebranch-and-pricetree,thenumberoftimesthesubproblem(i.e.,thestargenerationalgorithm)wasexecuted,thenumberofcandidatestarsproduced,andtheaveragetotaltimespentsolvingthesubproblem.Forthesakeofsimplicity,withtheexceptionoftheexecutiontimes,thevalueswereroundeduptothenextinteger.AscanbeseeninTable 5-3 ,theexecutiontimeofthesubproblemisinaverageabout70%ofthetotalrunningtimeofthealgorithm.Thisisexpectedbecauseeachofthesubproblemcallscorrespondstosolvingatotalofmnshortestpathproblemswithsideconstraints.Forthisreason,thetimerequiredtosolvetheRMPcompared 134

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Table5-3. StatisticsoftheSPformulation. SetnodesSubproblemcallsVariablesSubproblemtime 3D515300.003D1018910.003D15191920.003D202112770.003D252123930.013D302145400.014D528550.004D1010292540.014D1542945350.094D20441008080.194D2521334012452.024D3040157816217.135D55181230.005D10441115880.095D1530953312854.285D20217727922410442.585D257822934442383690.606D518553130.036D10344704145611.176D153869543643702690.067D524874760.067D102890467349661683.837D153513589676084699.188D5441537780.258D103915729896744453.219D517654018341.759D1031407782196816903.8810D55231528417712.6510D10396810930298397004.0211D5134438949425192.0211D1018687650351227048.9312D51731524913175737.1212D1020117911387777098.1415D5261211901525106582.7215D108214780390196328.9120D413229307635926641.7620D58466556620336625.83 totheoneneededtogeneratethenewcandidatestarsisalmostnegligible.Moreover,thispercentageisevenhigher(95%)fortheinstancesthatwherenotsolvedtooptimality.Thisisnotonlybecausethosearetheproblemswithlagersizes,butalsobecausethecorrespondingbranch-and-boundtreeshavemorenodesandtherefore,moresideconstraintsareintroducedtotheshortestpathproblems.Itisalsointerestingtonotethattheaverageexecutiontimeofeachoftheindividualmnsubproblemsisveryshort.Forexample,considertheexecutiontimeofthe20D5instances.Theaveragetimethattakesevaluatingeachoftheshortestpathsis6;625:83=(6;556205)=0:01seconds. 135

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CHAPTER6CONCLUDINGREMARKSInthischapter,wecommentontheresearchtasksthatwehavediscussedinChapters 2 through 5 .Wereviewthemethodologiesandtechniquesemployed,summarizethecomputationalresults,andsuggestfutureresearchdirections.InChapter 2 ,wepresentedageneraloverviewofthecriticalelementdetectionproblems,alongwithadetaileddescriptionofseveralvariationscommonlyfoundintheliterature.Wealsosummarizedsomeoftherecentadvancesandsolutiontechniquesfortheseproblems,includingheuristicalgorithms,mathematicalprogrammingapproaches,approximatedalgorithms,anddynamicprogrammingschemes.InChapter 3 ,westudiedacriticaledgedetectionproblemnamedtheminimumedgeblockerdominatingsetproblem(EBDP),whichistheproblemofremovingasubsetofedgesofminimumcardinalitysuchthattheminimumweightofalldominatingsetsintheremaininggraphisboundedbelowbyagivenintegerr.WebeganbyprovingthatthedecisionversionofEBDPisNP-hardforanyxedintegerr1.Then,weformulatedtheEBDPasamixed-integerprogram,anddevisedtheprojectionofsuchformulationontothespaceoftheedge-deletionvariables.Moreover,weidentiedfacet-inducinginequalitiesfortheconvexhulloffeasiblesolutionstoEBDP.Wesolvedtheproposedformulationviabranchandcut,wherenontrivialconstraintsareintroducedinalazyfashion.Wetestedtheperformanceofthisexactalgorithmonatest-bedofuniformrandominstancesandsomereal-lifepower-lawnetworks.ItturnedoutthatEBDPisrathercomputationallychallengingformoderatevaluesofr,whereasitcanbesolvedinareasonabletimeforverysmallorverylargevaluesofthisparameter.Wealsoobservedthatstrengtheningtheviolatedconstraintsdetectedateachnodeofthesearchtreebytheapproachimprovestheperformanceoftheproposedbranch-and-cutapproach.Forpotentialfuturestudies,itwouldbebenecialtoconsiderEBDPundertheassumptionthatthereisadeletioncost 136

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associatedwitheachedge.StudyingEBDPongraphssubjecttoprobabilisticvertex/edgefailuresisanotherinterestingchallenge.InChapter 4 ,wetackledtheproblemofdetectingcriticalcliquesofagraphwhosedeletionoptimallydisconnectsthegivengraph.Werstintroducedamathematicalformulationasanmixedintegerlinearprogram.Then,wepresentedaheuristictwo-stagedecompositionstrategythatrstidentiesacandidatecliquepartitionandthenusesthispartitiontotranslateandsolvetheproblemasageneralizedcriticalvertexdetectionproblem.Moreover,wereformulatedthisproblemasamixed-integerlinearprogramwithalargenumberofvariablesandconstraintsandsolveditusingabranch-price-and-cutapproach.Wealsointroducedapreprocessingstrategyforthecolumngenerationstagethatusesthedualinformationofthemasterproblemtoprunethesubproblemgraphinordertoproducenewcolumnseciently.Futureresearchmayinvolveextendingtheproposedapproachforsolvingothercriticalelementdetectionproblemsthatinvolveremovingsimilarcomplexgraphsubstructuressuchaspathsandtrees.Finally,inChapter 5 ,wepresentedareformulationofthemultidimensionalassignmentproblemwithdecomposablecosts,whereeachoftheassignmentsisassumedtobeastar.Weproposedacontinuousnonlinearformulationfortheproblem,anditslinearizationintoamixedintegerprogram.Inadditiontothat,weproposedaseriesofvalidinequalitiestostrengthentheformulation.Last,weimplementedabranch-and-priceframeworktosolveasetpartitioningreformulationoftheproblem.Alltheapproacheswerecompared,andthefamiliesofcutsintroducedprovedtobeveryecientattacklinglarge-scaleinstances.Ourworkstemmedfromawell-knowndataassociationproblemknownasthemulti-sensormulti-targettrackingproblem,forwhichtheMAPiswidelyused.Weshowthatourapproachisalsoaviableoptionforsolvingtheseproblems.Anadvantageofthisformulationisthefactthatthecenterofeverystarplaystheimportantroleofarepresentativemeasurement.Further,theeectofnoisysensorsisalleviated,comparedtoemployingothervariations.Asfarasfutureworkisconcerned,asseenin 137

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thecomputationalresults,thenumberofminimumsumcutsproducedisveryhigh,eventhoughtheyareeective.Amethodtosmartlychoosethebestofthesecutswouldresultinasmallermodelsize,whichcouldprovebenecial.Moreover,especiallyinlarge-scaleinstances,adecompositionschemecouldbeusedinordertoreducethesizeofeverysubproblem,makingitmanageableintime.Morespecically,Benders'decomposition,alongwithgraphpartitioningschemes,mightbeofinterest. 138

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APPENDIXPROOFOFCLAIM 1 ProofofClaim 1 . Supposen(P)
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kXi=1(dim(P)+1Xj=1j0ij)yi=0 (A{5) dim(P)+1Xj=1j=0 (A{6) wherePki=10ij=1forallj2f1;:::;dim(P)+1g.Nowconsidersystem dim(P)+1Xj=1j0ij=0;8i2f1;:::;kg: (A{7) NotethatanysolutiontoSystem( A{7 )isalsoasolutiontoSystem( A{5 )-( A{6 ).Since0isasolutiontoSystem( A{7 )andnumberofequationsinthissystem(k)isstrictlysmallerthanthenumberofvariablesinit(dim(P)+1),thenSystem( A{7 )hasinnitelymanysolutions,whichcontradictswiththeuniquenessofsolutiontoSystem( A{5 )-( A{6 ). 140

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REFERENCES Addis,B.,Summa,M.D.,Grosso,A.,2013.Identifyingcriticalnodesinundirectedgraphs:Complexityresultsandpolynomialalgorithmsforthecaseofboundedtreewidth.DiscreteAppliedMathematics161(16{17),2349{2360. Agarwal,Y.,Mathur,K.,Salkin,H.M.,1989.Aset-partitioning-basedexactalgorithmforthevehicleroutingproblem.Networks19(7),731{749. Ahuja,R.K.,Magnanti,T.L.,Orlin,J.B.,1993.NetworkFlows:Theory,Algorithms,andApplications.PrenticeHall,NewJersey,USA. Aiex,R.M.,Resende,M.G.C.,Pardalos,P.M.,Toraldo,G.,2005.Graspwithpathrelinkingforthree-indexassignment.INFORMSJournalonComputing17(2),224{247. Albert,R.,Jeong,H.,Barabasi,A.-L.,2000.Errorandattacktoleranceofcomplexnetworks.Nature406(6794),378{382. Aneja,Y.P.,Punnen,A.P.,1999.Multiplebottleneckassignmentproblem.EuropeanJournalofOperationalResearch112(1),167{173. Appa,G.,Magos,D.,Mourtos,I.,2006.Onmulti-indexassignmentpolytopes.LinearAlgebraanditsApplications416(2{3),224{241. Arulselvan,A.,Commander,C.W.,Elefteriadou,L.,Pardalos,P.M.,2009.Detectingcriticalnodesinsparsegraphs.ComputersandOperationsResearch36(7),2193{2200. Arulselvan,A.,Commander,C.W.,Pardalos,P.M.,Shylo,O.,2010.Managingnetworkriskviacriticalnodeidentication.RiskManagementinTelecommunicationNetworks.Springer,Heidelberg. Baird,D.,Ulanowicz,R.E.,1989.Theseasonaldynamicsofthechesapeakebayecosystem.EcologicalMonographs59(4),329{364. Balas,E.,Saltzman,M.J.,1989.Facetsofthethree-indexassignmentpolytope.DiscreteAppliedMathematics23(3),201{229. Balas,E.,Saltzman,M.J.,1991.Analgorithmforthethree-indexassignmentproblem.OperationsResearch39(1),150{161. Bandelt,H.J.,Crama,Y.,Spieksma,F.C.R.,1994.Approximationalgorithmsformulti-dimensionalassignmentproblemswithdecomposablecosts.DiscreteAppliedMathematics49(1{3),25{50. Bandelt,H.J.,Maas,A.,Spieksma,F.C.R.,2004.Localsearchheuristicsformulti-indexassignmentproblemswithdecomposablecosts.TheJournaloftheOperationalResearchSociety55(7),694{704. Barabasi,A.-L.,Albert,R.,1999.Emergenceofscalinginrandomnetworks.Science286(5439),509{512. 141

PAGE 142

Barnhart,C.,Johnson,E.L.,Nemhauser,G.L.,Savelsbergh,M.W.P.,Vance,P.H.,1998.Branch-and-price:Columngenerationforsolvinghugeintegerprograms.OperationsResearch46(3),316{329. Bauer,D.,Harary,F.,Nieminen,J.,Suel,C.L.,1983.Dominationalterationsetsingraphs.DiscreteMathematics47(0),153{161. Bazaraa,M.S.,Jarvis,J.J.,Sherali,H.D.,2009.LinearProgrammingandNetworkFlows.Wiley. Bazgan,C.,Toubaline,S.,Tuza,Z.,2011.Themostvitalnodeswithrespecttoindependentsetandvertexcover.DiscreteAppliedMathematics159(17),1933{1946. Bazgan,C.,Toubaline,S.,Vanderpooten,D.,2010.Complexityofdeterminingthemostvitalelementsforthe1-medianand1-centerlocationproblems.In:Wu,W.,Daescu,O.(Eds.),CombinatorialOptimizationandApplications.Vol.6508.SpringerBerlinHeidelberg,pp.237{251. Berthold,T.,Gamrath,G.,Gleixner,A.M.,Heinz,S.,Koch,T.,Shinano,Y.,2012.SolvingmixedintegerlinearandnonlinearproblemsusingtheSCIPOptimizationSuite.ZIB-Report12{17,ZuseInstituteBerlin,Takustr.7,14195Berlin. Bixby,R.E.,Gregory,J.W.,Lustig,I.J.,Marsten,R.E.,Shanno,D.F.,1992.Verylarge-scalelinearprogramming:Acasestudyincombininginteriorpointandsimplexmethods.OperationsResearch40(5),885{897. Bodlaender,H.,Grigoriev,A.,Grigorieva,N.,Hendriks,A.,2008.Thevalvelocationprobleminsimplenetworktopologies.In:Broersma,H.,Erlebach,T.,Friedetzky,T.,Paulusma,D.(Eds.),Graph-TheoreticConceptsinComputerScience.Vol.5344.SpringerBerlinHeidelberg,pp.55{65. Borgatti,S.P.,2006.Identifyingsetsofkeyplayersinasocialnetwork.Computational&MathematicalOrganizationTheory12,21{34. Borgatti,S.P.,Everett,M.G.,2006.Agraph-theoreticperspectiveoncentrality.SocialNetworks28(4),466{484. Burkard,R.E.,2002.Selectedtopicsonassignmentproblems.DiscreteAppliedMathematics123(1{3),257{302. Burkard,R.E.,Cela,E.,1999.Linearassignmentproblemsandextensions.In:Du,D.-Z.,Pardalos,P.M.(Eds.),HandbookofCombinatorialOptimization.SpringerUS,pp.75{149. Burkard,R.E.,Cela,E.,Pardalos,P.M.,Pitsoulis,S.,1998.Thequadraticassignmentproblem.In:Du,D.-Z.,Pardalos,P.M.(Eds.),HandbookofCombinatorialOptimization.SpringerUS,pp.241{338. 142

PAGE 143

Burkard,R.E.,Rudolf,R.,Woeginger,G.J.,1996.Three-dimensionalaxialassignmentproblemswithdecomposablecostcoecients.DiscreteAppliedMathematics65(1{3),123{139. Cheng,X.,Huang,X.,Li,D.,Wu,W.,Du,D.-Z.,2003.Apolynomial-timeapproximationschemefortheminimum-connecteddominatingsetinadhocwirelessnetworks.Networks42(4),202{208. Chummun,M.R.,Kirubarajan,T.,Pattipati,K.R.,Bar-Shlom,Y.,2001.Fastdataassociationusingmultidimensionalassignmentwithclustering.AerospaceandElectronicSystems,IEEETransactionson37(3),898{913. Church,R.L.,Scaparra,M.P.,Middleton,R.S.,2004.Identifyingcriticalinfrastructure:Themedianandcoveringfacilityinterdictionproblems.AnnalsoftheAssociationofAmericanGeographers94(3),491{502. Clemons,W.K.,Grundel,D.A.,Jecoat,D.E.,2004.Applyingsimulatedannealingtothemultidimensionalassignmentproblem.In:Grundel,D.A.,Murphey,R.,Pardalos,P.M.(Eds.),Theoryandalgorithmsforcooperativesystems.WorldScientic,pp.45{61. Corley,H.,Sha,D.Y.,1982.Mostvitallinksandnodesinweightednetworks.OperationsResearchLetters1(4),157{160. Crama,Y.,Spieksma,F.C.R.,1992.Approximationalgorithmsforthree-dimensionalassignmentproblemswithtriangleinequalities.EuropeanJournalofOperationalResearch60(3),273{279. Davis,T.A.,Hu,Y.,2011.Theuniversityoforidasparsematrixcollection.ACMTransactionsonMathematicalSoftware38(1),1{25. Deb,S.,Pattipati,K.R.,Bar-Shalom,Y.,1993.Amultisensor-multitargetdataassociationalgorithmforheterogeneoussensors.AerospaceandElectronicSystems,IEEETransactionson29(2),560{568. Deb,S.,Yeddanapudi,M.,Pattipati,K.,Bar-Shalom,Y.,1997.AgeneralizedSDassignmentalgorithmformultisensor-multitargetstateestimation.AerospaceandElectronicSystems,IEEETransactionson33(2),523{538. Desaulniers,G.,Desrosiers,J.,Spoorendonk,S.,2011.Cuttingplanesforbranch-and-pricealgorithms.Networks58(4),301{310. Desrosiers,J.,Lubbecke,M.E.,2010.Branch-price-and-cutalgorithms.In:Cochran,J.J.,Cox,L.A.,Keskinocak,P.,Kharoufeh,J.P.,Smith,J.C.(Eds.),WileyEncyclopediaofOperationsResearchandManagementScience.JohnWiley&Sons,Inc. Dessmark,A.,Jansson,J.,Lingas,A.,martaLundell,E.,Persson,M.,2006.Ontheapproximabilityofmaximumandminimumedgecliquepartitionproblems.InternationalJournalofFoundationsofComputerScience18,2007. 143

PAGE 144

DiSumma,M.,Grosso,A.,Locatelli,M.,2011.Complexityofthecriticalnodeproblemovertrees.ComputersandOperationsResearch38(12),1766{1774. DiSumma,M.,Grosso,A.,Locatelli,M.,2012.Branchandcutalgorithmsfordetectingcriticalnodesinundirectedgraphs.ComputationalOptimizationandApplications53(3),649{680. Ding,L.,Gao,X.,Wu,W.,Lee,W.,Zhu,X.,Du,D.-Z.,2011.Anexactalgorithmforminimumcdswithshortestpathconstraintinwirelessnetworks.OptimizationLetters5(2),297{306. Dinh,T.N.,Xuan,Y.,Thai,M.T.,Pardalos,P.M.,Znati,T.,2011.Onnewapproachesofassessingnetworkvulnerability:Hardnessandapproximation.Networking,IEEE/ACMTransactionsonPP(99). Edmonds,J.,Karp,R.M.,1972.Theoreticalimprovementsinalgorithmiceciencyfornetworkowproblems.JournaloftheACM(JACM)19(2),248{264. Feillet,D.,Dejax,P.,Gendreau,M.,Gueguen,C.,2004.Anexactalgorithmfortheelementaryshortestpathproblemwithresourceconstraints:Applicationtosomevehicleroutingproblems.Networks44(3),216{229. Fink,J.F.,Jacobson,M.S.,Kinch,L.F.,Roberts,J.,1990.Thebondagenumberofagraph.DiscreteMathematics86(1-3),47{57. Frederickson,G.N.,Solis-Oba,R.,1999.Increasingtheweightofminimumspanningtrees.JournalofAlgorithms33(2),244{266. Frieze,A.M.,Yadegar,J.,1981.Analgorithmforsolving3-dimensionalassignmentproblemswithapplicationtoschedulingateachingpractice.TheJournaloftheOperationalResearchSociety32(11),989{995. Gaofeng,H.,Lim,A.,2003.Ahybridgeneticalgorithmforthree-indexassignmentproblem.In:The2003CongressonEvolutionaryComputation,2003.CEC'03.Vol.4.pp.2762{2768. Garey,M.,Johnson,D.,Stockmeyer,L.,1976.Somesimpliednp-completegraphproblems.TheoreticalComputerScience1(3),237{267. Garey,M.R.,Johnson,D.S.,1990.ComputersandIntractability;AGuidetotheTheoryofNP-Completeness.W.H.Freeman&Co.,NewYork,NY,USA. Gilbert,K.C.,Hofstra,R.B.,1988.Multidimensionalassignmentproblems.DecisionSciences19(2),306{321. Grotschel,M.,Monma,C.,Stoer,M.,1995.Designofsurvivablenetworks.In:M.O.Ball,T.L.Magnanti,C.M.,Nemhauser,G.(Eds.),NetworkModels.Vol.7ofHandbooksinOperationsResearchandManagementScience.Elsevier,pp.617{672. 144

PAGE 145

Grotschel,M.,Wakabayashi,Y.,1990.Facetsofthecliquepartitioningpolytope.MathematicalProgramming47,367{387. Grubesic,T.H.,Matisziw,T.C.,Murray,A.T.,Snediker,D.,2008.Comparativeapproachesforassessingnetworkvulnerability.InternationalRegionalScienceReview31(1),88{112. Grubesic,T.H.,Murray,A.T.,2006.Vitalnodes,interconnectedinfrastructures,andthegeographiesofnetworksurvivability.AnnalsoftheAssociationofAmericanGeographers96(1),64{83. Grundel,D.A.,Oliveira,C.A.S.,Pardalos,P.M.,2004.Asymptoticpropertiesofrandommultidimensionalassignmentproblems.JournalofOptimizationTheoryandApplications122(3),487{500. Grundel,D.A.,Pardalos,P.M.,2005.Testproblemgeneratorforthemultidimensionalassignmentproblem.ComputationalOptimizationandApplications30(2),133{146. Gutin,G.,Karapetyan,D.,2009.Aselectionofusefultheoreticaltoolsforthedesignandanalysisofoptimizationheuristics.MemeticComputing1(1),25{34. Hewett,R.,2011.Towardidenticationofkeybreakersinsocialcyber-physicalnetworks.In:Systems,Man,andCybernetics(SMC),2011IEEEInternationalConferenceon.pp.2731{2736. Hopcroft,J.,Tarjan,R.,1973.Algorithm447:ecientalgorithmsforgraphmanipulation.Commun.ACM16,372{378. Houck,D.J.,Kim,E.,O'Reilly,G.P.,Picklesimer,D.D.,Uzunalioglu,H.,2004.Anetworksurvivabilitymodelforcriticalnationalinfrastructures.BellLabsTechnicalJournal8(4),153{172. Hu,F.-T.,Xu,J.-M.,2012.Onthecomplexityofthebondageandreinforcementproblems.JournalofComplexity28(2),192{201. Irnich,S.,Desaulniers,G.,2005.Shortestpathproblemswithresourceconstraints.In:Desaulniers,G.,Desrosiers,J.,Solomon,M.M.(Eds.),ColumnGeneration.SpringerUS,pp.33{65. Israeli,E.,Wood,R.K.,2002.Shortest-pathnetworkinterdiction.Networks40(2),97{111. Jenelius,E.,Petersen,T.,Mattsson,L.-G.,2006.Importanceandexposureinroadnetworkvulnerabilityanalysis.TransportationResearchPartA:PolicyandPractice40(7),537{560. Karapetyan,D.,Gutin,G.,2011a.Localsearchheuristicsforthemultidimensionalassignmentproblem.JournalofHeuristics17(3),201{249. 145

PAGE 146

Karapetyan,D.,Gutin,G.,2011b.Anewapproachtopopulationsizingformemeticalgorithms:Acasestudyforthemultidimensionalassignmentproblem.EvolutionaryComputation19(3),345{371. Karp,R.M.,2010.Reducibilityamongcombinatorialproblems.50YearsofIntegerProgramming1958-2008,219{241. Krebs,V.,2002.Uncloakingterroristnetworks.FirstMonday7(4). Krokhmal,P.A.,Grundel,D.A.,Pardalos,P.M.,2007.Asymptoticbehavioroftheexpectedoptimalvalueofthemultidimensionalassignmentproblem.Mathematicalprogramming109(2),525{551. Kuhn,H.W.,1955.Thehungarianmethodfortheassignmentproblem.Navalresearchlogisticsquarterly2(1{2),83{97. Kuroki,Y.,Matsui,T.,2009.Anapproximationalgorithmformultidimensionalassignmentproblemsminimizingthesumofsquarederrors.DiscreteAppliedMathematics157(9),2124{2135. Larsen,M.,2012.Branchandboundsolutionofthemultidimensionalassignmentproblemformulationofdataassociation.OptimizationMethodsandSoftware27(6),1101{1126. Lewis,J.M.,Yannakakis,M.,1980.Thenode-deletionproblemforhereditarypropertiesisnp-complete.JournalofComputerandSystemSciences20(2),219{230. Li,Y.,Thai,M.T.,Wang,F.,Yi,C.-W.,Wan,P.-J.,Du,D.-Z.,2005.Ongreedyconstructionofconnecteddominatingsetsinwirelessnetworks.WirelessCommunicationsandMobileComputing5(8),927{932. Lim,C.,Smith,J.C.,2007.Algorithmsfordiscreteandcontinuousmulticommodityownetworkinterdictionproblems.IIETransactions39(1),15{26. Lozano,L.,Medaglia,A.L.,2013.Onanexactmethodfortheconstrainedshortestpathproblem.Computers&OperationsResearch40(1),378{384. Lubbecke,M.E.,Desrosiers,J.,2005.Selectedtopicsincolumngeneration.OperationsResearch53(6),1007{1023. Lusseau,D.,Schneider,K.,Boisseau,O.,Haase,P.,Slooten,E.,Dawson,S.,2003.Thebottlenosedolphincommunityofdoubtfulsoundfeaturesalargeproportionoflong-lastingassociations.BehavioralEcologyandSociobiology54(4),396{405. Magos,D.,Mourtos,I.,2009.Cliquefacetsoftheaxialandplanarassignmentpolytopes.DiscreteOptimization6(4),394{413. Malhotra,R.,Bhatia,H.L.,Puri,M.C.,1985.Thethreedimensionalbottleneckassignmentproblemanditsvariants.Optimization16(2),245{256. 146

PAGE 147

Marsten,R.E.,Hogan,W.W.,Blankenship,J.W.,1975.Theboxstepmethodforlarge-scaleoptimization.OperationsResearch23(3),389{405. Matisziw,T.C.,Murray,A.T.,2009.Modelings-tpathavailabilitytosupportdisastervulnerabilityassessmentofnetworkinfrastructure.ComputersandOperationsResearch.36,16{26. Moreeld,C.L.,1977.Applicationof0-1integerprogrammingtomultitargettrackingproblems.AutomaticControl,IEEETransactionson22(3),302{312. Murphey,R.A.,Pardalos,P.M.,Pitsoulis,L.,1999.Aparallelgraspforthedataassociationmultidimensionalassignmentproblem.In:Pardalos,P.M.(Ed.),ParallelProcessingofDiscreteProblems.Vol.106ofTheIMAVolumesinMathematicsanditsApplications.SpringerNewYork,pp.159{179. Myung,Y.-S.,joonKim,H.,2004.Acuttingplanealgorithmforcomputingk-edgesurvivabilityofanetwork.EuropeanJournalofOperationalResearch156(3),579{589. Oliveira,C.A.S.,Pardalos,P.M.,2004.Randomizedparallelalgorithmsforthemultidimensionalassignmentproblem.AppliedNumericalMathematics49(1),117{133. Oosten,M.,Rutten,J.H.G.C.,Spieksma,F.C.R.,2007.Disconnectinggraphsbyremovingvertices:apolyhedralapproach.StatisticaNeerlandica61(1),35{60. Ortiz-Arroyo,D.,Hussain,D.M.,2008.Aninformationtheoryapproachtoidentifysetsofkeyplayers.In:Proceedingsofthe1stEuropeanConferenceonIntelligenceandSecurityInformatics.Springer-Verlag,pp.15{26. Ostergard,P.R.J.,Dec.2001.Anewalgorithmforthemaximum-weightcliqueproblem.NordicJ.ofComputing8(4),424{436. Pardalos,P.M.,Pitsoulis,L.S.E.,2000.Nonlinearassignmentproblems:algorithmsandapplications.Vol.7.Springer. Pasiliao,E.L.,Pardalos,P.M.,Pitsoulis,L.S.,2005.Branchandboundalgorithmsforthemultidimensionalassignmentproblem.OptimizationMethodsandSoftware20(1),127{143. Pentico,D.W.,2007.Assignmentproblems:Agoldenanniversarysurvey.EuropeanJournalofOperationalResearch176(2),774{793. Pierskalla,W.P.,1968.Themultidimensionalassignmentproblem.OperationsResearch16(2),422{431. Poore,A.B.,1994.Multidimensionalassignmentformulationofdataassociationproblemsarisingfrommultitargetandmultisensortracking.ComputationalOptimizationandApplications3(1),27{57. 147

PAGE 148

Poore,A.B.,RobertsonIII,A.J.,1997.Anewlagrangianrelaxationbasedalgorithmforaclassofmultidimensionalassignmentproblems.ComputationalOptimizationandApplications8(2),129{150. Pusztaszeri,J.-F.,Rensing,P.E.,Liebling,T.M.,1996.Trackingelementaryparticlesneartheirprimaryvertex:Acombinatorialapproach.JournalofGlobalOptimization9(1),41{64. RobertsonIII,A.J.,2001.Asetofgreedyrandomizedadaptivelocalsearchprocedure(GRASP)implementationsforthemultidimensionalassignmentproblem.ComputationalOptimizationandApplications19(2),145{164. Rousseau,L.M.,Gendreau,M.,Feillet,D.,2007.Interiorpointstabilizationforcolumngeneration.OperationsResearchLetters35(5),660{668. Ryan,D.M.,Foster,B.A.,1981.Anintegerprogrammingapproachtoscheduling.Computerschedulingofpublictransporturbanpassengervehicleandcrewscheduling,269{280. Salmeron,J.,Wood,K.R.,Baldick,R.,2004.Analysisofelectricgridsecurityunderterroristthreat.PowerSystems,IEEETransactionson19(2),905{912. Santos,L.,Coutinho-Rodrigues,J.,R.,C.J.,2007.Animprovedsolutionalgorithmfortheconstrainedshortestpathproblem.TransportationResearchPartB:Methodological41(7),756{771. Shen,S.,Smith,J.C.,2012.Polynomial-timealgorithmsforsolvingaclassofcriticalnodeproblemsontreesandseries-parallelgraphs.Networks60(2),103{119. Shen,S.,Smith,J.C.,Goli,R.,2012.Exactinterdictionmodelsandalgorithmsfordisconnectingnetworksvianodedeletions.DiscreteOptimization9(3),172{188. Shen,Y.,Nguyen,N.P.,Xuan,Y.,Thai,M.T.,Jun.2013.Onthediscoveryofcriticallinksandnodesforassessingnetworkvulnerability.IEEE/ACMTransactionsinNetworks.21(3),963{973. Shmoys,D.B.,1997.Cutproblemsandtheirapplicationtodivide-and-conquer.In:ApproximationalgorithmsforNP-hardproblems.PWSPublishingCo.,Boston,MA,USA,pp.192{235. Simaan,M.,Cruz,J.B.,1973.Onthestackelbergstrategyinnonzero-sumgames.JournalofOptimizationTheoryandApplications11,533{555. Spieksma,F.C.R.,2000.Multiindexassignmentproblems:complexity,approximation,applications.NonlinearAssignmentProblems:AlgorithmsandApplications7,1{12. Tao,Z.,Zhongqian,F.,Binghong,W.,2005.Epidemicdynamicsoncomplexnetworks.ProgressinNaturalScience16(5). 148

PAGE 149

ValeriodeCarvalho,J.M.,1999.Exactsolutionofbin-packingproblemsusingcolumngenerationandbranch-and-bound.AnnalsofOperationsResearch86(0),629{659. vanderZwaan,R.,Berger,A.,Grigoriev,A.,2011.Howtocutagraphintomanypieces.In:Ogihara,M.,Tarui,J.(Eds.),TheoryandApplicationsofModelsofComputation.Vol.6648.SpringerBerlinHeidelberg,pp.184{194. Ventresca,M.,2012.Globalsearchalgorithmsusingacombinatorialunranking-basedproblemrepresentationforthecriticalnodedetectionproblem.Computers&OperationsResearch39(11),2763{2775. Ventresca,M.,Aleman,D.,2014.Aderandomizedapproximationalgorithmforthecriticalnodedetectionproblem.Computers&OperationsResearch43(0),261{270. Veremyev,A.,Boginski,V.,Pasiliao,E.,2014a.Exactidenticationofcriticalnodesinsparsenetworksvianewcompactformulations.OptimizationLetters8(4),1245{1259. Veremyev,A.,Prokopyev,O.A.,Pasiliao,E.L.,2014b.Anintegerprogrammingframeworkforcriticalelementsdetectioningraphs.JournalofCombinatorialOptimization,1{41. Vogiatzis,C.,Pasiliao,E.L.,Pardalos,P.M.,2014.Graphpartitionsforthemultidimensionalassignmentproblem.ComputationalOptimizationandApplications58(1),205{224. Watts,D.J.,Strogatz,S.H.,1998.Collectivedynamicsofsmall-worldnetworks.nature393(6684),440{442. Wollmer,R.,1964.Removingarcsfromanetwork.OperationsResearch12(6),934{940. Wood,R.K.,1993.Deterministicnetworkinterdiction.MathematicalandComputerModeling17(2),1{18. Xu,J.-M.,2013.Onbondagenumbersofgraphs:asurveywithsomecomments.InternationalJournalofCombinatorics2013. Zachary,W.,1977.Aninformationowmodelforconictandssioninsmallgroups1.Journalofanthropologicalresearch33(4),452{473. Zenklusen,R.,2010.Matchinginterdiction.DiscreteAppl.Math.158(15),1676{1690. Zenklusen,R.,Ries,B.,Picouleau,C.,deWerra,D.,Costa,M.-C.,Bentz,C.,2009.Blockersandtransversals.DiscreteMathematics309(13),4306{4314. Zhu,X.,Yu,J.,Lee,W.,Kim,D.,Shan,S.,Du,D.-Z.,2010.Newdominatingsetsinsocialnetworks.JournalofGlobalOptimization48(4),633{642. 149

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BIOGRAPHICALSKETCH JoseL.WalteroswasborninBogota,Colombiain1983.Joseearnedhisbachelor'sandmaster'sdegreesinindustrialengineeringfromUniversidaddelosAndesin2005and2007,respectively.Aftergraduatingwithhismaster's,JosejoinedtheIndustrialEngineeringDepartmentatUniversidadthelosAndes,wereheworkedasanInstructorforthenextthreeyears.InAugust2010hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFloridaundertheguidanceofDistinguishedProfessorDr.PanosM.Pardalos.HeearnedhisDoctorofPhilosophyinindustrialandsystemsengineeringinAugust2014.Followinggraduation,hejoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityatBualoasafacultymember. 150