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The Vertex Separator Problem and Edge-Concave Quadratic Programming

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Title:
The Vertex Separator Problem and Edge-Concave Quadratic Programming
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1 online resource (80 p.)
Language:
english
Creator:
Hungerford, James T
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Hager, William Ward
Committee Members:
Jury, Michael Thomas
Bona, Miklos
Pardalos, Panagote M
Pilyugin, Sergei S
Davis, Timothy Alden
Rao, Anil

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Subjects / Keywords:
algorithm -- complexity -- continuous -- convex -- discrete -- multilevel -- optimization
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
The vertex separator problem on a graph G is to find the smallest set of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In this dissertation, we show how the vertex separator problem may be formulated as a continuous quadratic program in which the objective function is concave along the edges of the feasible set. The continuous formulation is compared to an edge-concave quadratic programming formulation of the edge separator problem for a graph, due to William Hager and Yaroslav Krylyuk. One remarkable property of edge-concave quadratic programs, which we establish, is that local optimality of any feasible point can be checked in polynomial time. The associated optimality conditions are derived by computing edge reductions of the standard first and second order optimality conditions for polyhedra. In the final chapter, these ideas are incorporated into a new multilevel algorithm for solving the vertex separator problem in which the continuous formulation is used as a local refinement tool. Computational results are presented.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
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.
Statement of Responsibility:
by James T Hungerford.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Hager, William Ward.

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Applicable rights reserved.
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MISSING IMAGE

Material Information

Title:
The Vertex Separator Problem and Edge-Concave Quadratic Programming
Physical Description:
1 online resource (80 p.)
Language:
english
Creator:
Hungerford, James T
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Hager, William Ward
Committee Members:
Jury, Michael Thomas
Bona, Miklos
Pardalos, Panagote M
Pilyugin, Sergei S
Davis, Timothy Alden
Rao, Anil

Subjects

Subjects / Keywords:
algorithm -- complexity -- continuous -- convex -- discrete -- multilevel -- optimization
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The vertex separator problem on a graph G is to find the smallest set of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In this dissertation, we show how the vertex separator problem may be formulated as a continuous quadratic program in which the objective function is concave along the edges of the feasible set. The continuous formulation is compared to an edge-concave quadratic programming formulation of the edge separator problem for a graph, due to William Hager and Yaroslav Krylyuk. One remarkable property of edge-concave quadratic programs, which we establish, is that local optimality of any feasible point can be checked in polynomial time. The associated optimality conditions are derived by computing edge reductions of the standard first and second order optimality conditions for polyhedra. In the final chapter, these ideas are incorporated into a new multilevel algorithm for solving the vertex separator problem in which the continuous formulation is used as a local refinement tool. Computational results are presented.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
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.
Statement of Responsibility:
by James T Hungerford.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Hager, William Ward.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045834:00001


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THEVERTEXSEPARATORPROBLEMANDEDGE-CONCAVEQUADRATICPROGRAMMINGByJAMEST.HUNGERFORDADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JamesT.Hungerford 2

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ACKNOWLEDGMENTS ThisdissertationwouldnothavebeenpossiblewithouttheencouragementandguidanceofmyadvisorWilliamW.Hager.IwouldalsoliketothankIlyaSafro,myresearchsupervisoratArgonneNationalLaboratoryduringmyinternshipthereintheSummerof2011.ThemultilevelalgorithmpresentedinChapter4wasdevelopedincollaborationwithIlyaandwouldnothavebeenpossiblewithouthim.TheresearchwhichledtothematerialinChapters2and3waspartiallysupportedbyagrantfromtheOfceofNavalResearch. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 5 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1Optimization .................................. 8 1.1.1ContinuousOptimization ........................ 9 1.1.2DiscreteOptimization .......................... 13 1.2EquivalencesBetweenDiscreteandContinuousOptimizationProblems 14 1.3OutlineofDissertation ............................. 17 2OPTIMALITYCONDITIONS ............................ 18 2.1EdgeReductionsofStandardOptimalityConditions ............ 19 2.2ComplexityofEdge-ConcaveQuadraticPrograms ............. 26 3THEVERTEXSEPARATORPROBLEM ...................... 29 3.1Introduction ................................... 29 3.2ContinuousEdge-ConcaveQuadraticProgrammingFormulation ..... 36 3.3LocalSolutions ................................. 43 4AMULTILEVELALGORITHMFORTHEVERTEXSEPARATORPROBLEM .. 53 4.1Introduction ................................... 53 4.2AContinuousFormulationforCoarseGraphs ................ 54 4.3OptimalityConditionsfortheCoarseQuadraticProgram .......... 60 4.4MultilevelAlgorithm .............................. 67 4.4.1Coarsening ............................... 68 4.4.2Solving .................................. 69 4.4.3Uncoarsening .............................. 69 4.4.4LocalRenement ............................ 70 4.4.5EscapingPoorLocalSolutions .................... 70 4.5PreliminaryComputationalResults ...................... 71 APPENDIX:NOTATION .................................. 74 REFERENCES ....................................... 75 BIOGRAPHICALSKETCH ................................ 80 4

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LISTOFTABLES Table page 3-1Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. ....................... 45 3-2Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. ....................... 45 4-1Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. ....................... 63 4-2Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. ....................... 64 4-3Illustrativecomparisonbetweenseparatorsobtained ............... 72 5

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LISTOFFIGURES Figure page 3-1AdescriptionofD\C(xxx,yyy) ............................. 46 4-1AdescriptionofD\C(xxx,yyy) ............................. 65 4-2Localrenement ................................... 70 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEVERTEXSEPARATORPROBLEMANDEDGE-CONCAVEQUADRATICPROGRAMMINGByJamesT.HungerfordAugust2013Chair:DouglasCenzerMajor:MathematicsThevertexseparatorproblemonagraphGistondthesmallestsetofverticeswhoseremovalbreaksthegraphintotwodisconnectedsubsetsofroughlyequalsize.Inthisdissertation,weshowhowthevertexseparatorproblemmaybeformulatedasacontinuousquadraticprograminwhichtheobjectivefunctionisconcavealongtheedgesofthefeasibleset.Thecontinuousformulationiscomparedtoanedge-concavequadraticprogrammingformulationoftheedgeseparatorproblemforagraph,duetoWilliamHagerandYaroslavKrylyuk.Oneremarkablepropertyofedge-concavequadraticprograms,whichweestablish,isthatlocaloptimalityofanyfeasiblepointcanbecheckedinpolynomialtime.Theassociatedoptimalityconditionsarederivedbycomputingedgereductionsofthestandardrstandsecondorderoptimalityconditionsforpolyhedra.Inthenalchapter,theseideasareincorporatedintoanewmultilevelalgorithmforsolvingthevertexseparatorprobleminwhichthecontinuousformulationisusedasalocalrenementtool.Computationalresultsarepresented. 7

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CHAPTER1INTRODUCTION 1.1OptimizationMathematicaloptimizationisthestudyofproblemsoftheform minf(xxx)subjecttoxxx2, (1) whereRniscalledthefeasiblesetandf:!Risareal-valuedfunctioncalledtheobjectivefunction.Thegoalofproblem( 1 )istondapointxxx2forwhichthevaluef(xxx)isminimum.(Sincemaximizingfisequivalenttominimizing)]TJ /F3 11.955 Tf 9.3 0 Td[(f,werestrictourattentiontotheminimizationcase.)Optimizationproblemsarisefrequentlyinoperationsresearch,computerscience,nance,statistics,andengineering[ 4 24 29 45 63 ].If=Rn,then( 1 )isanunconstrainedoptimizationproblem.If6=Rn,then( 1 )isaconstrainedoptimizationproblem.Apointxxx2Rnisfeasiblein( 1 )ifxxx2.Afeasiblepointxxxisaglobalsolutionifforeveryyyy2wehavef(xxx)f(yyy).AneighborhoodofxxxisasetoftheformB(xxx)=fyyy2Rn:jjxxx)]TJ /F3 11.955 Tf 11.42 0 Td[(yyyjj0.(Here,jjjjdenotestheEuclideannorm.)Alimitpointofisapointxxx2suchthatineveryneighborhoodB(xxx)thereexistssomeyyy2B(xxx)\suchthatyyy6=xxx.Alimitpointxxxisalocalsolution(orequivalently,alocalminimizer)to( 1 )ifthereexistsaneighborhoodB(xxx)suchthatforeveryyyy2B(xxx)\wehavef(xxx)f(yyy).Iff(xxx)
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foreveryt2[0,1].Afunctionfisconcaveif)]TJ /F3 11.955 Tf 9.3 0 Td[(fisconvex.Ingeneral,anoptimizationproblemmayhavemanymorelocalsolutionsthanglobalsolutions.However,inthespecialcasewherefandareconvex,everylocalsolutionisaglobalsolution(see[ 8 ],Section4.2.2): Theorem1.1. LetRnbeaconvexsetandletf:!Rbeaconvexfunction.Theneverylocalsolutionto( 1 )isaglobalsolution. 1.1.1ContinuousOptimizationWecalltheoptimizationproblem( 1 )continuousiff(x)iscontinuousandeveryfeasiblepointisalimitpointof.Thefollowingresultsarewellknown(see[ 51 ]). Theorem1.2. Let=Rn,f:Rn!R,andxxx2Rn. (A)Iffiscontinuouslydifferentiableinsomeneighborhoodofxxxandxxxisalocalsolutionto( 1 ),then rf(xxx)=000.(1) (B)Ifr2f(xxx)iscontinuousinaneighborhoodofxxx,andxxxisalocalsolutionto( 1 ),then( 1 )holdsandthematrixr2f(xxx)ispositivesemidenite. Theorem1.3. Let=Rn,f:Rn!R,andletxxx2Rn.Supposethatr2f(xxx)iscontinuousinsomeneighborhoodofxxx.Ifrf(xxx)=0andr2f(xxx)ispositivedenite,thenxxxisastrictlocalsolutionto( 1 ). Theorem1.4. Letf:Rn!Rbedifferentiableandconvex.Thenapointxxxisaglobalsolutionto( 1 )ifandonlyif( 1 )holds.ThelocaloptimalityconditionsofTheorems 1.2 and 1.3 maybegeneralizedtoconstrainedoptimizationproblemsundercertainassumptionsonthealgebraicandgeometricstructureofthefeasibleset.Belowwewillderivetheconditionsinthespecialcasewhereisapolyhedronandfissmooth.Foraderivationofoptimalityconditionsforgeneralconstrainedoptimizationproblems,wereferthereaderto[ 51 ].LetPdenotethepolyhedron P=fxxx2Rn:AxAxAxbbbg(1) 9

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forsomemnmatrixAAA2Rmnandsomevectorb2Rm,andconsidertheoptimizationproblem minf(x)subjecttoxxx2P, (1) wherefisacontinuouslydifferentiablefunctiondenedoversomeopensetcontainingP.Henceforth,weassumethatP6=;andthat( 1 )hasanoptimalsolution.Givenanyxxx2P,thesetofactiveconstraintsatxxxisdenedbyA(xxx)=fi2[1,m]:AAAixxx=big,whereAAAidenotesthei-throwofAAA.TheconeF(xxx)ofrst-orderfeasibledirectionsatxxxistheset F(xxx)=fddd2Rn:AAAiddd0foralli2A(xxx)g.(1)Observethatforeachddd2F(xxx),wehavexxx+ddd2Pfor0sufcientlysmall.Hence,ifxxxisalocalminimizer,then(f(xxx+ddd))]TJ /F3 11.955 Tf 11.95 0 Td[(f(xxx))=0when>0issufcientlysmall.Bytakingthelimitastendstozerofromtherightweobtainthefollowingrst-ordernecessaryoptimalitycondition[ 3 ,p.94]: Theorem1.5. Iffiscontinuouslydifferentiableandxxx2Pisalocalminimizerof( 1 ),then rf(xxx)ddd0foreveryddd2F(xxx).(1)Therst-ordernecessarycondition( 1 )isequivalenttothewell-knownKarushKuhnTuckerconditions:Thereexists2Rnsuchthat 000,bbb)]TJ /F3 11.955 Tf 11.43 0 Td[(AxAxAx000,T(bbb)]TJ /F3 11.955 Tf 11.43 0 Td[(AxAxAx)=0,andrf(xxx)+TAAA=000.(1)ReferencesfortheKKTconditionsinclude[ 3 28 46 51 ]. 10

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Nowsupposethatfistwicecontinuouslydifferentiableandxxx2Psatises( 1 ).ThecriticalconeC(xxx)isthesetdenedby C(xxx)=fddd2F(xxx):rf(xxx)ddd=0g=fddd2F(xxx):AAAiddd=0foralliforwhichi>0g. (1) Observethatifddd2C(xxx),thenrf(xxx)ddd=0,xxx+ddd2Pfor0sufcientlysmall,andwehavef(xxx+ddd)=f(xxx)+rf(xxx)ddd+1 22dddTr2f(xxx)ddd=f(xxx)+1 22dddTr2f(xxx)ddd,wherexxxliesbetweenxxxandxxx+ddd.Itfollowsthat dddTr2f(xxx)ddd=2(f(xxx+ddd))]TJ /F3 11.955 Tf 11.95 0 Td[(f(xxx)) 2.(1)Takingthelimitastendstozerofromtherightin( 1 ),weobtainthefollowingsecondordernecessarycondition: Theorem1.6. Iffistwicecontinuouslydifferentiableandxxx2Pisalocalminimizerof( 1 ),thenxxxsatises( 1 )inadditionto dddTr2f(xxx)ddd0foreveryddd2C(xxx),(1)Borwein[ 7 ]andContesse[ 13 ]showthatwhenfisquadratic,theconditions( 1 )and( 1 )arealsosufcientforlocaloptimality.Foramoregeneralfunction,asimplemodicationtotheargumentabovegivesthefollowingsecond-ordersufcientcondition[ 51 ,Thm.12.6]: Theorem1.7. Letfbetwicecontinuouslydifferentiable,andletxxx2P.Ifxxxisalocalminimizerof( 1 ),thenxxxsatises( 1 )inadditionto dddTr2f(xxx)ddd>0foreveryddd2C(xxx),ddd6=000.(1) 11

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Continuousoptimizationproblemsaretypicallysolvedusingiterativealgorithms.Startingfromsomeintialguessxxx0,eachiterationupdatesxxxkuntilapointxxxisfoundwhichsatises( 1 )ortheKKTconditions( 1 ).Inunconstrainedoptimization,theupdatestypicallytaketheformxxxk+1=xxxk+kpk,wherepkisthesearchdirectionandkisthestepsize.Onecommonwayofchoosingpkistosolveasubprobleminwhichtheobjectivefunctionismodelledbyalinearorquadraticfunctionwhichisoptimizedoveraspherefppp2R:jjpppjj2rgforsomeradiusr>0.Whenfismodelledbyarst-orderTaylorseries,weobtainthesearchdirectionpppk=rf(xxxk)knownasthesteepestdescentdirection.Ifxxxkissufcientlyclosetothesolutionandr2f(xxxk)ispositivedenite,thenminimizingasecond-orderTaylorseriesapproximationtofyieldsthesearchdirectionpk=r2f(xxxk))]TJ /F4 7.97 Tf 6.59 0 Td[(1rf(xxxk)knownastheNewtondirection.Inconjugategradientmethods,thesteepestdescentdirectionismodiedinordertomakesuccessivesearchdirectionsconjugate,ensuringafasterrateofconvergence.InQuasi-Newtonmethods,pktakestheform)]TJ /F3 11.955 Tf 9.3 0 Td[(B)]TJ /F4 7.97 Tf 6.58 0 Td[(1krf(xxxk)whereBkapproximatesthethehessianr2f(xxxk).Thesteplengthkintheseschemesisoftenchosentosatisfycurvatureandsufcientdecreaseconditionswhichguaranteeconvergenceoftheiteration(see,forinstance,theWolfeConditionsortheGoldsteinConditionsin[ 51 ]).Algorithmsforsolvingconstrainedoptimizationproblemsincludeactivesetmethods,interiorpointmethods,penaltyandaugmentedlagrangianmethods,andsequentialquadraticprogrammingmethods.Inactivesetmethods,thesearchdirectionisobtainedbysolvingasubprobleminwhichasubcollectionoftheactiveconstraintsatthecurrentiterateareheldatequalityandtheremainingconstraintsareignored.Primal-dualinteriorpointmethodscomputepkbyapplyingNewton'sMethodtoaperturbedversionoftheKKTconditions,wherekistypicallychosensothatsuccessiveiterateslieintheinteriorofthefeasibleset.Inpenaltyandaugmentedlagrangian 12

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methods,eachsuccessiveiterateisthesolutiontoanunconstrainedsubprobleminwhichtheconstraintshavebeenmovedintotheobjectivefunctionintheformofapenaltyterm.Apenaltyparamaterisgraduallyincreaseduntilthesolutiontothesubproblemconvergestothesolutiontotheoriginalproblem.Finally,sequentialquadraticprogrammingmethodsarebasedonsolvingaseriesofconstrainedquadraticprogramswhichapproximatetheorginalproblemwithincreasinglygreateraccuracy. 1.1.2DiscreteOptimizationWecalltheoptimizationproblem( 1 )discreteifcontainsnolimitpoints.Belowwegiveanexampleofadiscreteoptimizationproblemcalledthe0)]TJ /F3 11.955 Tf 13.12 0 Td[(1knapsackproblem: mincccTxxxsubjectto`aaaTxxxuandxxx2f0,1gn. (1) Here,ccc,aaa2Rnand`,u2R.Problem( 1 )arisesinresourceallocationwhenwewishtoallocatenresourcesinordertominimizeacostfunctioncccTxxx.TheconstraintonaaaTxxxisknownastheknapsackconstraint,andenforcestheconditionthatthetotalamountofresourcesallocatedliesbetween`andu.InChapter3,wewillstudyadiscreteoptimizationproblemknownasthevertexseparatorproblemonagraph.TheproblemistopartitiontheverticesofanetworkintothreesetsA,B,andS,suchthattherearenoedgesbetweenAandB,andthenumberofverticesinSisasmallaspossible.In[ 1 ],SouzaandBalasstudiedalinearintegerprogrammingformulationofthevertexseparatorproblem: maxxxx,yyy2BncccT(xxx+yyy) (1) subjecttoxxx+yyy111,xi+yj1forevery(i,j)2E`a111Txxxua,`b111Tyyyub.HereBn=f0,1gnisthecollectionofbinaryvectorswithncomponents,Eisthesetofedgesofthenetwork,ciistheweightofvertexi,111isthevectorwhoseentriesareall1, 13

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andxxxandyyyaretheincidencevectorsforAandBrespectively;thatis,xi=1ifi2Aandxi=0otherwise.Sincethefeasiblesetsofdiscreteoptimizationproblemsdonotcontainlimitpoints,thenotionofalocalsolutionwhichwedenedearlierdoesnotapply.Instead,localsolutionsaretypicallydenedonanadhocbasis.Forexample,alocalsolutionto( 1 )maybedenedasanyfeasiblepointxxxsuchthatf(xxx)f(yyy)foreveryfeasiblepointyyy2\N(xxx),whereN(xxx)istheneighborhoodofxxxdenedby N(xxx)=fyyy2f0,1gn:`aaaTyyyuand9jsuchthatyi=xi8i6=jg.(1)Withthisdenitionofneighborhood,checkinglocaloptimalityinproblem( 1 )isafairlyeasytask,requiringatmostncomparisons.However,itisoftenverydifculttondglobalsolutionstodiscreteoptimizationproblems,sincethetoolsofcalculusareessentiallyinapplicableandofferalmostnoguidanceonndinggoodlocalsolutions.Techniquesforsolvingdiscreteoptimizationproblemsincludebranchandbound/cutalgorithms,dynamicprogramming,tabusearch,continuousrelaxations,andheuristicssuchassimulatedannealingandantcolonies[ 67 ]. 1.2EquivalencesBetweenDiscreteandContinuousOptimizationProblemsOneapproachtosolvingadiscreteoptimizationproblemwithintegralityconstraints,suchas( 1 )or( 1 ),istosolveacontinuousrelaxation.Acontinuousrelaxationisacontinuousoptimizationproblemobtainedbyignoringtheintegralityconstraintsandextendingtheobjectivefunctiontoalargersetsuchthateverypointxxx2isalimitpoint.Forexample,thefollowingprogramisacontinuousrelaxationof( 1 )obtainedbyignoringtheintegralityconstraintonxxx,allowingeachxitorangefreelybetween0and1: minxxx2RncccTxxxsubjectto`aaaTxxxuand000xxx111. (1) 14

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Itisoftenmucheasiertosolveacontinuousrelaxationthanitistosolvetheoriginaldiscreteoptimizationproblem.Inthecaseof( 1 ),itscontinuousrelaxation( 1 )isalinearprogram,andisthereforesolvableinpolynomialtime(see[ 36 40 ]).Incontrast,( 1 )isNP-hard[ 23 ].Onceasolutiontoacontinuousrelaxationisobtained,onecantypicallymovetoanearbypointwhichisfeasibleintheoriginaldiscreteset.In( 1 ),forexample,adiscretepointmaybefoundbysimplymovingcomponentsofxxxtotheirupperorlowerbounds,assumingsuchmovesmaintainfeasibility.Ifoneisabletondaglobalsolutiontoacontinuousrelaxation,movingtoanearbydiscretepointmaygiveanearoptimalsolutiontotheoriginaldiscreteproblem(althoughthisisfarfromalwaysbeingthecase;see[ 67 ],Chapter2).However,insomespecialcasesthecontinuousrelaxationhasaglobalsolutionwhichisalsofeasibleinthediscreteoptimizationproblem.Inthiscase,thecontinuousrelaxationisinasenseequivalenttothediscreteproblem.Themostwellknownexampleofthisisinlinearprogramming.Inthelate1940s,itwasobservedbyDantzigandKantorovichthatafeasibleandboundedlinearprogramwhichhasatleastoneextremepointinitsfeasiblesethasanextremepointsolution.ThisresultissometimesknownastheFundamentalTheoremofLinearProgramming.Hence,theproblemofminimizingalinearfunctionovertheextremepointsofalinearlyconstrainedsetisinsomesenseequivalenttotheproblemofminimizingthefunctionovertheentireset.In1999,Tardellageneralizedthisresulttothemuchwiderclassofedge-concaveoptimizationproblems.InordertostateTardella'sresultprecisely,wemustreviewsomeconceptsfromthetheoryofpolyhedra.LetPbethepolyhedrondenedin( 1 ).AfaceofPisanon-emptysetoftheformH=fxxx2P:AAAIxxx=bbbIgforsomeIf1,2,...,mg.ThedimensionofthefaceHisonelessthanthemaximumnumberofafnelyindependentpointsinHandisdenoteddim(H).Ifdim(H)=0,then 15

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HisavertexofP.Ifdim(H)=1,thenHisanedgeofP.IfHisanedgeandJistheindexsetofconstraintsthatareactiveatmorethanonepointontheedge,thenthesetofsolutionstoAAAJxxx=bbbJisalinecontainingtheedge,andnull(AAAJ)isthecollectionofvectorswiththesamedirectionasthatoftheline.Werefertoanynonzerovectorddd2null(AAAJ)asanedgedirection.NotethatifdddisanedgedirectionforP,then)]TJ /F3 11.955 Tf 8.77 0 Td[(dddisanedgedirectionforP.AsetDisanedgedescriptionofPifforeachedgeofP,thereisaparalleldirectioninD. Denition1.2. LetPbeapolyhedronandletf:P!Rbeafunction.fisedge-concaveoverPifateverypointxxx2P,fisconcavealongthedirectionsparalleltotheedgesofP;thatis,forsomeedgedescriptionDofP,wheneverxxx,yyy2Paresuchthatyyy=xxx+tdddforsomeddd2Dandsomet2R,wehavethat f(xxx+(1)]TJ /F10 11.955 Tf 11.96 0 Td[()yyy)f(xxx)+(1)]TJ /F10 11.955 Tf 11.96 0 Td[()f(yyy),(1)forevery2[0,1].InthecasewherefistwicecontinuouslydifferentiableoveranopensetcontainingP,fisedge-concaveifandonlyif dddTr2f(xxx)ddd0foreveryddd2D,(1)forsomeedgedescriptionDofP.(Weleavetheproofofthisfactasanexerciseforthereader.)Considerthefollowingdiscreteoptimizationproblem: minf(xxx)subjecttoxxx2V, (1) whereV=fxxx2P:xxxisavertexofPg.ThefollowingtheoremfollowsdirectlyfromProposition2.1andCorollary3.1.1of[ 64 ]: Theorem1.8(Tardella). SupposethatPhasatleastonevertex,andletf:P!RbeanyfunctionwhichisboundedfrombelowonP.Iffisedge-concaveoverP,then 16

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fattainsitsminimumvalueatavertexofP.Hence,inthiscaseproblem( 1 )isequivalenttothediscreteoptimizationproblem( 1 ).Notethat,sinceeverylinearfunctionisedge-concave,theFundamentalTheoremofLinearProgrammingfollowsfromTheorem 1.8 asaspecialcase.Theorem 1.8 alsogeneralizesaresultbyRosenberg[ 60 ]whichstatesthatacomponentwiselinearfunctionattainsitsminimiumvalueatavertexofahypercube. 1.3OutlineofDissertationTheoutlineofthisdissertationisasfollows.InChapter2,wederiveanewrst-ordernecessaryconditionandasecond-ordersufcientconditionforapointtobealocalminimizerofafunctionoverapolyhedron.Theseconditionsarederivedfromthestandardoptimalityconditionsviaanedgedescriptionofthefeasibleset.Anewsufcientoptimalityconditionisobtainedinthecasewheretheobjectiveisedge-concave,andnecessaryandsufcientconditionsareobtainedinthecasewherefisedge-concaveandquadratic.Theseconditionsareusedtoestablishpolynomialcomplexityforedge-concavequadraticprograms.InChapter3,wewillconsideraparticulardiscreteoptimizationproblemknownasthevertexseparatorproblemonagraph.WewillshowhowTheorem 1.8 maybeusedtotransformthisproblemintoacontiuousedge-concavequadraticprogram.LocalsolutionstothecontinuousprogramareanalyzedusingtheoptimalityconditionsderivedinChapter2.Chapter4developsamultilevelalgorithmforthevertexseparatorproblemwhichmakesuseofthecontinuoustechniquesdevelopedinChapters2and3. 17

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CHAPTER2OPTIMALITYCONDITIONSThestandardrstandsecond-orderoptimalityconditionsforminimizingafunctionoverapolyhedron( 1 1 ,and 1 ),arestatedintermsoftheconesF(xxx)andC(xxx).SinceF(xxx)andC(xxx)areinnitesets,theseconditionsareingeneraldifculttoverify.Infact,asshownin[ 50 54 ],checkinglocaloptimalityforanindenitequadraticprogrammingproblemcanbeNP-hard.Inthischapter,wewillderivenewrst-ordernecessaryoptimalityconditionsandnewsecond-ordersufcientconditionswhichreducetoexaminingnitesubsetsoftheconesF(xxx)andC(xxx)correspondingtoanedgedescriptionofthepolyhedronP.Necessarysecond-orderconditionsarederivedforthecasewhentheobjectivefunctionisconcaveintheedgedirections.Inimportantspecialcases,suchaspolyhedrathatincludeboxconstraints,weshowthatthesizeoftheedgedescriptionisboundedbyapolynomialinn.Consequently,localoptimalityforaquadratic,edge-concaveobjectivefunctioncanbecheckedinpolynomialtime.Edge-directionswereintroducedinDantzig'ssimplexmethod[ 14 ]forlinearprogramminginordertomovefromonebasicfeasiblesolutiontoanother.Aspointedoutin[ 53 ],theuseofedge-directionsinoptimizationhasbeenexploredmorerecentlyinanumberofworksincluding[ 27 34 52 64 ].Edge-directionshavebeenstudiedincombinatorialoptimizationinthecontextofvertexenumerationproblems[ 27 52 ],andinidentifyingconditionsunderwhichdiscreteoptimizationproblemsareequivalenttocontinuousoptimizationproblems[ 34 64 ].In[ 64 ]Tardellashowedthatiftheobjectivefin( 1 )isconcavealongalledge-directions,thenproblem( 1 )hasavertexminimizer.Asimilarresult,inwhichPisreplacedbyacompact,convexsetwasobtainedbyHwangandRothblum[ 34 ].InSection 2.1 ,weshowhowtoderiveedgedescriptionsofF(xxx)andC(xxx)fromanedgedescriptionofP.Theseedgedescriptionsareexploitedtoobtainanewedgereducedversionoftherst-ordernecessaryoptimalitycondition,andthesecond-order 18

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sufcientoptimalitycondition.Whentheobjectivefunctionisedge-concave,weobtainanewsecond-ordernecessaryoptimalitycondition.InSection 2.2 weobtainanestimateonthenumberofedgesofapolyhedronthatincludesaboxconstraint.Thisestimatetogetherwiththeedgereducednecessaryandsufcientoptimalityconditionsimplypolynomialcomplexityforedge-concavebox-constrainedquadraticprograms. 2.1EdgeReductionsofStandardOptimalityConditionsLetPbethepolyhedrondenedby P=fxxx2Rn:AxAxAxbbbg,(2)forsomemnmatrixAAA2Rmnandsomevectorb2Rm,andconsidertheoptimizationproblem minf(xxx)subjecttoxxx2P, (2) wherefisacontinuouslydifferentiablefunctiondenedoversomeopensetcontainingP.Throughoutthischapter,weassumethatrank(AAA)=n.AswewillseeintheproofofLemma 2.1 ,thisassumptionisusedtoguaranteethatPcontainsatleastonevertex.RecallfromChapter1thatasetDisanedgedescriptionofPifforeachedgeofP,thereisaparalleldirectioninD.Additionally,if)]TJ /F3 11.955 Tf 8.77 0 Td[(ddd2Dwhenddd2D,thenwesaythatDisareectiveedgedescriptionofP. Lemma2.1. LetPbethepolyhedrondenedin( 2 )whereAAAhasfullcolumnrank,andletDbeareectiveedgedescriptionofP.Foranyxxx2PandIA(xxx),theconedenedbyKI(xxx)=fddd2F(xxx):AAAIddd=000ghasthepropertythat KI(xxx)=span+(KI(xxx)\D).(2) 19

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Proof. First,supposethatxxxisavertexofP,inwhichcaseF(xxx)containsnolines.SinceKI(xxx)F(xxx),KI(xxx)alsocontainsnolines.ByaresultofKlee[ 41 ,Corollary2.3],thereexistvectorsddd1,ddd2,...,dddl2KI(xxx)suchthatforeachi,dddiisanedge-directionofKI(xxx)and KI(xxx)=span+(ddd1,ddd2,...,dddl).(2)WewillnowshowthateachedgedirectiondddiforKI(xxx)isalsoanedgedirectionofP.LetEbeanedgeofKI(xxx)associatedwiththeedgedirectiondddi.SinceKI(xxx)isafaceofF(xxx),EisalsoanedgeofF(xxx).Therefore,xxx+EcontainsanedgeofP.Hence,dddiisalsoanedgedirectionofP.SinceDisreective,itfollowsthateachdddiisapositivemultipleofanelementi2D.SinceKI(xxx)isacone,i2KI(xxx).Hence,( 2 )withdddireplacedbyiimpliesthat( 2 )holdsinthecasethatxxxisavertexofP.NowsupposethatxxxisnotavertexofP.SinceKI(xxx)isaconvexcone,wehaveKI(xxx)=span+(KI(xxx))span+(KI(xxx)\D).Wewillshowthereversecontainment.Letddd2KI(xxx).ConsiderthefollowingfaceofP:F=fxxx2P:AAAA(xxx)xxx=bbbA(xxx)g.Sincerank(AAA)=n,thereexistsavertexvvvofthepolyhedronF.Thisvertexisfoundbymakingaseriesofmovesintheface;eachmoveisinthenullspaceassociatedwiththeactiveconstraintswhilemaintainingnonorthogonalitytoatleastoneoftherowsofAAAassociatedwithaninactiveconstraint.Eachmovestopsatapointwhereapreviouslyinactiveconstraintbecomesactive.Sincerank(AAA)=n,weeventuallyreachavertexwherenconstraintsareactive.ForthisvertexwehaveA(vvv)A(xxx)andrank(AAAA(vvv))=n.Sinceddd2KI(xxx)F(xxx),thereexistssomet>0suchthatvvv+(xxx)]TJ /F3 11.955 Tf 11.35 0 Td[(vvv)+tddd2P.Thisimpliesthat(xxx)]TJ /F3 11.955 Tf 11.35 0 Td[(vvv)+tddd2F(vvv).Furthermore,AAAIddd=000sinceddd2KI(xxx),andAAAI(xxx)]TJ /F3 11.955 Tf 10.84 0 Td[(vvv)=000sinceIA(xxx)A(vvv).HenceAAAI((xxx)]TJ /F3 11.955 Tf 10.85 0 Td[(vvv)+tddd)=0and 20

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therefore(xxx)]TJ /F3 11.955 Tf 11.42 0 Td[(vvv)+tddd2KI(vvv).Thus, tddd2(vvv)]TJ /F3 11.955 Tf 11.43 0 Td[(xxx)+KI(vvv).(2)Wewillshowthattherightsideof( 2 )iscontainedinspan+(KI(xxx)\D).SinceA(vvv)A(xxx),wehaveF(vvv)F(xxx),andthereforeKI(vvv)KI(xxx).AndsincevvvisavertexofF,itisalsoavertexofP.Earlierweshowedthat( 2 )holdsatanyvertexofP.Therefore,replacingxxxbyvvvin( 2 ),wehave KI(vvv)=span+(KI(vvv)\D)span+(KI(xxx)\D).(2)ThelastinclusionissinceA(xxx)A(vvv)whichimpliesthatKI(vvv)KI(xxx).Nextweshowthat (vvv)]TJ /F3 11.955 Tf 11.42 0 Td[(xxx)2span+(KI(xxx)\D).(2)SincevvvisavertexofF,theconeFF(vvv)ofrst-orderfeasibledirectionsforFatvvvcontainsnolines.Sincexxx)]TJ /F3 11.955 Tf 12.23 0 Td[(vvv2FF(vvv),itagainfollowsfromtheresultofKlee[ 41 ,Corollary2.3]thatxxx)]TJ /F3 11.955 Tf 11.48 0 Td[(vvvisapositivelinearcombinationofedgedirectionsD0DforF.SinceDisreective,wemaychooseD0tobereective.Hence,wehave vvv)]TJ /F3 11.955 Tf 11.42 0 Td[(xxx=)]TJ /F3 11.955 Tf 9.3 0 Td[((xxx)]TJ /F3 11.955 Tf 11.42 0 Td[(vvv)2span+(D0).(2)Sinceanyedge-directionofFliesinnull(AAAA(xxx))KI(xxx),wehaveD0=KI(xxx)\D0.Therefore,by( 2 ) (vvv)]TJ /F3 11.955 Tf 11.42 0 Td[(xxx)2span+(KI(xxx)\D0)span+(KI(xxx)\D),(2)whichestablishes( 2 ).Combining( 2 )with( 2 )and( 2 ),wehavetddd2span+(KI(xxx)\D).Sincet>0,thisimpliesddd2span+(KI(xxx)\D).SincedddwasanarbitraryvectorinKI(xxx),theproofiscomplete. 21

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Corollary2.1. LetPbethepolyhedrondenedin( 2 )whereAAAhasfullcolumnrank,andletDbeareectiveedgedescriptionofP.Foranyxxx2P,wehave F(xxx)=span+(F(xxx)\D).(2)Furthermore,ifxxxsatisestherst-orderoptimalityconditions( 1 )forproblem( 2 ),then C(xxx)=span+(C(xxx)\D).(2) Proof. Theidentity( 2 )followsimmediatelyfromLemma 2.1 bytakingI=;.Ifxxxsatises( 1 ),thendeneI=fi:i>0g.Bycomplementaryslackness,IA(xxx).Andbythedenition( 1 )forthecriticalcone,wehaveC(xxx)=KI(xxx).Hence,( 2 )alsofollowsfromLemma 2.1 Theidentities( 2 )and( 2 )arerelatedtoaconformalcircuitdecomposition[ 34 59 ].ConsiderthepolyhedronQdenedbyQ=fxxx2Rn:BBBxxx=bbbandxxx0g,whereBBB2Rmnandbbb2Rm.Foranyxxx2QwehaveF(xxx)=fddd2Rn:BBBddd=000anddj0wheneverxj=0g.AcircuitofthematrixBBBisanon-zerovectorddd2null(BBB)suchthatjjdddjj1=1,andsupp(ddd)isinclusion-minimal.Thatis,ifddd02null(BBB)andsupp(ddd0)supp(ddd),thenddd0=dddforsomescalar.Aspointedoutin[ 53 ],itfollowsfrom[ 59 ,ex.10.14,p.506]thatanynon-zerovectorddd2null(BBB)hasaconformalcircuitdecomposition;thatis,thereexist 22

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scalarsi>0andcircuitsdddiofBBBsuchthat ddd=Xiidddi,(2)whereforeachiwehavedijdj>0forallj2supp(dddi).Henceifddd2F(xxx),thendddi2F(xxx)foreachi.IfeverycircuitofBBBisparalleltoanedgeofQ,then( 2 )followsfromthedecomposition( 2 ).Asimilarargumentproves( 2 ).However,itisnottrueingeneralthateverycircuitofBBBisparalleltoanedgeofQ(see[ 34 ]),andconversely,someedgesmaynotbecircuits.Ourproofof( 2 )and( 2 )isvalidforanypolyhedronPoftheform( 2 ),aslongasthematrixAAAhasfullcolumnrank.WhenthecolumnsofAAAarelinearlydependent,Lemma 2.1 maynothold;forexample,considerthecasewherePisahalf-space.TheassumptionthatAAAhasfullcolumnrankinLemma 2.1 isusedtoensuretheexistenceofanextremepoint(avertex)ofP.However,Corollary 2.1 isfalseifthepolyhedronPisreplacedbyageneralconvexsetcontaininganextremepoint.Forexample,considertheunitsphereX=f(x1,x2)2R2:x21+x221g.EverypointontheboundaryofXisanextremepoint.However,Xcontainsnoextremedirections.ThereforeD=;.So( 2 )wouldimplythatF(xxx)=;foreveryxxx2X.Thisisclearlyfalse;forinstanceifxxx=()]TJ /F3 11.955 Tf 9.3 0 Td[(1,0),thenF(xxx)=fddd2R2:d10g.Thus,Lemma 2.1 andCorollary 2.1 arepropertiesofpolyhedra,notgeneralconvexsets. Proposition2.1. Iff:P!Riscontinuouslydifferentiable,AAA2Rmnhasfullcolumnrank,andDisareectiveedgedescriptionofP,thenafeasiblepointxxx2Psatisestherst-orderoptimalityconditions( 1 )ifandonlyif rf(xxx)ddd0foreveryddd2F(xxx)\D.(2) 23

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Proof. Obviously,( 1 )implies( 2 ).Conversely,supposexxxsatises( 2 )andletddd2F(xxx).ByCorollary 2.1 ,wehaveddd=kXi=1idddi,forsomevectorsdddi2F(xxx)\Dandsomescalarsi>0.Thus( 2 )impliesrf(xxx)ddd=kXi=1irf(xxx)dddi0,whichyields( 1 ). Sincetheconditions( 1 ),( 1 ),and( 2 )areallequivalent,wewillrefertotheseconditionscollectivelyastherst-orderoptimalityconditionsfor( 2 ).Next,weconsidersecond-orderoptimalityconditions. Proposition2.2. Supposef:P!Ristwicecontinuouslydifferentiable,AAAhasfullcolumnrank,DisareectiveedgedescriptionofP,andxxx2Psatisestherst-orderoptimalityconditions. 1. xxxisalocalminimizerof( 2 )if (ddd1)Tr2f(xxx)ddd2>0foreveryddd1,ddd22C(xxx)\D.(2)Iffisquadratic,thenthestrictinequalityin( 2 )canbereplacedbyaninequality. 2. Ifxxxisalocalminimizerof( 2 )and dddTr2f(xxx)ddd0foreveryddd2D,(2)then (ddd1)Tr2f(xxx)ddd20foreveryddd1,ddd22C(xxx)\D.(2) Proof. Supposethatxxxsatisestherst-orderoptimalityconditions.Toprovepart1,weneedonlyshowthat( 2 )impliestheusualsecond-ordersufcientcondition( 1 ).ByCorollary 2.1 ,foranyddd2C(xxx)wehaveddd=kXi=1idddi, 24

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forsomevectorsdddi2C(xxx)\Dandsomescalarsi>0.ThereforedddTr2f(xxx)ddd=kXi=1idddiTr2f(xxx)kXi=1idddi=kXi,j=1ijdddir2f(xxx)dddj>0,if( 2 )holds.Consequently,( 1 )holdsandxxxisalocalmaximizer.Iffisquadraticandthestrictinequality( 2 )isreplacedbyaninequality,thenthesameargumentyields( 1 ),whichissufcientforlocaloptimalitywhenfisquadratic[ 7 13 ].Thiscompletestheproofofpart1.Next,weconsiderpart2.Thesecond-ordernecessarycondition( 1 )statesthat dddTr2f(xxx)ddd0foreveryddd2C(xxx).(2)If( 2 )holds,then dddTr2f(xxx)ddd=0foreveryddd2C(xxx)\D.(2)SinceC(xxx)isaconvexcone,itfollowsthatforanyddd1,ddd22C(xxx)\D,wehaveddd1+ddd22C(xxx).Therefore,( 2 )and( 2 )imply0)]TJ /F3 11.955 Tf 4.95 -9.68 Td[(ddd1+ddd2)Tr2f(xxx))]TJ /F3 11.955 Tf 4.95 -9.68 Td[(ddd1+ddd2=2(ddd1)Tr2f(xxx)ddd2,whichyields( 2 ). ThefollowingcorollaryisanimmediateconsequenceofProposition 2.2 Corollary2.2. LetPbethepolyhedrondenedin( 2 )whereAAAhasfullcolumnrank,andletDbeareectiveedgedescriptionofP.Iffisquadraticand( 2 )holds,thenxxxisalocalminimizerof( 2 )ifandonlyif( 2 )(therst-ordercondition)and( 2 )(thesecond-ordercondition)hold.Remark.Wenowobservethatiffisquadratic,( 2 )holds,andxxxsatisestherst-orderoptimalitycondition( 2 ),thenwhenanyoftheconditions( 2 )areviolated,thereisaeasilycomputeddescentdirection.Inparticular,ifQQQ=r2fthenbya 25

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Taylorexpansion,wehave f(xxx+ddd)=f(xxx)+1 22dddTQdQdQd(2)foranyddd2C(xxx)and>0,sincerf(xxx)ddd=0by( 1 ).IfdddTQdQdQd<0forsomeddd2C(xxx)\D,thendddisadescentdirection.Otherwise,dddTQdQdQd0foreveryddd2C(xxx)\D.Soby( 2 ),dddTQdQdQd=0foreveryddd2C(xxx)\D.If(ddd1)TQQQddd2<0forsomeddd1andddd22C(xxx)\D,then(dddi)TQQQdddi=0fori=1,2.Sinceddd1+ddd22C(xxx),itfollowsfrom( 2 )thatf(xxx+(ddd1+ddd2))=f(xxx)+2(ddd1)TQdQdQd20.Thisshowsthatddd=ddd1+ddd22C(xxx)isadescentdirection. 2.2ComplexityofEdge-ConcaveQuadraticProgramsWedeneaminimaledgedescriptionofPtobeanedgedescriptionwhichdoesnotproperlycontainanyotheredgedescription.UndertheassumptionsofCorollary 2.2 ,thecomputationalcomplexityofcheckingwhetheragivenpointisalocalmaximizerisproportionaltothesizeofaminimaledgedescriptionsquared.SupposethenumberofcolumnsofAAAisheldxed,butthenumberofrowsofAAAisallowedtovary.Atrivialupperboundonthesizeofaminimaledgedescriptionis)]TJ /F4 7.97 Tf 9.71 -4.38 Td[(mn)]TJ /F4 7.97 Tf 6.59 0 Td[(1.Sincethisisapolynomialinmwhennisxed,itfollowsthatlocaloptimalitycanbecheckedinpolynomialinmtime.Ifmisxedandnisallowedtovary,thenwecannotapplyCorollary 2.2 sincethecolumnsofAAAarelinearlydependentwhenn>m.Ontheotherhand,wecanapplyCorollary 2.2 whenthepolyhedronincludesaboxconstraint: PB=fxxx2Rn:AxAxAxbbband`xxxuuug.(2)HereAAA2Rmnisanymatrix,`2Rn,anduuu2Rn.Duetotheboxconstraint,theconstraintmatrixassociatedwiththepolyhedronhasfullcolumnrank.Hence,we 26

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considertheoptimizationproblem maxf(xxx)subjecttoxxx2PB.(2)Wenowshowthatfortheboxconstrainedpolyhedronwheremisxedandnisallowedtoincrease,thesizeofaminimaledgedescriptionisboundedbyapolynomialinn. Corollary2.3. Ifmisheldxed,f:Rn!Risquadratic,andforsomeedgedescriptionDofPB,wehavedddT(r2f)ddd0foreveryddd2D,thenlocaloptimalityofanyfeasiblepointinproblem( 2 )canbecheckedintimethatisboundedbyapolynomialinn. Proof. Toprovethisresult,weobtainaboundonthesizeofaminimaledgedescriptionforPB.AnyedgeofPBiscontainedinthesolutionsetofalinearsystemoftheformMxMxMx=cccwhereMMM2R(n)]TJ /F4 7.97 Tf 6.59 0 Td[(1)nhasrankn)]TJ /F3 11.955 Tf 12.53 0 Td[(1.Anedgedirectionisanynonzerovectorinnull(MMM);MMMwillbecalledtheedgematrix.NotethatmultiplyingarowofMMMandthecorrespondingcomponentofcccby)]TJ /F3 11.955 Tf 9.3 0 Td[(1hasnoeffectoneithernull(MMM)orthesolutionsetofMxMxMx=ccc.WhenbuildinganedgematrixcorrespondingtosomeedgeofPB,uptomoftherowsofMMMcanbetakenfromrowsofAAAwhiletheremainingrowsaretakenfromtheidentitymatrix;theselatterrowscorrespondtotheconstraintsxi`iorxiuithatareactiveontheedge.Foranedgealongwhicheitherxi=`iorxi=ui,thecorrespondingrowoftheedgematrixwillbethei-throwoftheidentitymatrix.Hence,whenconstructingtheedgematrix,itdoesnotmatterwhetherxiisattheupperboundoratthelowerbound,allthatmattersiswhetherthei-thboxconstraintisactive.Forn>m,anupperboundonthenumberofedgedirectionsisgivenbytheexpression mXi=1nn)]TJ /F3 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(imi.(2) 27

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HereirepresentsthenumberofrowsofAAAtobeinsertedintotheedgematrixandn)]TJ /F3 11.955 Tf 12.14 0 Td[(1)]TJ /F3 11.955 Tf 12.14 0 Td[(iisthenumberofrowsoftheidentitymatrixtobeinsertedintotheedgematrix.Thereare)]TJ /F4 7.97 Tf 5.48 -4.38 Td[(midifferentcollectionsofrowsofAAAthatcouldbeplacedintheedgematrix.AndforeachselectionoftherowsinAAA,anothern)]TJ /F3 11.955 Tf 12.58 0 Td[(1)]TJ /F3 11.955 Tf 12.58 0 Td[(irowsareselectedfromtheidentitymatrix.Thereare)]TJ /F4 7.97 Tf 15.36 -4.38 Td[(nn)]TJ /F4 7.97 Tf 6.58 0 Td[(1)]TJ /F4 7.97 Tf 6.58 0 Td[(idifferentcollectionsofrowsfromtheidentitymatrix.Sincemisxed,theexpression( 2 )isapolynomialinn. Notethatwecouldhaveeither`i=orui=+1,butnotboth,andCorollary 2.3 stillholds.Thereasonisthatthecolumsoftheconstraintmatrixremainlinearlyindependentwheneitheroneoftheconstraints`ixiorxiuiisdropped(butnotboth).In[ 64 ]Tardellagivesaconditionthatensuresedge-concavityforafunctiondenedoverapolyhedronPBthatincludesaboxconstraint.Iff:X!RwhereXRnisaconvexset,thenfisk)]TJ /F3 11.955 Tf 11.71 0 Td[(concaveoverXiff(xxx+(1)]TJ /F10 11.955 Tf 11.72 0 Td[()yyy)f(xxx)+(1)]TJ /F10 11.955 Tf 11.72 0 Td[()f(yyy)forevery2[0,1]andforeveryxxx,yyy2Xsuchthatxi=yiforatleastn)]TJ /F3 11.955 Tf 12.16 0 Td[(kindicesi2[1,n].ThefollowingresultfollowsfromProposition2.4of[ 64 ]: Proposition2.3(Tardella). Supposef:P!Ristwicecontinuouslydifferentiable,rank(AAA)=k)]TJ /F3 11.955 Tf 12.22 0 Td[(1forsomek,andDisanedgedescriptionofPB.Iffisk)]TJ /F3 11.955 Tf 12.21 0 Td[(concaveoverPB,thendddTr2f(xxx)ddd0foreveryddd2D.WecombineCorollary 2.2 ,Corollary 2.3 ,andProposition 2.3 toobtainthefollowingresult: Corollary2.4. Supposethatrank(AAA)=k)]TJ /F3 11.955 Tf 12.72 0 Td[(1forsomek,theobjectivefunctionfisquadratic,fisk)]TJ /F3 11.955 Tf 12.46 0 Td[(concaveoverPB,andDisareectiveedgedescriptionofPB.Thenafeasiblepointxxxofproblem( 2 )islocallyoptimalifandonlyif( 2 )and( 2 )hold.Hence,undertheseassumptions,localoptimalityin( 2 )canbecheckedintimeboundedbyapolynomialinn. 28

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CHAPTER3THEVERTEXSEPARATORPROBLEM 3.1IntroductionLetGbeasimple,undirectedgraphwithverticesV=f1,2,...,ng,withedgesEVV,andwithnonnegativevertexweightsc1,c2,...,cn,notallzero.(Notethatinthepresentchapter,weusethewordvertextorefertoanodeinagraphandedgetorefertoaconnectionbetweentwonodes.WhilethesewordswerealsodenedinChapter2inthecontextofpolyhedra,theintendedmeaningshouldalwaysbeclearfromcontext.)Sincethegraphisundirected,(i,j)2Eifandonlyif(j,i)2E,andsincethegraphissimple,theelementsofEaredistinctand(i,i)62Eforanyi2V.AvertexseparatorofGisasetofverticesSwhoseremovalbreaksthegraphintotwodisconnectedsetsofverticesAandB.Thatis,(AB)\Eisempty.TheVertexSeparatorProblem(VSP)istominimizethesumoftheweightsofverticesinSwhilerequiringthatAandBsatisfysizeconstraints: minA,BVXi2Sci (3) subjecttoS=Vn(A[B),A\B=;,(AB)\E=;,`ajAjua,`bjBjub.HerejAjdenotesthenumberofelementsinthesetA,and`a,ua,`b,andubaregivenintegerparametersthatdescribetheexibilityinthesizeofthesetsAandB.Theseparametersshouldbesuchthat0`auanub`b0.Weassumethat( 3 )isfeasible.If`a,`b1,thenthisimpliesGisnotcomplete;thatis,forsomei6=j2V,wehave(i,j)=2E.VertexseparatorshaveapplicationstoVLSIchipdesign[ 39 42 66 ],toniteelementmethods[ 47 ],toparallelprocessing[ 19 ],to 29

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thecomputationofll-reducingorderingsforsparsematrixfactorizations[ 25 ],andtonetworksecurity.ThedifcultyofsolvingtheVSPandthesizeoftheoptimalsolutionarestronglytiedtothestructureofthegraph.Forinstance,everytreehasanoptimalvertexseparatorconsistingofexactlyonevertex.Forplanargraphs,LiptonandTarjanshowedthataseparatorofsizeO(p n)canbefoundinlineartime[ 43 ].However,forgeneralgraphs(andevenplanargraphs)theVSPisNP-hard[ 9 22 ].Hence,heuristicalgorithmshavebeendevelopedforobtainingapproximatesolutions;forexample,see[ 18 20 38 ].In[ 1 11 16 ],theauthorsstudiedthefollowingexactintegerprogrammingformulationoftheVSP: maxxxx,yyy2BncccT(xxx+yyy) (3) subjecttoxxx+yyy111,xi+yj1forevery(i,j)2E`a111Txxxua,`b111Tyyyub.HereBn=f0,1gnisthecollectionofbinaryvectorswithncomponents,ciistheweightofvertexi,111isthevectorwhoseentriesareall1,andxxxandyyyaretheincidencevectorsforAandBrespectively;thatis,xi=1ifi2Aandxi=0otherwise.Therefore,minimizingtheweightoftheseparatorisequivalenttomaximizingtheweightcccT(xxx+yyy)oftheverticesoutsidetheseparator.Theinequalityxxx+yyy111isthepartitionconstraint,whichensuresthatAandBaredisjoint.Theconditionxi+yj1when(i,j)2Eistheseparationconstraint,whichensuresthat(AB)\E=;.Finally,thebalanceconstraints`a111Txxxuaand`b111TyyyubrestrictthesizeofthesetsAandB.In[ 1 ]theauthorsstudiedtheprogram( 3 )inthecasewhere`a=`b=1.Validinequalitiesfortheintegerpolytopeof( 3 )wereobtainedandtheprogramwassolvedonavarietyofsmall(n200)probleminstancesusingabranchandcutalgorithm.In[ 11 ]animprovedalgorithmwaspresentedwhichmadeuseofLagrangianrelaxation, 30

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improvingthepoolofcuttingplanesandprovidingbetterprimalboundsforthenodesinthesearchtree.Inthecurrentpaper,wedevelopconditionsunderwhichtheVSPisequivalenttothefollowingcontinuousbilinearquadraticprogramforsomechoiceoftheparameter>0: minxxx,yyy2Rn)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy (3) subjectto000xxx111,000yyy111,`a111Txxxua,and`b111Tyyyub,whereAAAistheadjacencymatrixforthegraphG;thatis,aij=1if(i,j)2Eandaij=0otherwise.WewillshowthatthetermxxxT(AAA+III)yyyintheobjectivefunctionamountstoapenaltytermforenforcingtheseparationconstraintthattherearenoedgesconnectingAandB.Weshowthat( 3 )isequivalenttotheVSPifthefollowingconditionsaresatised: (C1)ciforalli. (C2)Thetotalweightofanoptimalvertexseparatorislessthanorequalto nXi=1ci!)]TJ /F10 11.955 Tf 11.96 0 Td[((`a+`b).(3)Therstconditionissatisedbytaking=maxfci:1ing.Inpractice,thesecondconditionisofteneasilysatised.Inthecommoncasewhereci=1foralliand`a=`b=1,theexpression( 3 )reduceston)]TJ /F3 11.955 Tf 12.15 0 Td[(2.Hence,inthiscase,(C2)issatisedaslongas( 3 )isfeasible,sinceAandBmusteachcontainatleastonevertex.TheequivalencebetweentheVSPand( 3 )isinthefollowingsense:Foranysolutionof( 3 ),thereisanassociated,easilyconstructedbinarysolution.Moreover,when(C1)and(C2)hold,thereexistsabinarysolutionforwhichthepenaltytermvanishesandtheseparationconstraintissatised.Forsuchasolution,anoptimal 31

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separatorfortheVSPisgivenby A=fi:xi=1g,B=fi:yi=1g,andS=fi:xi=yi=0g.(3)Insomeapplicationssuchasniteelementmethods,parallelprocessing,andsparsematrixfactorizations,itisimportanttoobtainanapproximatesolutiontotheVSPquickly.Inthiscase,wecanapplymultileveltechniquessuchasthosedescribedin[ 32 38 61 62 ]andstandardoptimizationalgorithmssuchasthegradientprojectionalgorithm(see[ 3 ])toquicklycomputeanapproximatesolutionof( 3 ).AmultilevellevelalgorithmispresentedinChapter 4 .Inotherapplicationswhereweneedtosolve( 3 )exactly,branchandboundtechniquescanbeapplied.Forillustration,in[ 31 ]theauthorsdevelopabranchandboundalgorithmforthecloselyrelatededgeseparatorproblem.Thecontinuousformulationoftheedgeseparatorproblemisthesameas( 3 ),butwiththeadditionalconstraintxxx+yyy=111.In[ 31 ]weshowthatabranchandboundalgorithmappliedtothecontinuousformulationoftheedgeseparatorproblemisparticularlyeffectiveforsparsegraphs.Asnotedearlier,ourcontinuousformulationoftheVSPisinsomesenseanexactpenaltymethod.Inmostexactpenaltymethodsforsolvingbinaryminimizationproblems,thepenaltyfunctionischosenbothtomaketheobjectivefunctionconcave,guaranteeinganextremepointsolution[ 2 ],andtoforcetheextremesolutiontobebinary[ 26 56 ].Ourpenaltyformulationdiffersinthesetwocrucialaspects.Inparticular,iffistheobjectivefunctionin( 3 ):f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy,thentheHessianisr2f=0B@000BBBBBB0001CA,BBB=AAA+III. 32

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Ifi,1in,aretheeigenvaluesofBBB,theni,1in,aretheeigenvaluesoftheHessian.Hence,fisneitherconvexnorconcave,andthenumberofpositiveeigenvaluesoftheHessianisequaltothenumberofnegativeeigenvalues.Nonetheless,wewillshowthatfisedge-concaveoverthefeasiblesetin( 3 ).Consequently,theexistenceofanextremepointminimizerfollowsfromTheorem 1.8 .Furthermore,weshowthateveryextremepointoftheconstraintpolyhedronin( 3 )isbinary,andif(C1)and(C2)hold,thenthereexistsabinaryminimizerof( 3 )suchthatxxxT(AAA+III)yyy=0.Theideaofexploitingconcavitytoobtainanextremepointsolutiontoacontinuousminimizationproblemhasalonghistory.TherstresultofthistypewasduetoBauer[ 2 ]:Theminimumvalueofaconcavefunctionoveracompact,convexsetisattainedatanextremepointoftheset.(Alsosee[ 58 ,Thm.32.3]and[ 33 ]).Inaseminalpaper,Raghavachari[ 56 ]appliedthisresulttothefollowingbinarylinearprogram: minxxx2BncccTxxxsubjecttoBBBxxxbbb.(3)Hereccc2Rnisanarbitraryvector,bbb2Rm,andBBB2Rmn.Observethatthebinaryconstraintxxx2BnisequivalenttothequadraticconstraintxxxT(111)]TJ /F3 11.955 Tf 11.78 0 Td[(xxx)=0,where1112Rnisthevectorofallones.Raghavacharishowedthatforsufcientlylarge0,( 3 )isequivalenttothefollowingcontinuouspenalizedproblem: minxxx2RncccTxxx+xxxT(111)]TJ /F3 11.955 Tf 11.43 0 Td[(xxx)subjecttoBBBxxxbbb,000xxx111.(3)Sincethepenalizedobjectivein( 3 )isconcave,Bauer'stheoremguaranteesthat( 3 )hasanextremepointsolution.Furthermore,whenissufcientlylarge,Raghavacharishowsthatthepenaltytermiszero,whichimpliesthatthesolutionisbinary.Raghavachari'sresultwasgeneralizedtononlinearbinaryprogrammingproblemsbyNiccolucciandGianessi[ 26 ].LowerboundsonwereobtainedbyRosenandKalantari[ 35 ]andbyBorchardt[ 6 ].Morerecently,Zhu[ 68 ]foundimprovedlower 33

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boundsoninthecasewheretheobjectiveisquadratic.AlternativechoicesofthepenaltyfunctionwereconsideredbyRinaldiandLucidi[ 44 57 ].AsnotedbyMurrayandNg[ 49 ],oneproblemwithconcaveformulationsofdiscreteminimizationproblemsistheastronomicalnumberoflocalminimizersthatarecreatedbymakingtheproblemconcave.Whenacontinuousoptimizationalgorithmisappliedtotheseprograms,theiteratestypicallyconvergetoastationarypoint,andhenceanyoneoftheselocalminimacantraptheiterates.Ontheotherhand,sincethepenaltyfunctionin( 3 )isnon-concave,thenumberoflocalminimaisreduced.Moreover,byTheorem 2.3 ,theprogram( 3 )hasthepropertythatlocalminimalitycanbedeterminedinpolynomialtime,whileforageneralindenitequadraticprogram,determiningwhetherastationarypointisalocalminimizerisNP-hard[ 50 54 ].Twootheroptimizationproblemsforgraphshavebeenformulatedascontinuousquadraticprograms.TherstwasthemaximumcliqueproblemanalyzedbyMotzkinandStraus[ 48 ],andthesecondwastheedgeseparatorproblemanalyzedbyHagerandKrylyuk[ 29 30 ].In[ 48 ]MotzkinandStrausshowthatifthemaximumsizeofacliqueinthegraphGisk,thenthefollowingquadraticprogramhasoptimalobjective(k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)=(2k): maxxxx2RnxxxTAxAxAxsubjecttoxxx000and111Txxx=111.(3)Moreover,thereexistsamaximumcliqueCandasolutionof( 3 )suchthatxi=1=kifandonlyifi2Candxi=0otherwise.Oneimportantdifferencebetweentheformulations( 3 )and( 3 )isthatadesiredsolutionof( 3 )isnotavertexofthefeasibleset,whileadesiredsolutionof( 3 )isavertexofthefeasibleset.Theedgeseparatorproblem,oftencalledthegraphpartitioningproblem,istopartitionV=A[BsoastominimizethenumberofedgesbetweenAandB,subjecttoaconstraint`jAju.Itwasshownin[ 29 ]thatthefollowingcontinuousquadraticprogrammingproblemisequivalenttotheedgeseparatorproblem: min(111)]TJ /F3 11.955 Tf 11.43 0 Td[(xxx)T(AAA+III)xxxsubjectto000xxx111,`111Txxxu. 34

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Inotherwords,associatedwithanysolutionof( 3 ),thereisaneasilycomputedbinarysolution,andforanybinarysolutionof( 3 ),anoptimalsolutionoftheedgeseparatorproblemisA=fi:xi=0gandB=fi:xi=1g.Observethat( 3 )isequivalenttotheprogram( 3 )with`b=0,withub=n,andwiththeadditionalconstraintxxx+yyy=111.Hence,theedgeseparatorproblemisthevertexseparatorproblemwithanadditionalconstraint.Necessaryandsufcientoptimalityconditionsfor( 3 )aregivenin[ 29 30 ],whileabranchandboundalgorithmforsolving( 3 )exactlyisgivenin[ 31 ].Recently,PovhandRendl[ 55 ]developedacompletelypositivelinearprogrammingformulationforaminimumcutprobleminwhichVispartitionedintothreesetsofgivensizesuchthatthenumberofedgesbetweentwoofthesesetsisminimized.Thegraphpartitioningproblemisaspecialcaseoftheirminimumcutproblemcorrespondingtosettingthesizeofoneofthesetsto0.Theyreformulatetheminimumcutproblemasalinearprogramovertheconeofcompletelypositivematrices.Moreover,Burer[ 10 ]showsthatanymixedintegerquadraticprogram(suchas( 3 )or( 3 ))canbeformulatedasacompletelypositivelinearprogram.Whileformulatingadiscreteproblemasacompletelypositiveprogramdoesnotimprovetheproblem'stractability,itdoesprovideanavenueforobtainingtractablerelaxations(seeforinstance[ 55 ]).Asurveyofcopositiveprogrammingisgivenin[ 5 ].InSection 3.2 wegiveaconstructiveproofoftheequivalencebetweenthecontinuousbilinearprogram( 3 )andtheintegerprogram( 3 ).Theproofbasicallyprovidesaneasilyimplementedalgorithmtotransformasolutionofthecontinuousprogram( 3 )intoasolutionofthebinaryvertexseparatorproblem( 3 ).InSection 3.3 ,weuseCorollary 2.2 toderiveeasilycheckedlocaloptimalityconditionsfor( 3 ).Therelationshipbetweenlocalsolutionsinthecontinuousprogramandintheoriginaldiscreteproblemisalsoanalyzed. 35

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3.2ContinuousEdge-ConcaveQuadraticProgrammingFormulationLetGbeanundirectedgraphonvertexsetV=f1,2,...,ngwithvertexweightsci0notallzero.Let0`auanand0`bubnbeintegers.WestartbyconsideringtheintegerprogrammingformulationoftheVSP( 3 ),whichwerestatehereforconvenience: maxxxx,yyy2BncccT(xxx+yyy) (3) subjecttoxxx+yyy111,xi+yj1forevery(i,j)2E`a111Txxxua,`b111Tyyyub.RecallthatxxxandyyyaretheincidencevectorsforAandBrespectively;thatis,xi=1ifi2Aandxi=0otherwise.Observethatforanyxxxandyyy2Bn,thepartitionconstraintA\B=;in( 3 )ortheconditionxxx+yyy111in( 3 )isequivalentto xxxTyyy=0.(3)Furthermore,ifAAAistheadjacencymatrixforthegraphGdenedbyaij=1if(i,j)2E0if(i,j)=2E,thentheproductxxxTAAAyyycountsthenumberofedgesbetweenAandB:xxxTAyAyAy=nXi=1nXj=1xiaijyj=Xxi=1Xyj=1aij=Xi2AXj2Baij=j(AB)\Ej.Hence,theseparationconstraint(AB)\E=;in( 3 )orxi+yj1in( 3 )isequivalenttothecondition xxxTAAAyyy=0.(3) 36

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Sincetheentriesinxxx,yyy,andAAAareallnon-negative,( 3 )and( 3 )bothholdifandonlyifxxxT(AAA+III)yyy=0.Therefore,thevertexseparatorproblem( 3 )or( 3 )isequivalenttothefollowingbinaryprogram: minxxx,yyy2Bn)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy) (3) subjecttoxxxT(AAA+III)yyy=0,`a111Txxxua,and`b111Tyyyub.(Notethatherewehavereplacedmaxbyminandtakenthenegativeoftheobjectivefunction,inordertoputthisprobleminthecontextofChapters1and2.)Inparticular,if(xxx,yyy)isasolutionto( 3 ),thenasolutiontothevertexseparatorproblem( 3 )is A=fi:xi=1g,B=fi:yi=1g,andS=fi:xi=yi=0g.(3)Nowconsiderthefollowingcontinuouspenalizedproblem: minxxx,yyy2Rn)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy (3) subjectto000xxx111,000yyy111,`a111Txxxua,and`b111Tyyyub.Forany0theoptimalobjectivefunctionvalueinthecontinuousprogram( 3 )isatleastassmallastheoptimalvalueinthebinaryvertexseparatorproblem( 3 )sincetherearemoreconstraintsinthebinaryproblemthatis,thevariablesintherelaxedproblem( 3 )canhaveanyvaluebetween0and1,whilein( 3 )itisrequiredthatxxxT(AAA+III)yyy=0.Wenowshowthatforanappropriatechoiceof,therelaxedproblem( 3 )andthebinaryproblem( 3 )areequivalent. Theorem3.1. Supposethevertexseparatorproblem( 3{1 )isfeasible.Let2R. 1. Thecontinuousprogram( 3{14 )hasabinarysolution. 2. Everystrictlocalminimizerof( 3{14 )isbinary. 37

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3. If(C1)and(C2)hold,thenthereexistsabinarysolution(xxx,yyy)toproblem( 3{14 )suchthat xxxT(AAA+III)yyy=0.(3)Moreover,if>maxfci:i2Vg,theneverybinarysolutionsatises( 3{15 ). 4. Foranybinarysolution(xxx,yyy)of( 3{14 )satisfyingthecondition( 3{15 ),anoptimalsolutiontothevertexseparatorproblem( 3{1 )isA=fi:xi=1g,B=fi:yi=1g,andS=fi:xi=yi=0g. Proof. Wepartitiontheproofintothefourpartsofthetheorem.Part1.Weprovethefollowingstrongerresult: (P)Forany(xxx,yyy)feasiblein( 3 ),thereexistsaneasilyconstructedfeasiblepoint[^xxx,^yyy]2B2nsuchthatf(^xxx,^yyy)f(xxx,yyy).Ifxxxandyyyarebinary,then(P)holdsandwearedone.Otherwise,eitherxxxoryyyisnotbinary.Withoutlossofgenerality,supposethatxxxisnotbinary.Case1:Supposethatxxxhasexactlyonenon-binarycomponentxk.Since`aanduaareintegers,wecannothave111Txxx=`aor111Txxx=uasincethesumofn)]TJ /F3 11.955 Tf 11.19 0 Td[(1binarynumbersandanumberintheopeninterval(0,1)cannotbeaninteger.Hencethepoint(xxx+teeek,yyy)isfeasibleforallsufcientlysmalljtj.Letfdenotetheobjectivefunctionof( 3 ): f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy(3)Sincefisbilinear,rf(xxx+teeek,yyy)=f(xxx,yyy)+tdforsomescalard.Ifd0,thenchooset<0suchthatxk+t=0.Ifd<0,thenchooset>0suchthatxk+t=1.Thistakesustoafeasiblepoint(^xxx,yyy)suchthat^xxx2Bnandf(^xxx,yyy)f(xxx,yyy).Case2:Supposexxxhasatleasttwonon-binarycomponentsiandj.Forsufcientlysmalljtjthepoint(xxx+t(eeei)]TJ /F3 11.955 Tf 11.4 0 Td[(eeej),yyy)isfeasiblein( 3 )sincetheconstraint`a111Txxxua 38

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isunaffected.Again,sincefisbilinear,rf(xxx+t(eeei)]TJ /F3 11.955 Tf 11.43 0 Td[(eeej),yyy)=f(xxx,yyy)+tdforsomescalard.Ifd0,thenchoosethelargestt<0suchthatxi+t=0orxj)]TJ /F3 11.955 Tf 10.79 0 Td[(t=1.Ifd<0,thenchoosethesmallestt>0suchthatxi+t=1orxj)]TJ /F3 11.955 Tf 12.5 0 Td[(t=0.Thepoint(xxx,yyy)=(xxx+t(eeei)]TJ /F3 11.955 Tf 12.04 0 Td[(eeej),yyy)isfeasible,ithasatleastonemorebinarycomponentthan(xxx,yyy),anditsatisesf(xxx,yyy)f(xxx,yyy).Wemayiteratethisprocedure,makingnon-binarycomponentsofxxxbinarywithoutincreasingtheobjectivevalue,untilwereachapoint(^xxx,yyy)suchthateither^xxx2Bnor^xxxhasasinglenon-binarycomponent,whichreducestoCase1.Thesameprocedurecanbeappliedtoyyytocompletetheproofof(P)andpart1.Part2.ThisfollowsfromtheproofofPart1.Ifastrictlocalminimizerisnotbinary,thentheconstructioninPart1showshowtomovetoabinarypointwithoutincreasingtheobjectivefunctionvalue.Hence,astrictlocalminimizermustbebinary.Part3.Suppose(C1)and(C2)hold.Let(xxx,yyy)beabinarysolutiontoproblem( 3 ).Sincexxx,yyy,andAAAarenonnegative,wehavexxxT(AAA+III)yyy0.Supposethat xxxT(AAA+III)yyy>0.(3)Sincetheobjectivefunctionin( 3 )isthenegativeoftheweightofthenodesoutsideaseparator,itfollowsthat minimumin( 3 )=minimumweightofvertexseparator)]TJ /F4 7.97 Tf 18.24 14.94 Td[(nXi=1ci (3) )]TJ /F10 11.955 Tf 28.56 0 Td[((`a+`b),wheretheinequalityisdueto(C2).Sincetheobjectivefunctionvaluef(xxx,yyy)intherelaxedproblem( 3 )isatleastassmallastheminimumin( 3 ),itfollowsfrom 39

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( 3 )that)]TJ /F10 11.955 Tf 9.3 0 Td[((`a+`b)f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy>)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy))]TJ /F10 11.955 Tf 21.92 0 Td[(111T(xxx+yyy).Thestrictinequalityisdueto( 3 ),andthelastinequalityisduetotheconstrainton.Therefore,wemusthaveeither111Txxx>`aor111Tyyy>`b.Assumewithoutlossofgeneralitythat111Txxx>`a.Sincexxxisbinaryand`aisaninteger,wehave111Txxx`a+1.Sincetheentriesinxxx,yyy,andAAAareallnon-negativeintegers,( 3 )impliesthatthereexistsanindexisuchthat(AAA+III)iyyy1andxi=1(recallthatsubscriptsonamatrixcorrespondtotherows).If^xxx=xxx)]TJ /F3 11.955 Tf 12.14 0 Td[(eeei,then(^xxx,yyy)isfeasibleinproblem( 3 )since111T^xxx=111Txxx)]TJ /F3 11.955 Tf 11.96 0 Td[(1`a.Furthermore, f(^xxx,yyy)=f(xxx,yyy)+ci)]TJ /F10 11.955 Tf 11.96 0 Td[((AAA+III)iyyyf(xxx,yyy)+ci)]TJ /F10 11.955 Tf 11.95 0 Td[(f(xxx,yyy),(3)since(AAA+III)iyyy1andci.Since(xxx,yyy)wasoptimalin( 3 ),sois(^xxx,yyy).Wecancontinuetosetcomponentsofxxxandyyyto0untilreachingabinaryfeasiblepoint(^xxx,^yyy)forwhich^xxxT(AAA+III)^yyy=0andf(xxx,yyy)=f(^xxx,^yyy).If>maxfci:i2Vg,then( 3 )isastrictinequality,whichcontradictstheoptimalityof(xxx,yyy)in( 3 ).Hence,xxxT(AAA+III)yyy=0,andtheproofofPart3iscomplete.Part4.Abinarysolutionoftherelaxedproblem( 3 )satisfying( 3 )isfeasibleinthediscreteproblem( 3 ).Moreover,theobjectivefunctionvaluesintherelaxedproblemandinthediscreteproblemarethesame.Hence,(xxx,yyy)isoptimalinthediscreteversion( 3 )ofthevertexseparatorproblem,andasolutionofthevertexseparatorproblemisgivenby( 3 ).ThiscompletestheproofofPart4. 40

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WenowexplaintheconnectionbetweenTheorem 3.1 andTheorem 1.8 ,whichyieldssomedeeperinsightsontherelationshipbetweenthediscreteandcontinuousformulationsofthevertexseparatorproblem.Inparticular,wewillshowthatPart1ofTheorem 3.1 followsfromedge-concavityoftheobjectivefunctionof( 3 ).ThepolyhedronQwhichdenesthefeasiblesetinproblem( 3 )isQ=f[xxx,yyy]2R2n:000xxx111,000yyy111,`a111Txxxua,`b111Tyyyubg.Toprovethatproblem( 3 )hasabinarysolution,wewillshowthattheverticesofQarepreciselythebinaryfeasiblepointsandthattheedge-concavityconditionofTheorem 1.8 issatised.Forconvenience,werefertotheconstraints000xxx111and000yyy111asthecomponentconstraintsofQ,andtheconstraintson111Txxxand111TyyyasthebalanceconstraintsofQ. Lemma3.1. ThesetofverticesofQisQ\B2n. Proof. If[xxx,yyy]2Q\B2n,then2ncomponentconstraintsareactiveat[xxx,yyy];sincetheseconstraintsarelinearlyindependent,[xxx,yyy]isavertexofQ.Conversely,supposethat[xxx,yyy]isavertexofQ.TheconstraintsdeningQcanbewrittenintheformBxBxBxbbbwhereBBBistotallyunimodularandbbbisinteger.Asaconsequence,theverticesofQareintegervectors.Duetothecomponentconstraints,theverticesarebinaryvectors.Anotherproofoftheconverse,whichisexploitedinthenextlemma,isasfollows:Atavertex[xxx,yyy]ofQ,2nlinearlyindependentconstraintsareactive.Duetotheseparableformoftheconstraints,therearenactivelinearlyindependentconstraintsoneachofxxxandyyy.Ifthenactiveconstraintsonxxxarecomponentconstraints,thenxxxisbinary.Iftherearen)]TJ /F3 11.955 Tf 12.67 0 Td[(1activecomponentconstraintsatxxxalongwithaconstraintoftheform111Txxx=`aor111Txxx=ua,thenthesinglefractionalcomponentofxxxmustbebinarysince`aanduaareinteger.Asimilarargumentappliestoyyy. 41

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LetDR2nbedenedby D=n[i,j=1,i6=jf[eeei,000],[000,eeei],[000,eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej],[eeei)]TJ /F3 11.955 Tf 11.43 0 Td[(eeej,000]g.(3) Lemma3.2. DisanedgedescriptionofQ.Thatis,foreveryedgeEofQ,thereexistssomeddd2DsuchthatdddisparalleltoE. Proof. LetEbetheedgeofQdenedbyE=f[xxx(t),yyy(t)]:t2[0,1]g,where[xxx(t),yyy(t)]=vvv1+t(vvv2)]TJ /F3 11.955 Tf 11.43 0 Td[(vvv1)andvvv1=(xxx1,yyy1)andvvv2=(xxx2,yyy2)areverticesofQ.Werstshowthateitherxxx1=xxx2oryyy1=yyy2.Supposetothecontrarythatxxx16=xxx2andyyy16=yyy2.Thenthereexistindicesiandjsuchthatxxx1i6=xxx2iandyyy1j6=yyy2j.Sincevvv1andvvv2arebinarybyLemma 3.1 ,itfollowsthat 0
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followsthatxi(t)+xj(t)isindependentoft.Hence,asxi(t)increases,xj(t)decreases.Sincexi(t)andxj(t)areconvexcombinationsof0and1,itfollowsthattheedgeisparallelto(eeei)]TJ /F3 11.955 Tf 11.7 0 Td[(eeej,000)2D.Ifxxx1=xxx2,thenasimilarargumentshowsthatEisparalleltoeither(000,eeei)2Dor(000,eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej)2Dforsomeiandj. Proposition3.1. Forany,theproblem( 3 )hasabinarysolution. Proof. ByLemma 3.2 ,Din( 3 )isanedgedescriptionofQ.Letf:Q!Rdenotetheobjectivefunctionof( 3 )denedin( 3 ).ItcanbecheckedthatdddTr2f(xxx,yyy)ddd=0forallddd2D.Hence,fisconcavealongtheedgesofQ,andbyTheorem 1.8 ,fattainsitsminimumoverQatavertexofQ.ByLemma 3.1 ,thisvertexisbinary. 3.3LocalSolutionsInSection 3.2 ,wedevelopedanequivalencebetweenthevertexseparatorproblemandthefollowingquadraticprogram: minxxx,yyy2Rn)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy (3) subjectto000xxx111,000yyy111,`a111Txxxua,and`b111Tyyyub.Inthepresentsection,wewillstudythelocalsolutionsto( 3 ).WebeginbyusingthetheoryofSection 2.1 toderiveeasilycheckednecessaryandsufcientconditionsforlocaloptimality.Letfbetheobjectivefunctionin( 3 ):f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy.Since( 3 )involves4lowerboundsand4upperbounds,thestandardstatementoftheKKTconditions( 1 )involves8multipliers,8inequalityconstraints,and8complementaryslacknessconditions.Amorecompactwayofexpressingthese16conditionsisasfollows:If(xxx,yyy)isalocalminimizerof( 3 ),thenthereexistmultipliers 43

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aandb2Rnandaandb2Rsuchthat 264rxxxf(xxx,yyy)ryyyf(xxx,yyy)375+264ab375+264a111b111375=000,(3)wherea2M(xxx),b2M(yyy),a2L(xxx,`a,ua),b2L(yyy,`b,ub)withM(zzz)=f2Rn:izimaxfi,0gforall1ing,andL(zzz,`,u)=f2R:111Tzzzmaxfu,`gg.AsshowninSection 3.2 ,thefollowingsetisareectiveedgedescriptionofthepolyhedronassociatedwith( 3 ):D=n[i,j=1,i6=jf[eeei,000],[000,eeei],[000,eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej],[eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej,000]g(reectivityistriviallyveried).Sincer2f=0B@000(AAA+III)(AAA+III)0001CA,itcanbecheckedthatdddT(r2f)ddd0foreveryddd2D.Therefore,byCorollary 2.2 ,afeasiblepoint(xxx,yyy)of( 3 )isalocalminimizerifandonlyiftherst-orderoptimalityconditions( 3 )holdand (ddd1)T(r2f)ddd20foreveryddd1,ddd22C(xxx,yyy)\D.(3)Tables 3-1 and 3-2 giveallthedifferentpossiblevaluesfor(ddd1)T(r2f)ddd2,whereddd1andddd2areedgedirections,intermsofHHH=AAA+III.SinceDisdescribedintermsof6differentvectors,thereare36products(ddd1)T(r2f)ddd2correspondingtothe66differentpairsamongthevectorsdescribingD.However,15oftheseproductsareknownby 44

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symmetry.Theremaining21productsareshowninTables 3-1 and 3-2 .Theblankentriescorrespondtoentriesknownfromsymmetry. Table3-1.Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. (eeek,000)(000,eeek)()]TJ /F3 11.955 Tf 8.76 0 Td[(eeek,000)(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek) (eeei,000)0hik0)]TJ /F10 11.955 Tf 9.3 0 Td[(hik(000,eeei)0)]TJ /F10 11.955 Tf 9.3 0 Td[(hik0()]TJ /F3 11.955 Tf 8.77 0 Td[(eeei,000)0hik(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)0 Table3-2.Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. (eeek)]TJ /F3 11.955 Tf 11.43 0 Td[(eeel,000)(000,eeek)]TJ /F3 11.955 Tf 11.43 0 Td[(eeel) (eeei,000)0(hki)]TJ /F3 11.955 Tf 11.96 0 Td[(hli)(000,eeei)(hki)]TJ /F3 11.955 Tf 11.95 0 Td[(hli)0()]TJ /F3 11.955 Tf 8.76 0 Td[(eeei,000)0(hli)]TJ /F3 11.955 Tf 11.95 0 Td[(hki)(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)(hli)]TJ /F3 11.955 Tf 11.96 0 Td[(hki)0(eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej,000)0(hik)]TJ /F3 11.955 Tf 11.96 0 Td[(hil)]TJ /F3 11.955 Tf 11.96 0 Td[(hjk+hjl)(000,eeei)]TJ /F3 11.955 Tf 11.43 0 Td[(eeej)0 Suppose(xxx,yyy)satisestherst-orderoptimalityconditions( 3 ).Thecondition( 3 )statesthat(xxx,yyy)isalocalminimizerifandonlyiftheentriesofTables 3-1 and 3-2 arenon-negativewheneverthevectorsinthecorrespondingrowandcolumnlieinC(xxx,yyy).Forexample,if(xxx,yyy)isalocalminimizer,thenwhenever(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek)lieintheconeC(xxx,yyy)forsomeiandk,wemusthavehik0(since>0).Thisimpliesthataik=0.Moreover,sincehii=1,itfollowsthatboth(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)cannotbecontainedinC(xxx,yyy)atalocalminimizer.Inordertoverifyeithertherstorsecond-orderoptimalityconditions,weneedtodeterminewhenavectorintheedgedescriptionDliesinF(xxx,yyy)orC(xxx,yyy).Fromthedenitions( 1 )and( 1 ),wehaveF(xxx,yyy)=8>>>>>>><>>>>>>>:(ddd1,ddd2)2R2n:111Tddd10if111Txxx=ua,d1i08iwithxi=1,111Tddd10if111Txxx=`a,d1i08iwithxi=0,111Tddd20if111Tyyy=ub,d2i08iwithyi=1,111Tddd20if111Tyyy=`b,d2i08iwithyi=09>>>>>>>=>>>>>>>;, 45

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(eeei,000)2C(xxx,yyy),111Txxx`axi>0ai=a=0,111Txxx>`aa=0i2A0(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)2C(xxx,yyy),111Tyyy>`byi>0bi=b=0,111Tyyy>`bb=0i2B0(eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej,000)2C(xxx,yyy),xi<1andxj>0ai=aj=0,i2A0andj2A0(000,eeei)]TJ /F3 11.955 Tf 11.42 0 Td[(eeej)2C(xxx,yyy),yi<1andyj>0bi=bj=0,i2B0andj2B0.Figure3-1. AdescriptionofD\C(xxx,yyy) C(xxx,yyy)=ddd2F(xxx,yyy):111Tddd1=0ifa6=0,di=08iwithai6=0111Tddd2=0ifb6=0,di=08iwithbi6=0.Nowdenethefollowingsets:A0=fi:xi>0,ai=0gandA0=fi:xi<1,ai=0g,B0=fi:yi>0,bi=0gandB0=fi:yi<1,bi=0g.Figure 3-1 showswheneachelementofDalsoliesinC(xxx,yyy).CombiningtheinformationinTables 3-1 and 3-2 andFigure 3-1 ,wewillestablishthefollowingtheorem. Theorem3.2. If(xxx,yyy)isfeasiblein( 3 ),then(xxx,yyy)isalocalminimizerof( 3 )ifandonlyif(V1){(V5)hold: (V1)Therst-orderconditions( 3 )hold. (V2)Supposea=0. 46

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a. If111Txxx`a,i2A0,j2B0,andk2B0,thenhijhik. (V3)Supposeb=0. a. If111Tyyy`b,i2B0,j2A0,andk2A0,thenhijhik. (V4)Supposea=b=0. a. If111Txxx>`aand111Tyyy`b,thenA0\B0=;andhij=0wheneveri2A0andj2B0. (V5)Ifi2A0,j2A0,k2B0,andl2B0,thenhik+hjlhil+hjk. Proof. First,supposethattheconditions(V1)(V5)aresatised.Wewishtoshowthat( 3 )holds,whichimpliesthat(xxx,yyy)isalocalminimizer.ConsideranyentryinTable 3-1 or 3-2 thatispotentiallynegative,suchas)]TJ /F10 11.955 Tf 9.3 0 Td[(hik.Referringtotherowandcolumnedgedirectionsinthetable,weonlyneedtoconsiderthispossibilityif(a)both(000,eeei)and()]TJ /F3 11.955 Tf 8.77 0 Td[(eeek,000)2C(xxx,yyy)or(b)both(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek)2C(xxx,yyy).If(b)holds,thenbyFigure 3-1 ,wehave111Txxx`b,b=0,andk2B0.By(V4b),i6=kandhik=0.If(a)holds,thenbyFigure 3-1 ,wehave111Txxx>`a,a=0,k2A0,111Tyyy
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ofFigure 3-1 ,wehave(eeei,000)and(000,eeej)]TJ /F3 11.955 Tf 11.47 0 Td[(eeek)2C(xxx,yyy).By( 3 )andTables 3-1 and 3-2 ,hji)]TJ /F3 11.955 Tf 12.33 0 Td[(hki0.Bysymmetry,thisisequivalenttohijhik.Thisestablishes(V2a).Thustheproofoftheconverseproceedsasfollows:Whenanyofthehypothesesin(V2)(V5)arefullled,weuseFigure 3-1 todeterminetwoedgedirectionsddd1andddd2thatmustlieinC(xxx,yyy).WeuseTable 3-1 and 3-2 toobtaintheproduct(ddd1)T(r2f)ddd2,andweuse( 3 )toobtainarelationbetweenelementsofHHHwhichcorrespondstotheconclusionappearingin(V2)(V5). Next,weexploretherelationshipbetweenlocaloptimalityin( 3 )andlocaloptimalityin( 3 ). Denition3.1. Let(A,S,B)beafeasiblepartitioninproblem( 3 ).Wecalltheactofmovingavertexfromonesetinthepartitiontoanothersetavertexmove.Let(mi)ki=1beasequenceofvertexmovesandlet(^A,^S,^B)bethepartitionarrivedataftermakingthemoves(mi)ki=1.Wesaythatthesequenceofvertexmovesisfeasibleif(^A,^S,^B)isfeasiblein( 3 ).Moreover,ifthereexistsanedge(i,j)2Esuchthati2Aandj2^B,ori2Bandj2^A,thenwecall(mi)ki=1acoupledsequenceofvertexmoves.Otherwise,wesaythat(mi)ki=1isuncoupled. Lemma3.3. Let(A,S,B)beafeasiblepartitioninproblem( 3 ),let(mi)ki=1beafeasiblesequenceofvertexmoves,andlet(^A,^S,^B)bethepartitionarrivedataftermakingthemoves(mi)ki=1.Letxxx,^xxx,yyy,and^yyybetheincidencevectorsforthesetsA,^A,B,and^B,respectively,andletdddx,dddy2Rnbedenedby (dddx,dddy)=(^xxx,^yyy))]TJ /F3 11.955 Tf 11.96 0 Td[((xxx,yyy).(3)Thenthesequenceofmoves(mi)ki=1isuncoupledifandonlyif dddTx(AAA+III)dddy=0.(3) 48

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Proof. First,weclaimthateachterminthesumdddTx(AAA+III)dddy=nXi=1nXj=1dxiaijdyjisnonpositive.Toseethis,supposebywayofcontradictionthatthereexistiandjsuchthatdxiaijdyj>0.Firstofall,thisimpliesthataij=1;hence(i,j)2E.Secondly,bydenitionof(dddx,dddy)wemusthavethateitherdxi=1anddyj=1ordxi=)]TJ /F3 11.955 Tf 9.3 0 Td[(1anddyj=)]TJ /F3 11.955 Tf 9.3 0 Td[(1.Intherstcasewehavethat^xi=1and^yj=1,whichimpliesthati2^Aandj2^B.Butsince(i,j)2E,thiscontradictsthefactthat(^A^B)\E=;(whichisaconsequenceof(^A,^S,^B)beingfeasiblein( 3 )).Similarly,inthesecondcasewehavei2Aandj2B,contradictingthefeasibilityof(A,S,B).Thisprovestheclaim.Hence,( 3 )holdsifandonlyif 8(i,j)2Eeitherdxi=0ordyj=0.(3)Wewillshowthatthesequence(mi)ki=1isuncoupledifandonlyif( 3 )holds.Firstsupposethatthesequence(mi)ki=1isuncoupledandlet(i,j)2E.Ifdxi=0,thenthereisnothingtoshow.So,supposethatdxi6=0.Then^xi6=xi,whichimpliesthateitheri2Aori2^A.Supposethati2A.Thenweknowthatj=2Bsince(AB)\E=;.Moreover,since(mi)ki=1isuncoupled,byDenition 3.1 i2Aimpliesthatj=2^B.Henceyj=^yj=0,implyingdyj=0.Asimilarargumentshowsthatdyj=0inthecasewherei2^A.Since(i,j)2Ewasarbitrary,theproofoftheforwarddirectioniscomplete. 49

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Nowsupposethat( 3 )holds.Inordertoshowthat(mi)ki=1isuncoupled,wemustshowthatforeveryedge(i,j)2Ebothofthefollowingimplicationshold: i2A)j=2^Bandi2B)j=2^A.(3)Solet(i,j)2E.Supposethati2A.Thenxi=1.By( 3 )wehavethatdxi=0ordyj=0.Soifdxi=0,then^xi=xi=1,andhencei2^A.Sosince(^A^B)\E=;wehavethatj=2^B.Asimilarargumentshowsthatj=2^Bwhendyj=0.Thisprovestherstimplicationin( 3 ).Inordertoprovethesecondimplication,leti2B.Thenyi=1.Since(i,j)2E,wealsohave(j,i)2E.Soby( 3 )wehavethatdxj=0ordyi=0.Ifdyi=0,then^yi=yi=1,andhencei2^B.Sosince(^A^B)\E=;wehavethatj=2^A.Asimilarargumentshowsthatj=2^Awhendxj=0.Thisprovesthesecondimplicationin( 3 ).Hence,( 3 )holds.Since(i,j)2Ewasarbitrary,(mi)ki=1isanuncoupledsequenceofmoves.Thisprovesthebackwardsdirectionandsotheproofiscomplete. Proposition3.2. Let(A,S,B)beafeasiblepartitionintheproblem( 3 )andletxxxandyyybetheincidencevectorsforAandB,respectively.If(xxx,yyy)satisestherst-orderoptimalityconditions( 3 ),thentheredoesnotexistanuncoupledfeasiblesequenceofvertexmoveswhichimproves(reduces)thevalueoftheobjectivefunctionin( 3 ). Proof. Let(mi)ki=1beanyuncoupledfeasiblesequenceofvertexmoves,let(^A,^S,^B)bethepartitionarrivedataftermakingthemoves.Let^xxxand^yyybetheincidencevectorsforthesets^Aand^B,respectively,anddenedddxanddddyby( 3 ).Suppose(xxx,yyy)satisestherst-orderoptimalityconditions( 3 ).Thenby( 1 ),wehavethatrf(xxx,yyy)(dddx,dddy)0.Andsince(mi)ki=1isuncoupled,byLemma 3.3 wehavedddTx(AAA+III)dddy=0. 50

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Hence,f(^xxx,^yyy)=f(xxx,yyy)+rf(xxx,yyy)(dddx,dddy)+(dddx,dddy)Tr2f(xxx,yyy)(dddx,dddy)f(xxx,yyy)+(dddx,dddy)Tr2f(xxx,yyy)(dddx,dddy)=f(xxx,yyy)+dddTx(AAA+III)dddy=f(xxx,yyy).Sinceboth(^A,^S,^B)and(A,S,B)arefeasiblein( 3 ),both(xxx,yyy)and(^xxx,^yyy)satisfy( 3 ).Thus,ifW(A[B)denotesthesumoftheweightsofverticesinA[B,thenW(^A[^B)=cccT(^xxx+^yyy)=)]TJ /F3 11.955 Tf 9.3 0 Td[(f(^xxx,^yyy))]TJ /F3 11.955 Tf 21.91 0 Td[(f(xxx,yyy)=cccT(xxx+yyy)=W(A[B),whichimpliesW(^S)W(S).Thus,wehaveshownthatforanarbitraryuncoupledfeasiblesequenceofvertexmoves,theobjectivefunctionin( 3 )isnotimproved.Thiscompletestheproof. Corollary3.1. Let(A,S,B)beafeasiblepartitioninproblem( 3 )andletxxxandyyybetheincidencevectorsforAandB,respectively.If(xxx,yyy)satisestherst-orderoptimalityconditions( 3 ),butdoesnotsatisfythesecond-orderoptimalityconditions(V2)(V5),thenthereexistsacoupledfeasiblesequenceofvertexmoveswhichimproves(reduces)theobjectivefunctionvaluein( 3 ). Proof. Ifthesecond-orderconditionsarenotsatisedatthepoint(xxx,yyy),then(xxx,yyy)isnotaglobalsolutionto( 3 ).Hence,thereexistssomefeasiblepartition(^A,^S,^B)suchthatW(^S)
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Corollary 2.2 ,whenthesecond-ordercondition( 3 )isviolated,thereexistvectors(ddd1x,ddd1y),(ddd2x,ddd2y)2C(xxx,yyy)\Dsuchthat(ddd1x+ddd2x,ddd1y+ddd2y)isadescentdirectionforf.Hence,if(xxx(t),yyy(t)):=(xxx,yyy)+t[(ddd1x,ddd1y)+(ddd2x,ddd2y)],thenforsufcientlysmallt>0(xxx(t),yyy(t))isfeasibleand f(xxx(t),yyy(t))
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CHAPTER4AMULTILEVELALGORITHMFORTHEVERTEXSEPARATORPROBLEM 4.1IntroductionForlargegraphs,neitherdiscreteheuristicsnorcontinuousmethodsareguaranteedtoyieldaglobal(orevenagoodlocal)solutiontotheVertexSeparatorProblem.Inessence,thereasonisthatthesemethodsareonlycapableofmakinglocalimprovementstothepartitionandareblindtothegraph'sglobalstructure.Themodernapproachtosolvinglargescalegraphpartitioningproblemsisthatofamultilevelalgo-rithm.Multilevelalgorithmswereoriginallydevelopedforsolvinglargelinearsystemsresultingfromthediscretizationofellipticpartialdifferentialequations[ 65 ].Recentlyhowever,theyhaveprovenusefulinsolvingavarietyoflargescalegraphlayoutandgraphpartitioningproblems(seeforinstance[ 32 61 62 ]).Thebasicideaofamultilevelalgorithmistocreateahierarchyofincreasinglycoarsergraphswhichapproximatetheoriginalgraphbutwithfewerdegreesoffreedom.Theproblemissolvedforcoarsestgraphandthesolutionisuncoarsenedandpropagatedbackupthehierarchytoobtainasolutionfortheoriginalgraph.Thegoalofmultilevelalgorithmsisnottoobtainexactsolutions,whichwouldbeunreasonableformanylargegraphs,butinsteadtoobtainapproximatesolutionsquickly,typicallyintimewhichislinearinthenumberofedgesjEj.Inthischapter,wegiveamultilevelalgorithmforsolvingthevertexseparatorproblem.Thealgorithmcoarsensthegraphbymatchingverticesandcontractingedges.Verticesarematchedaccordingtothedegreeoftheircoupling,whichwemeasureusingtheheavyedgedistance(see[ 37 ]).Theuncoarseningprocessisastraightforwardunmatchingofvertices.Multilevelalgorithmsongraphsperhapsdiffermostwidelyinthewaytheyrenesolutionsduringtheuncoarseningprocess.Thealgorithmsof[ 32 ]and[ 37 ]makelocalimprovementstoapartitionusingvariantsoftheKernighan-Linand 53

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Fiduccia-Mattheysesvertexexchangealgorithms.In[ 62 ],renementsaremadebyapplyingGauss-Seidel-likerelaxationtoalinearsysteminvolvingthegraphLaplacian,togetherwithawindowminimizationprocedureandsimulatedannealing.Inouralgorithm,renementsaremadeprimarilybysolvingcontinuousquadraticprogramsoftheform( 4 ).BasedonargumentsgiveninSection 3.3 ,localsolutionstotheseprogramsinsomesensecorrespondtolocalsolutionstotheVSP.InSection2,werelookatthecontinuousformulationoftheproblemgiveninChapter3andshowhowitcanbemodiedforsuitablityatthecoarserlevelsinthealgorithm.Section4developsnecessaryandsufcientlocaloptimalityconditionsforthequadraticprogramarisinginthecoarserlevels.InSection5,wepresentthemultilevelalgorithm.SomenumericalresultsarepresentedinSection6. 4.2AContinuousFormulationforCoarseGraphsIntheproofofTheorem 3.1 ,weoutlinedastep-by-stepprocedureformovingfromanycontinuoussolutionof( 3 )toabinarysolutionwhichsatises( 3 ).Thus,theVSPmaybesolvedinthefollowingway:First,obtainaglobalsolutiontothequadraticprogram( 3 ).Second,movetoabinarysolutionsatisfying( 3 ).Finally,constructanoptimalpartitionvia( 3 ).WhenGhasasmallnumberofvertices,thedimensionofthequadraticprogram( 3 )issmall,andthisapproachmaybeveryeffective.However,sincetheobjectivefunctionin( 3 )isnon-convex,asjVjbecomeslargethenumberoflocalsolutionsto( 3 )growsquicklyandobtainingaglobalsolutiontothequadraticprogrambecomesincreasinglydifcult.InordertondgoodapproximatesolutionswhenGislarge,inSection4wewillincorporatetheapproachjustdescribedintoamultilevelframework.Thebasicideaistocoarsenthegraphintoasmallergraphhavingasimilarstructuretotheoriginalgraph;theVSPisthensolvedforthecoarsegraph,andthesolutionisuncoarsenedtogiveasolutionfortheoriginalgraph. 54

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Atthecoarserlevelsinthealgorithm,eachvertexrepresentsanaggregateofverticesfromtheoriginalgraph.Hence,theappropriatewayofmeasuringthesizesofthesetsAandBatthecoarselevelsistotakethesumofthesizesoftheaggregatesinthosesets.ThisleadstothefollowingformulationofthecoarseVSP: minA,BVW(S) (4) subjecttoS=Vn(A[B),A\B=;,(AB)\E=;,`aPi2Aiua,and`bPi2Biub.Here,eachi2Visavertexaggregateinthecoarsegraph,i>0denotesthesizeoftheithaggregate,andW(S)denotesthesumoftheweightsciofverticesinS.Solving( 4 )atthecoarselevelsensuresthatwhenthegraphisuncoarsened,theuncoarsenedsolutionisfeasibleintheVSP( 3 ).Althoughintheorythecoarseproblemmaynothaveafeasiblesolutionevenwhentheneproblemdoes,inpracticethisdoesnottypicallyhappen,sincethecoarseningusuallypreservestheoverallstructureofthegraph.Inlightoftheequivalencebetween( 3 )and( 3 ),itisnaturaltoconsidersolvingthefollowingcontinuousprogramatthecoarserlevels: minxxx,yyy2Rn)]TJ /F3 11.955 Tf 8.76 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy (4) subjectto000xxx111,000yyy111,`aTxxxua,and`bTyyyub.Here,ndenotesthenumberofverticesinthecoarsegraph,AAAandcccaretheadjacencymatrixandweightvectorforthecoarsegraph,andisthevectorwhoseithcomponentisi.However,theprogram( 4 )isingeneralonlyanapproximateformulationof( 4 ),aswewillnowsee. Denition4.1. Letxxx,yyy2Rn.Thepoint(xxx,yyy)2R2nisalmost-binaryifxxxandyyyeachhaveatmostonenon-binarycomponent. 55

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Theorem4.1. Suppose( 4 )isfeasible.Let2R. 1. Theprogram( 4 )hasanalmost-binarysolution(xxx,yyy)2R2n. 2. Supposemaxfci:i2Vg.Let(xxxB,yyyB)beabinaryfeasiblepointof( 4 )andlet=jf(xxxB,yyyB))]TJ /F3 11.955 Tf 12.58 0 Td[(f(xxx,yyy)j,where(xxx,yyy)isanyoptimalsolutionto( 4 ).Let(A,S,B)beanyoptimalpartitionin( 4 )andsupposethatW(S)islessthanorequalto W(V))]TJ /F3 11.955 Tf 11.95 0 Td[([+ min(`a+`b+2max)],(4)wheremaxandminarethemaximumandminimumentriesof.Thenthereex-istsapiecewiselinearpathfrom(xxxB,yyyB)toabinaryfeasiblepoint(xxx,yyy)satisfying( 3 )suchthatjW(S))-254(W(S)j,where(A,S,B)isthepartitiondenedby( 3 ). Proof. Part1.Let(xxx,yyy)beanyfeasiblepointof( 4 ).Denotetheobjectivefunctionin( 4 )byf:f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy.Weprovethefollowingstrongerresult: (P)Forany(xxx,yyy)feasiblein( 4 ),thereexistsaneasilyconstructedfeasiblepoint[^xxx,^yyy]2R2nwhichisalmost-binaryandsatisesf(^xxx,^yyy)f(xxx,yyy).Ifxxxandyyyeachhaveatmostonenon-binarycomponent,thenwearedone.Otherwise,withoutlossofgeneralityweconsiderthecasewherexxxhastwodistinctnon-binarycomponentskandl;thatis,00).Inthecasewhered<0,wemayinccreasetuntileitherxk=1orxl=0.Ineithercase,thenumberofnon-binarycomponentsinxxxisreducedbyatleastone;moreover,feasiblityismaintainedsince 56

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Txxx(t)=Txxx+t(k k)]TJ /F9 7.97 Tf 11.88 5.12 Td[(l l)=Txxx,andtheobjectivevaluedoesnotincreasebythechoiceofthesignoft.Wemaycontinuemovingcomponentstoboundsinthismanneruntileitherxxxbecomesbinary,orithasasinglenon-binarycomponent.Thesameproceduremaybeappliedtoyyy.Hence,wearriveatafeasiblepoint[^xxx,^yyy]suchthat^xxxand^yyyeachhaveatmostonenon-binarycomponentandf(^xxx,^yyy)f(xxx,yyy).ThiscompletestheproofofPart1.Part2.Let(xxx,yyy)beanyoptimalsolutionto( 4 )andlet(A,S,B)beanyoptimalpartitionin( 4 ).Let(xxxB,yyyB)beabinaryfeasiblepointof( 4 ).First,supposethatxxxTB(AAA+III)yyyB=0.Let=f(xxxB,yyyB))]TJ /F3 11.955 Tf 11.99 0 Td[(f(xxx,yyy),let(xxx,yyy)=(xxxB,yyyB),anddenethepartition(A,S,B)by( 3 ).Then, W(S)=W(V))-222(W(A[B)=W(V)+f(xxx,yyy)W(V)+f(xxx,yyy)+W(V))-222(W(A[B)+ (4) =W(S)+,where( 4 )followsfromthefactthatthepairofincidencevectorsassociatedwiththepartition(A,S,B)isafeasiblepointin( 4 )withobjectivevalueW(A[B),and(xxx,yyy)isoptimalin( 4 ).So,wearedone.Next,supposethat xxxTB(AAA+III)yyyB>0.(4)WeclaimthatwecandropacomponentofeitherxxxBoryyyBfromonetozerowithoutincreasingtheobjectivefunctionvalueandwithoutlosingfeasibilityin( 4 ).Since,byassumption,W(S)isboundedfromaboveby( 4 ),wehavethatf(xxx,yyy)W(A[B))]TJ /F3 11.955 Tf 21.91 0 Td[([+ min(`a+`b+2max)]. 57

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Hence, f(xxxB,yyyB)=f(xxx,yyy)+)]TJ /F10 11.955 Tf 30.39 8.08 Td[( min(`a+`b+2max).(4)Ontheotherhand,bythechoiceofandby( 4 )wehave f(xxxB,yyyB)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxxB+yyyB)+xxxTB(AAA+III)yyyB (4) )]TJ /F3 11.955 Tf 28.03 0 Td[(cccT(xxxB+yyyB))]TJ /F10 11.955 Tf 28.56 0 Td[(111T(xxxB+yyyB).Combining( 4 )and( 4 )gives min(`a+`b+2max)111T(xxxB+yyyB),whichimplies`a+`b+2maxmin111T(xxxB+yyyB)T(xxxB+yyyB).Therefore,wemusthavethat`a+maxTxxxBor`b+maxTyyyB.Assumewithoutlossofgeneralitythat`a+maxTxxxB.SincetheentriesinAAA,xxxB,andyyyBareallnonnegativeintegers,( 4 )impliesthatthereexistssomeindexisuchthatxBi=1and(AAA+III)iyyyB1.Considertheupdate(xxx,yyy)=(xxxB,yyyB))]TJ /F3 11.955 Tf 11.62 0 Td[((eeei,000).Then(xxx,yyy)isfeasible,sinceTxxx=TxxxB)]TJ /F10 11.955 Tf 11.95 0 Td[(iTxxxB)]TJ /F10 11.955 Tf 11.95 0 Td[(max`aandtheboxconstraintsarenotviolated.Moreover,bytakingaTaylorexpansionoffabout(xxxB,yyyB),wehavef(xxx,yyy)=f(xxxB,yyyB)+ci)]TJ /F10 11.955 Tf 11.95 0 Td[((AAA+III)iyyyBf(xxxB,yyyB)+ci)]TJ /F10 11.955 Tf 11.95 0 Td[(f(xxxB,yyyB),wheretherstinequalityfollowsfromthefactthat(AAA+III)iyyyB1andthesecondinequalityfollowsfromthechoiceof. 58

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Hence,aslongas( 4 )holds,wemaycontinuetodropcomponentsofeitherxxxBoryyyBtozerowithoutincreasingtheobjectivefunctionvalueorlosingfeasiblity.Eventuallywearriveatbinaryfeasiblepoint(xxx,yyy)satisfying( 3 ).Letting(A,S,B)bethepartitiondenedby( 3 ),wehaveW(A[B)=)]TJ /F3 11.955 Tf 9.29 0 Td[(f(xxx,yyy))]TJ /F3 11.955 Tf 21.92 0 Td[(f(xxxB,yyyB)=)]TJ /F3 11.955 Tf 9.3 0 Td[(f(xxx,yyy))]TJ /F10 11.955 Tf 11.95 0 Td[(W(A[B))]TJ /F10 11.955 Tf 11.96 0 Td[(.Thus,W(S)=W(V))-221(W(A[B)W(V))-221(W(A[B)+=W(S)+.Thiscompletestheproof. Remark2. TheassumptioninPart2oftheTheoremthatW(S)isboundedby( 4 )istypicallynottoorestrictive.Forexample,inmanyapplicationswehaveci=1foreachiinthenestgraph,andwemaytake`a=`b=1.IfwecoarsenthegraphusingtheschemepresentedinSection4,thenatthekthlevel(wherek=1indicatestheoriginalgraph)wehavemaxk,min=1,andk.Hencetheupperboundreduceston)]TJ /F3 11.955 Tf 12.64 0 Td[([+2k(1+k)].Themaximumvalueofkistypicallylessthanlog2(n).Hence,assumingissufcientlysmall,itisreasonabletoexpectthatthereisasolutionto( 4 )whichsatisesthebound( 4 ).Theproofofpart1ofTheorem 4.1 wasconstructive;thatis,weshowedhowtomovefromanygivensolutiontoanalmost-binarysolution.However,sincefisedge-concave(wewillprovethisfactinthenextsection),theexistenceofanalmostbinarysolutionfollowsfromTheorem 1.8 andthefactthateveryvertexofthefeasiblesetof( 4 )isanalmost-binarypoint(sinceavertexpointisactiveatexactly2nlinearlyindependentconstraints,atleastn)]TJ /F3 11.955 Tf 12.62 0 Td[(1componentsofeachofxxxandyyymustlieatanupperorlowerbound).However,unlikeinprogram( 3 ),theremayexistextremepointsofthefeasiblesetof( 4 )whicharenotbinary(considerthecasewheren=2,`a=1,ua=2,`b=1,ub=3,=(1,2),and(xxx,yyy)=(0,0.5,1,1)).Hence,( 4 )does 59

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notnecessarilyhaveabinarysolution.Ontheotherhand,givenabinaryfeasiblepoint,Theorem 4.1 showsthatunderreasonableassumptionsonthesizeoftheweightoftheoptimalseparator,onecanndapartitionfeasiblein( 4 )bydroppingcomponentsofxxxandyyyuntil( 3 )holds,andthenconstructingthepartition( 3 );andmoreover,thequalityoftheresultingpartitionisessentiallythesameasthequalityofthebinarysolutionwestartwith.Onewayofndingabinaryfeasiblepointwithalargeobjectivevalueistorstndanoptimal(ornearoptimal)almost-binarysolution,andthentomovethenon-binarycomponentstotheirupperorlowerbounds.However,itmayhappenthatanyattempttomovetoanearbybinarypointeitherresultsinalossoffeasiblityoranunacceptableincreaseintheobjectivefunctionvalue.Inthiscase,itispreferrabletorefrainfrommovingtoabinarypointandfromconstructingapartitionatthecurrentlevel.Instead,thenon-binarysolutioncanbeusedtoconstructaninitialguesstothecontinuousprogramatthenextnerlevel.Eventually,anelevelwillbereachedinwhichonecansafelymovetoabinaryfeasiblepointwithoutenduringalossoffeasibilityordeteriorationinsolutionquality(assumingthesolutionsateachlevelaresufcientlyclosetooptimal). 4.3OptimalityConditionsfortheCoarseQuadraticProgramInthemultilevelalgorithmwewillpresentinthenextsection,program( 4 )isusedrsttosolvethecoarsestproblem,andthentoperiodicallyrenethethesolutionduringtheuncoarseningprocess.Thegoaloftherenementphaseatagivencoarselevelistomakelocalimprovementstoaninitialpartitionuntilonearrivesatatpartitionwhichisinsomesenselocallyoptimal.Onecommonapproachtoreningsolutionstographpartitioningproblemsistomakeaseriesofswaps,exchangingverticesinonesetwithverticesinadifferentset(seeforinstance[ 32 37 ]).Ifthequalityofthepartitionimprovesafterperformingthe 60

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swaps,thenthenewpartitionisacceptedandtherenementphaserepeats.Otherwise,thepartitionisregardedasbeinglocallyoptimal.Inthealgorithmwegiveinthecurrentpaper,insteadofswappingverticesbetweensets,renementsaremadebyndinglocalsolutionstothecontinuousprogram( 4 ).Moreprecisely,ateachlevelintheuncoarseningprocessaninitialpartitionistranformedintoaninitialguessfortheprogram( 4 ).Alocalsolutiontothequadraticprogramisfound,andthissolutionisthentransformedbackintoapartition.Whenanoptimizationalgorithmisappliedtotheprogram( 4 ),theiteratestypicallyconvergetoastationarypoint;thatis,apointwhichsatisestherst-orderoptimality(orKKT)conditions.Forageneralquadraticprogram,determiningwhetherastationarypointisalocalminimizerisanNP-hardproblem[ 54 ].However,justasinthecaseofprogram( 3 ),theobjectivefunctionin( 4 )isedge-concaveoverthefeasiblesetof( 4 ).Hence,Corollary 2.3 impliesthatlocaloptimalityin( 4 )canbecheckedinpolynomialtime.Inthissection,wewillderiveoptimalityconditionsfortheprogram( 4 )whichareanalagoustothosegiveninTheorem 3.2 forproblem( 3 ).Letfbetheobjectivefunctionin( 4 ):f(xxx,yyy)=)]TJ /F3 11.955 Tf 8.77 0 Td[(cccT(xxx+yyy)+xxxT(AAA+III)yyy.Sincethefeasiblesetin( 4 )involves4lowerboundsand4upperbounds,thestandardstatementoftheKKTconditions( 1 )involves8multipliers,8inequalityconstraints,and8complementaryslacknessconditions.Amorecompactwayofexpressingthese16conditionsisasfollows:If(xxx,yyy)isalocalminimizerof( 4 ),thenthereexistmultipliersaandb2Rnandaandb2Rsuchthat 264rxxxf(xxx,yyy)ryyyf(xxx,yyy)375+264ab375+264a111b111375=000,(4) 61

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wherea2M(xxx),b2M(yyy),a2L(xxx,`a,ua),andb2L(yyy,`b,ub),withM(zzz)=f2Rn:izimaxfi,0gforall1ingandL(zzz,`,u)=f2R:Tzzzmaxfu,`gg. Proposition4.1. Thefollowingsetisareectiveedgedescriptionforthefeasiblesetin( 4 ):D=n[i,j=1i6=jf[eeei,000],[000,eeei],[000,1 ieeei)]TJ /F3 11.955 Tf 15.09 8.08 Td[(1 jeeej],[1 ieeei)]TJ /F3 11.955 Tf 15.09 8.08 Td[(1 jeeej,000]g. Proof. LetEbeanedgeofthefeasiblesetin( 4 )connectingdistinctvertices(xxx1,yyy1)and(xxx2,yyy2).Thentheremustexist2n)]TJ /F3 11.955 Tf 12.61 0 Td[(1linearlyindependentconstraintswhichareactiveatboth(xxx1,yyy1)and(xxx2,yyy2).Sincethereareatotalofnlinearlyindependentconstraintsoneachofxxxandyyy,wemayassumewithoutlossofgeneralitythatxxx1andxxx2areactiveatthesamesetofnlinearlyindependentconstraints(hence,xxx1=xxx2)andtherearen)]TJ /F3 11.955 Tf 12.18 0 Td[(1linearlyindependentconstraintsonyyywhichareactiveatbothyyy1andyyy2.Thisonlypossibleifatleastn)]TJ /F3 11.955 Tf 12.47 0 Td[(2componentsofyyy1andyyy2areequalandlieatupperorlowerboundsofthebox000yyy111.Ifexactlyn)]TJ /F3 11.955 Tf 12.21 0 Td[(2componentsareequal,thenbothTyyy1andTyyy2mustbeactiveatthesamebound.Inthiscase,T(yyy2)]TJ /F3 11.955 Tf 11.7 0 Td[(yyy1)=0,sowemusthaveyyy2=yyy1+t(1 ieeei)]TJ /F4 7.97 Tf 14.88 4.71 Td[(1 jeeej)forsomet2Randsomeiandj.Hence,inthiscaseEisparalleltothevector(000,1 ieeei)]TJ /F4 7.97 Tf 14.56 4.71 Td[(1 jeeej)2D.Inthecasewheren)]TJ /F3 11.955 Tf 11.66 0 Td[(1componentsofyyy1andyyy2areequal,wehavethatyyy2=yyy1+teeeiforsomet2Randsomei.Hence,inthiscaseEisparalleltothevector(000,eeei)2D.Asimilaranalysismaybecarriedoutinthecasewhereyyy1=yyy2andxxx1andxxx2haven)]TJ /F3 11.955 Tf 11.96 0 Td[(2componentsincommon. 62

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Hence,everyedgeofthefeasiblesetin( 4 )isparalleltoavectorinD.Therefore,Disanedgedescriptionofthefeasibleset.SinceDisclearlyclosedundertakingnegatives,itisreective.Thiscompletestheproof. Sincer2f=0B@000(AAA+III)(AAA+III)0001CA,itcanbecheckedthatdddT(r2f)ddd0foreveryddd2D.Therefore,byTheorem 2.2 afeasiblepoint(xxx,yyy)of( 4 )isalocalminimizerifandonlyiftherst-orderoptimalityconditions( 4 )holdand (ddd1)T(r2f)ddd20foreveryddd1,ddd22C(xxx,yyy)\D.(4)Tables 4-1 and 4-2 giveallthedifferentpossiblevaluesfor(ddd1)T(r2f)ddd2,whereddd1andddd2areedgedirections,intermsofHHH=AAA+III.SinceDisdescribedintermsof6differentvectors,thereare36products(ddd1)T(r2f)ddd2correspondingtothe66differentpairsamongthevectorsdescribingD.However,15oftheseproductsareknownbysymmetry.Theremaining21productsareshowninTables 4-1 and 4-2 .Theblankentriescorrespondtoentriesknownfromsymmetry. Table4-1.Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. (eeek,000)(000,eeek)()]TJ /F3 11.955 Tf 8.76 0 Td[(eeek,000)(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek) (eeei,000)0hik0)]TJ /F10 11.955 Tf 9.3 0 Td[(hik(000,eeei)0)]TJ /F10 11.955 Tf 9.3 0 Td[(hik0()]TJ /F3 11.955 Tf 8.77 0 Td[(eeei,000)0hik(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)0 Suppose(xxx,yyy)satisestherst-orderoptimalityconditions( 4 ).Thecondition( 4 )statesthat(xxx,yyy)isalocalminimizerifandonlyiftheentriesofTables 4-1 and 4-2 arenonnegativewheneverthevectorsinthecorrespondingrowandcolumnlieinC(xxx,yyy).Forexample,if(xxx,yyy)isalocalminimizer,thenwhenever(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek)lieintheconeC(xxx,yyy)forsomeiandk,wemusthavehik0(since>0).Thisimplies 63

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Table4-2.Valuesof(ddd1)T(r2f)ddd2forddd1,ddd22D. (1 keeek)]TJ /F4 7.97 Tf 14.76 4.71 Td[(1 leeel,000)(000,1 keeek)]TJ /F4 7.97 Tf 14.75 4.71 Td[(1 leeel) (eeei,000)0(1 khki)]TJ /F4 7.97 Tf 14.75 4.71 Td[(1 lhli)(000,eeei)(1 khki)]TJ /F4 7.97 Tf 14.75 4.71 Td[(1 lhli)0()]TJ /F3 11.955 Tf 8.77 0 Td[(eeei,000)0(1 lhli)]TJ /F4 7.97 Tf 15.62 4.71 Td[(1 khki)(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)(1 lhli)]TJ /F4 7.97 Tf 15.62 4.71 Td[(1 khki)0(1 ieeei)]TJ /F4 7.97 Tf 14.86 4.71 Td[(1 jeeej,000)0(1 ikhik)]TJ /F4 7.97 Tf 18.47 4.71 Td[(1 ilhil)]TJ /F4 7.97 Tf 19.44 4.71 Td[(1 jkhjk+1 jlhjl)(000,1 ieeei)]TJ /F4 7.97 Tf 14.85 4.71 Td[(1 jeeej)0 thataik=0.Moreover,sincehii=1,itfollowsthatboth(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeei)cannotbecontainedinC(xxx,yyy)atalocalminimizer.Inordertoverifythesecond-orderoptimalitycondition( 4 ),weneedtodeterminewhenavectorintheedgedescriptionDliesinC(xxx,yyy).Fromthedenitions( 1 )and( 1 ),wehaveF(xxx,yyy)=8>>>>>>><>>>>>>>:(ddd1,ddd2)2R2n:Tddd10ifTxxx=ua,d1i08iwithxi=1,Tddd10ifTxxx=`a,d1i08iwithxi=0,Tddd20ifTyyy=ub,d2i08iwithyi=1,Tddd20ifTyyy=`b,d2i08iwithyi=09>>>>>>>=>>>>>>>;,C(xxx,yyy)=ddd2F(xxx,yyy):Tddd1=0ifa6=0,d1i=08iwithai6=0Tddd2=0ifb6=0,d2i=08iwithbi6=0.Nowdenethefollowingsets:A0=fi:xi>0,ai=0gandA0=fi:xi<1,ai=0g,B0=fi:yi>0,bi=0gandB0=fi:yi<1,bi=0g.Figure 4-1 showswheneachelementofDalsoliesinC(xxx,yyy).CombiningtheinformationinTables 4-1 and 4-2 andFigure 4-1 ,wewillestablishthefollowingtheorem. Theorem4.2. Afeasiblepoint(xxx,yyy)for( 4 )isalocalminimizerifandonlyifthefollowingconditionshold: 64

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(eeei,000)2C(xxx,yyy),Txxx`axi>0ai=a=0,Txxx>`aa=0i2A0(000,)]TJ /F3 11.955 Tf 8.76 0 Td[(eeei)2C(xxx,yyy),Tyyy>`byi>0bi=b=0,Tyyy>`bb=0i2B0(1 ieeei)]TJ /F4 7.97 Tf 14.85 4.71 Td[(1 jeeej,000)2C(xxx,yyy),xi<1andxj>0ai=aj=0,i2A0andj2A0(000,1 ieeei)]TJ /F4 7.97 Tf 14.85 4.71 Td[(1 jeeej)2C(xxx,yyy),yi<1andyj>0bi=bj=0,i2B0andj2B0.Figure4-1. AdescriptionofD\C(xxx,yyy) (V1)Therst-orderconditions( 4 )hold. (V2)Supposea=0. a. IfTxxx`a,i2A0,j2B0,andk2B0,thenkhijjhik. (V3)Supposeb=0. a. IfTyyy`b,i2B0,j2A0,andk2A0,khijjhik. (V4)Supposea=b=0. a. IfTxxx>`aandTyyy`b,thenA0\B0=;andhij=0wheneveri2A0andj2B0. 65

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(V5)Ifi2A0,j2A0,k2B0,andl2B0,then1 ikhik+1 jlhjl1 ilhil+1 jkhjk. Proof. First,supposethattheconditions(V1)(V5)aresatised.Wewishtoshowthat( 4 )holds,whichimpliesthat(xxx,yyy)isalocalminimizer.ConsideranyentryinTable 4-1 or 4-2 thatispotentiallynegative,suchas)]TJ /F10 11.955 Tf 9.3 0 Td[(hik.Referringtotherowandcolumnedgedirectionsinthetable,weonlyneedtoconsiderthispossibilityif(a)both(000,eeei)and()]TJ /F3 11.955 Tf 8.77 0 Td[(eeek,000)2C(xxx,yyy)or(b)both(eeei,000)and(000,)]TJ /F3 11.955 Tf 8.77 0 Td[(eeek)2C(xxx,yyy).If(b)holds,thenbyFigure 4-1 ,wehaveTxxx`b,b=0,andk2B0.By(V4b),i6=kandhik=0.If(a)holds,thenbyFigure 4-1 ,wehaveTxxx>`a,a=0,k2A0,Tyyy
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Remark3. If(V1)holdsforsomefeasiblepoint(xxx,yyy),thenwhenanyoftheconditions(V2)(V5)isviolated,thereisaneasilycomputabledescentdirection.Forexample,supposetheassumptionsin(V2)hold,buttheconclusionisviolated,thenlettingddd1=(eeei,000)andddd2=(000,1 jeeej)]TJ /F4 7.97 Tf 15.62 4.71 Td[(1 keeek),wemusthavethatddd1,ddd22C(xxx,yyy)and (ddd1)Tr2f(xxx,yyy)ddd2<0.(4)Sincefislinearalongbothddd1andddd2,( 4 )impliesthat(ddd1+ddd2)Tr2f(xxx,yyy)(ddd1+ddd2)=2(ddd1)Tr2f(xxx,yyy)ddd2<0.Sinceddd1andddd2bothlieinC(xxx,yyy),sodoestheirsum(sinceC(xxx,yyy)isacone).Hence,f((xxx,yyy)+t(ddd1+ddd2))=f(xxx,yyy)+t2 2(ddd1+ddd2)Tr2f(xxx,yyy)(ddd1+ddd2)<0.Therefore,ddd1+ddd2isadescentdirection. 4.4MultilevelAlgorithmGivenalargegraphG,thebasicideaofamultilevelalgorithmisthefollowing:CoarsenGintoasmallergraphhavingfewerverticeswhilepreservingtheoverallstructureofG;solvetheproblemforthecoarsegraph;anduncoarsenandrenethesolutiontoobtainasolutionfortheoriginalgraph.Thecoarseningprocesstypicallyconsistsofseverallevelsoverwhichahierarchyofincreasinglysmallergraphsisgenerated: G=G0,G1,...,Gs.(4)AftertheproblemissolvedforthecoarsestgraphGs,thesolutionispropagatedbackupthehierarchyandperiodicallyreneduntilasolutionisobtainedfortheoriginalgraphG0.Thecompleteprocessofcoarsening,solving,anduncoarseningagraphiscalledaV-cycle.WewilldescribeoneV-cycleofouralgorithmbelow. 67

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4.4.1CoarseningDuringthecoarseningprocess,verticesareaggegatedinsuchawayastopreservetheoverallstructureofthegraphwhileconsiderablyreducingitssize.Acrucialconsiderationincoarseningagraph,therefore,isindeterminingwhichverticestoaggregatesoasnottochangetheglobalstructureofthegraph.Onecommonapproach,whichwewilladopt,istodeneadistancedonthesetofvertices.Twoverticesiandjarematchedandaggregatedifthedistanced(i,j)betweenthemissmall.FortheVSP,themetricdshouldbedenedinsuchawaythatthedistancebetweeniandjissmallwheniandjhaveahighlikelihoodofappearinginthesamesetintheoptimalpartition(A,S,B).Inthisdissertationwewillconsideraspecicdistanceknownastheheavyedgedistance,whichwewilldenelater.Forthetimebeinghowever,assumethatwehavesomesuitabledistance,andcallitd.LetGl=(Vl,El)besomegraphinthehierarchy( 4 ),where0l
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Thecoarseningprocessoutlinedabovepresupposesadistanced:VlVl!R0[f1g.Thespecicchoiceofdwewillconsiderisoftencalledtheheavyedgedistance,anisdenedbydheavy(i,j)=8><>:1=i,j,if(i,j)isanedge1,otherwise.Hence,thedistancebetweenapairofverticesinthenegraphisequalto1iftheverticesareconnectedbyanedge,and1otherwise.Atthecoarserlevels,thedistancebetweenvertexaggregatesissmallwhenthenumberofedgesconnectingverticesindifferentaggregatesislarge. 4.4.2SolvingThecoarseningprocessrepeatsuntilweobtainagraphGshavingfewerthan100verticesorfewerthan10edges.ThecoarseVSP( 4 )issolvedforGseitherbysolvingthecontinuousprogram( 4 )orbyusingheuristics,suchasthosedevelopedin[ 37 ].Anexactsolutionisdesirable,butnotnecessary.Wedenotethesolutionobtainedby(As,Ss,Bs). 4.4.3UncoarseningLetGlbesomecoarsegraphinthehierarchy( 4 )with1ls,andlet(Al,Sl,Bl)denotethesolutiontothecoarseVSP( 4 )atlevell.AninitialpartitionforthegraphGl)]TJ /F4 7.97 Tf 6.58 0 Td[(1isobtainedbysimplyunmatchingverticesinGl;thatis,unmatchedverticeslieinthesameset(A,S,orB)astheirrepresentativesinthecoarsegraph.Inthecasewhereapartitioncouldnotbeobtainedatlevell(duetoreasonsgiveninSection 4.2 )thecontinuoussolutionto( 4 )canbeuncoarsenedbyassigningvaluestoentriesofthevectorsxxxl)]TJ /F4 7.97 Tf 6.59 0 Td[(1andyyyl)]TJ /F4 7.97 Tf 6.59 0 Td[(1equaltothevaluesoftheirrepresentativesinthecoarsegraph.Assumingthesolutionatlevellissufcientlystrong,andassumingthecoarseningGl)]TJ /F4 7.97 Tf 6.58 0 Td[(1!GleffectivelypreservedtheglobalstructureofGl)]TJ /F4 7.97 Tf 6.59 0 Td[(1,theinitialsolutionatlevel 69

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(xxx(0),yyy(0)) (xxxl)]TJ /F4 7.97 Tf 6.59 0 Td[(1,yyyl)]TJ /F4 7.97 Tf 6.58 0 Td[(1). fork=0,1,2,...,K Let(^xxx(k+1),^yyy(k+1))beasolutiontomaxxxx,yyycccTl)]TJ /F4 7.97 Tf 6.59 0 Td[(1(xxx+yyy))]TJ /F10 11.955 Tf 11.95 0 Td[(xxxT(AAAl)]TJ /F4 7.97 Tf 6.58 0 Td[(1+III)yyysubjectto000xxx111,000yyy111,`aTxxxua,`bTyyyub,xxx=xxx(k)ifkiseven,andyyy=yyy(k)ifkisodd. (xxx(k+1),yyy(k+1)) (xxx(k),yyy(k))+k(^xxx(k+1),^yyy(k+1)),forsomek2(0,1). endforFigure4-2. Localrenement l)]TJ /F3 11.955 Tf 12.17 0 Td[(1shouldbeagoodapproximatesolutionforthegraphGl)]TJ /F4 7.97 Tf 6.58 0 Td[(1.Itislikely,however,thatlocalimprovementsinthissolutioncanstillbemadetofurtherminimizetheweightoftheseparatorset. 4.4.4LocalRenementLocalimprovementstothesolutionaremadebyalternatelyxingthevariablesxxxandyyyandsolving( 4 )overtheremainingfreevariables(seeFigure 4-2 .)Noticethatwhenxxxoryyyisheldxed,problem( 4 )issimplyalinearprogram,andcanbesolvedveryefciently.Afterthelinearprogramissolved,astepistakeninthedirectionofthesolution.ThisprocessrepeatsforaxednumberKofiterations.Inpractice,thenalpointarrivedatafterthisprocessistypicallyastationarypointoftheprogram( 4 ).ThelocaloptimalityconditionsofTheorem 4.2 maybecheckedinordertoverifythatthestationarypointisalocalminimizer.Ifthetestfails,thenadescentdirectioniscomputedasintheRemarkfollowingTheorem 4.2 andtherenementisrepeated. 4.4.5EscapingPoorLocalSolutionsSincetheVSPisanNP-hardproblem,thecontinuousprogram( 4 )mayhavemanylocalminimizerswhenthegraphGislarge.Thus,onewouldndthat 70

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implementingtheschemewehavedescribedabovewouldnotbeentirelysatisfactory,sinceanyoneoftheselocalminimizershasthepotentialoftrappingtheiteratesandpreventingfurtherimprovementinthepartition.Theimportanceofescapingfrompoorlocalsolutionsisaddressedinmanymultilevelalgorithms.Onecommontechniqueforescapinglocalsolutionsissimulatedannealing(seeforinstance[ 62 ]).Inthecurrentalgorithm,weproposeatechniquewhichmakesinnovativeuseofthequadraticprogram( 4 ).Attheendoftherenementprocess,thevalueoftheparameterin( 4 )isdecreasedandtherenementisrepeatedusingthenewsolutionasaninitialguess.IntheexperimentsofSection 4.5 ,isreducedinincrementsof0.1untileitheritreacheszero,orthecurrentsolutionchangesaftertherenements.Ifisreducedtozerowithoutanychangeinthesolution,thenthecurrentsolutionisacceptedandthegraphisuncoarsenedtothenextnerlevel.Ontheotherhand,ifthecurrentsolutionchangesafterapplyingtherenementswithsomereducedvalueof,thenisreturnedtoitsoriginalvalueofmaxfci:i2Vl)]TJ /F4 7.97 Tf 6.59 0 Td[(1gandtherenementsarerepeatedusingthenewsolutionasaninitialguess.Thisproceduremaybeinterprettedfromthepointofviewofgraphtheoryasfollows.Asisreduced,thepenaltytermxxxT(AAAl)]TJ /F4 7.97 Tf 6.59 0 Td[(1+III)yyyisde-emphasized,andtheconditionthattheremustbenoedgesbetweenAandBisinsomesenserelaxed.Hence,partitionswhichwouldhavepreviouslybeenrejectedforviolatingtheseparationcondition,arenowacceptedifdoingsoimprovesthevalueoftherelaxedobjectivefunction.Hence,theiteratesareencouragedtoleavethecurrentlocalsolutionandexplorenewdirectionswhichwouldhavepreviouslybeenprohibited.Oncetheiteratesescapethelocalsolution,isreturnedtoitsoriginalvalueandtheseparationrequirementbetweenAandBisonceagainenforced. 4.5PreliminaryComputationalResultsThemultilevelalgorithmwasprogrammedinC++andthegraphstructureswerestoredusingtheLemonGraphLibrary[ 17 ].Atthecoarsestlevel,theVSPwassolved 71

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usingthefunctionNDMETISfromtheMETISgraphpartitioningpackage[ 37 ].ThelinearprogramsarisingintherenementstageweresolvedusingNAPHEAP,aCprogramforsolvingtheseparableconvexquadraticknapsackproblem,developedbyWilliamHager,JamesHungerford,andTimothyDavis[ 15 ].SinceoneoftheprincipalapplicationsoftheVSPisincomputingll-reducingorderingsforthecolumnsofasparsematrixbeforeitisfactored,ourpreliminarytestsetconsistedofseveralsymmetricmatricesfromtheUniversityofFloridaSparseMatrixLibrary( http://www.cise.ufl.edu/research/sparse/matrices ).Thematriceshaddimensionsrangingfromn=1000ton=5000anddensitiesrangingfrom210)]TJ /F4 7.97 Tf 6.58 0 Td[(3to4.610)]TJ /F4 7.97 Tf 6.59 0 Td[(2.Fromeachmatrix,agraphwasconstructedbyidentifyingthecolumnswithvertices,andconnectingtwoverticesiandjbyanedgewheneverthe(i,j)thentryofthematrixwasnonzero.Inallgraphs,verticeswereassumedtobeunweighted;thatis,ci=1foreveryi. Table4-3.IllustrativecomparisonbetweenseparatorsobtainedusingeitherMETISorthemultilevelalgorithmwithcontinuousrenementsandheavyedgedistancebasedmatching. ProblemDimensionSparsityQPMETIS 1138 BUS1138.00401093ELT4720.00125063AIRFOIL14253.00133941BCSPWR061454.003149BCSSTK081074.01216363BCSSTK132003.0214217219BCSSTM271224.03833434BLCKHOLE2132.00376767C-182169.00362821CAN 10541054.01192830CEGB30243024.00904238DELAUNAY N101024.00583027 Table 4-3 comparesthesizesofthevertexseparatorsfoundbyouralgorithmwiththosefoundbythemultilevelsolverNDMETIS.Column1ofthetablegivesthenameofthegraph,followedbythenumberofverticesandthegraphsparsity(jEj n(n)]TJ /F4 7.97 Tf 6.58 0 Td[(1)=2).In 72

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bothouralgorithmandinNDMETIS,weusedtheparametervalues`a=`b=1andua=ub=2n 3forallgraphs(see( 3 )).TheessentialdifferencebetweenouralogrithmandNDMETISisinthelocalrenementphase.WhereasNDMETISrenessolutionsusingavariantoftheKernighan-Linalgorithm(vertexexchanges),ouralgorithmmakesrenementsusingthecontinuousprogram( 4 ).Thus,thepreliminaryresultsofTable 4-3 suggestthatthecontinuousprogram( 4 )maybeaneffectivelocalrenementtool.Inthefuture,itwillbeworthinvestigatingtheextenttowhichNDMETISmaybenetbyemployingcontinuousrenementsinadditiontousingtheKernighan-Linalgorithm. 73

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APPENDIX:NOTATION 000and111denotevectorswhoseentriesareall0andall1respectively.Theirdimensionshouldbeclearfromcontext. rf(xxx)denotesthegradientoff,arowvector,andr2f(xxx)denotestheHessianoff. ForasetZ,jZjisthenumberofelementsinZ. RisthesetofrealnumbersandB=f0,1gisthesetofbinarynumbers. Ifxxxandyyy2Rn,then[xxx,yyy]denotesthevectorinR2nobtainedbystackingxxxaboveyyy. eeei2Rndenotesthei-thcolumnofthennidentitymatrix. IfAAA2RmnandIf1,...,mg,thenAAAIdenotesthesubmatrixcorrespondingtotherownumbersi2I. Ifxxx2Rn,thenthesupportofxxxisdenedbysupp(xxx)=fi2[1,n]:xi6=0g. ThepositivespanofasetS,denotedspan+(S),isthesetoflinearcombinationsofvectorsinSwithnonnegativecoefcients. jSjdenotesthenumberofelementsinS. ThenullspaceofAAAisdenotednull(AAA). IfSisasetofindicesassociatedwithweightsci2R,thenW(S)denotesthesumPi2Sci. 74

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BIOGRAPHICALSKETCH JamesHungerfordwasborninPlantation,Florida.HeearnedhisB.S.andPh.D.inmathematicsattheUniversityofFlorida.Inhissparetime,heenjoysreadingandplayingthekeyboard. 80