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vBinaryhenDecisionhaftlicDiagramsUnivandSaarbr?cInectegerakult?tenProdesgrammingMarkusDissertationenzurh-TErlangunghniscdesFGradesderdesersit?tDoktorsSaarlandesderonIngenieurwissenscBehlehaftenkder2007Naturwissenscf?rDatumridesenKaderbolloErnstquiums:ter:22.brand,DezeKurtmenbf?reGutacrDr.2007hDekersit?tanProf.derMax-PlancNaturwissenscSaarbr?chaftlicPDh-TMax-PlancecSaarbr?chniscenhenhFProf.akult?tFI:edricProf.EisenDr.UnivThorstenPHerfetornVDr.orsitzender:Mehlhorn,Prof.k-InstitutDr.Informatik,GerhardkWBeisitzer:eikum,Dr.Max-PlancAlthaus,k-Institutk-Institutf?rInformatik,Informatik,kSaarbr?ckalidAbstractwIn0/1thists.wthatorkorwoutpeeshoourwIPs.hoandwaBinaryvDecisionumeraDiagramsy'scannobbasedeB&Cuseditedastheaenpsolutionooptimalweerfuledtoololdesforof0/1CutInotegereProgramminghandIPsrelatedeproutineolyhedralcomputationalproblems.isWsmalleerticesdevondielopolytopanitsoutput-sensitivanecounalgorithmumerateforsbobuildingFadevthresholdfreelyBDD,ewhicwhichexistingrepre-thesenotspthehfeasibleto0/1mesolutionstoofIPs.atlinearelconstraingeneratet,forandhgivBDDs.eplemenabasedparallelnandork.-opshoerationapproacforellthresholdsolvBDDshardtovbuildofthecorrespBDDngforpae,0/1umerateIfacets,Pnd.optimalInoradditiontweneallconstrusolutionctotlinearajectiv0/1function.IPurthermoreforendelopingthetheaoptiailablmatolazovevhariableerformsordercoandforcomputingenthetivnariable0/1orderingoinspBrancec&truismdaofstate-of-the-artaththresholddBDD.solvF0/1orWthepreseninavvestigationapproacoftothevpinequalitiesolyhedral0/1structurewhicofisaon0/1WIPim-wtedeBDDshoseparationwihoawframewBDDsOurcanresultsbweourappliedhtowcounsuttooreenbutumerate0/1all0/1derKurzzusammenfassungderInscdieserwArbBDDseitert,zeigendwir,nktenwieenBinarytiert.DecisiontopsDiagramseiner(BDDs)enalsoeindiem?chhIPs.tigesungenWwirerk-gutzeugPf?enrendieoptimale0z?hlt/1hinausGanzzahligehProgrammierungt(0/1estehendeIP)vundeitsm??igzugeh?rigeheutzutagepL?senolyscedriscGenerierunghePPro-BDD-basierteblemeFeingesetztzeigen,wkleinererdenIPskl?onmenen.FWirundenZielfunktiontndetwicoptimalenkdelnDar?beinenenoutput-sensitivfreienTAlgorithmazoveuskzumhesBauendeseinesnThreshold0/1BDDs,hderertrit.dieCutzul?ssigenMetho0/1ahlL?sungenoneinerblinearenenUngleicAnsatzhUngleicung0/1darstellt,derundUbhabesceinemhreiborkenReceineunserparalleleL?-undzugleic-Opwierigererationist.f?roThyresholdz?hltBDDs,derumudenriBDDseinef?racetteneinumeriert0/1zuIPlinearenzueinebauen.L?sungDesoWalleeiterenL?sungenkoonstruierenerwirumeriert.einer0/1habIPwirzumasFindenerh?ltlicdereoptimalenoVlariablenordnenungwicundelt,zumelcBerecbhnenCodesf?rVEaumerierungrionaPubgesclenordnwindigkung?bSpBrancektrums&einesistThresholddieBDDs.deZurWUnzumtersucvh0/1ungWirdere-phreibolyeineneuartigendzurrisczul?ssigerhenhStrukturf?reinIess,0/1aufIPsbasiert.zeigennserewir,SeparierungsroutinewieenmaninmitB&CHilferamewvimplemenonUnsereBDDshenresultateadasslAnsatzlezum0/1senEcundkhenhdes0/1dazugeh?rigengeeignet0/1oAurthermorecfruitfulknoforwledgmengotsKarrenManyyProf.pmeoplek-Institutinuencedparticulm(notyunkwdiscussions.orkhisontthisthethesis.ears.Firsteopleofcreatingallen,amIythankhmAlthausyinadvisorspProf.thankDr.theFIritzforEisenbbrandandfortohisatadvice,yencouragemenallttheandInformatikguidancehduringlivthet.lastr,ytoears.forIterestingamconcerningthankfultopics.forthankgivingStefanmefortheinopporkortunittimeyaddition,tornstwpartsorkuscriptincommenhisgratefulformerKurtDiscretesOptimizationtgarouthispgivingatossibilitthenMax-Plancwke-InManstitutthanksf?rtoInformatik.pIatwMax-Plancasf?rmostforfortunatesuctoameetandProf.elyDr.vironmenBerndInBecakIergratefulandAndreasRalfbauerWimmermanfrominthediscussionsAlbonly)ert-Ludwigs-Univresearcersit?trelatedFFreiburg.ITheyErnstinandtroFducedemetheirtotereststhemeldwofandbinaryendingdecisionondiagramsInandIarousedEmAlthausyreadinginofterestsmaninandcomhelpfulbiningts.BDDsamwithtoinDr.tegerMehlhornprogramming.hiIinstanwcommitmenantotecometorefereethankfthemthesisforforthemevperyyfruitfulcollabishorationyduringorkthethlastMPI.y.V.Con25ten.ts.1.In.tro.duction.1.1.14.1Motiv.ation..........ts...............inclusion.of...p.......3.3.2...22...........a.4.olytop..............1.1.2.Outline....coun.............3.3.1...............with.....Exact.......0/1.......order.....h.t.w-p..........2.1.34.2.1Sources........of.......4.2.3.........threshold...4.3.........ter.......36...........heuristics.............algorithm..3.2.Preliminaries.5.2.1.Binary.DecisionSizeDusediag.rams......23...............23.teger.............V.ectrum.function.......ol.problems.b.een5and2.2eP.olyhedral.problems..4.2...................h.................4.2.2.face...............olytop....7.2.3.In.teger.Programming..4.2.4.for.........V.................4.3.1.a.e..........................9.3.Binary.Decision20DiagramsPre-construction11.3.1.W.eigh.ted.thresh.o.l.d.BDDs......21.Sifting...............................3.3.3.reduction.un.constrain..........11.3.1.1.Basic.construction..3.3.4.minimization.............................3.3.5.In.Programming..................12.3.1.2.Output-sensitiv.e3.3.6buildingariable.sp.of.threshold.............28.P.y.edral.29.Relation.e.w.BDD-p.e.o.olytop......14.3.2.Syn.thesis..30.Optimization...................................33.Ane.ull..............................17.3.2.134SequenDimensiontialaand.-op.eration........................35.P.e....................17.3.2.2.P.arallel.and35-opCerticateerationcorrectness.a.BDD.............35.0/1.ertex.ting............................19.3.336VCenariabofle0/1orderolytop.............................VI.ContentsBibliograph4.4.0/1.V.ertex.en.umeration....Separation.o.64.tigh...5.5.............A...relaxation.BDD.5.3.1.....................Results37.4.5.azove..CoLinearmp.utational.resultsp.....5.2.4.t.strengthening.....um.........................tation.......5.5.238.4.6.F.acet.en.umeration......Summary.......60.solving.........61.plane.h.BDD-SEP.........via.the.d.Heuristic.with.........the.er.v42.4.6.1.Pro.jection.of5.4extended.o.w-p.olytop.e............results...............5.5.1........43.4.6.2.Gift-wrapping.with.ahmarkBDD..............72...................Zusammenfassung.90..........44.4.6.35.2.2Computationalviaresultsa.Program.................5.2.3.cutting.a.proac.f.r.................63.Separation.Lagrangean46and5subgradienInmethoteger.Programming5.355for5.1inequalitiesUsingaBDDs.in.Branc.h.&.Cut....66.Increasing.n.b.of.t.ertices.................67.Lifting................57.5.1.1.Learning..................68.Computational...............................70.Implemen..............58.5.2.Separation.with.BDDs........71.Benc.sets...............................5.5.3................60.5.2.1.P.olynomial.time.solv.abilit.y.of.B72DD-SEP79.81.y..een1t1oInsevtrotheductionor1.1,Motivsization0/1BinaryFDewithcisionhDiagrBDDsamstation(BDDstford?short)wareesathrdatastructureoptimizationrepreseni.e.tedaboybuildapairwisedionerecwhictedthanacyclicQuestiongraph,erwhicintermehokaimsfromatoppaecmanyompepractonsistsandproblemsecient0/1represen(0/1tationlinearofjectivbvowoleanafunctions.theSincefortheirandsignicanonttialextensionThisinwill1986vinsevthethefamousatespapeerforbn,ycBryananw,tconnection[inBry86t]inthey3.haovtheeonstructsreceivdesedaphaaslotBDDoffattenmantioncominexistseldsgerlikam-eformcomputationalsetlogicstsandlinearhardwfunctionareofvtoerication.TheTheyyarebusedforasproblemanwing.industrialthresholdstrengthhtoaratelyol,usee.g.-opinsetVLSIadesignun[isMT98a].termediateOneeclasscanofaBDDsthataretimestheatso-calledBDD.thrproblemesholdnextBDDsIs.dierAothresholdandBDDtrepresenthattsexplo-indaBDDscompactewtilaeyattheetsetandofprogramming0/1pvview.ectorsarewhicinhdirection.arecfeasiblegerforammingadgivofenoldlinearasconstrainnot.ofAgrsrthereesentationistheanesholdobcviousorel?aortiyoinnbinatorialtotheretheaKnapsacintekprproblem,grandmingthIP)uulation,satoof0/1constrainintogethertegeraprogrammingobinegeneral,andwrestrictionethewariablesere0attracted1.bnaturalyathisfclassrofuildingBDDs.BDDThesucclassicalaalgoisrifollothmFirstforabuildingBDDaeacthresholdconstrainBDDsep-(see,e.g.then[aWandeg00erator])theisofininprinciplesequensimilarfashion,totildynamicBDDprogrammingleft.fwoyrinsolvingBDDsabKnapsacconstructedkhproblemha(seeee.g.represen[size,Scish86eral]).largerItthisofanalrecursivThiseeremethomotivd,ourwhicquestion.h2.ensurestheraauniqueenrepresenapprtationachofthethe-opoutpautiobsuchytheapplyingecertainsionrulesausewhilebybuildingdiatethecBDD.bInavoideparticular,UnisomonorpwhiclosubgraphsedwillthebbewdetectedBDDsafter0/1btegereingonlybuiltoneandointhenofdeletedButoremergedalsoagain.terestedThistheraisesositeourQuestionrstHowquestion.anQuestioninte1.prCangranbalgorithmappliebtoeeldgiven,thrwhichhonlyBDDs?cd
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