Constraint Satisfaction The Approximability of Minimization Problems Sanjeev Khanna Madhu - PDF document

ConstraintSatisfaction:TheApproximabilityofMinimizationProblemsSanjeevKhannaMadhuSudanLucaTrevisanAbstractThispapercontinuestheworkinitiatedbyCreignou[5]andKhanna,SudanandWilliamson[15]whoclassifymaximizationproblemsderivedfromBooleanconstraintsatisfaction.Herewestudytheapproximabilityofmin-imizationproblemsderivedthence.Aprobleminthisframeworkischaracterizedbyacollectionof“con-straints”(i.e.,functions)andaninstanceofaproblemisconstraintsdrawnfromap-pliedtospeciedsubsetsofBooleanvariables.Westudythetwominimizationanalogsofclassesstudiedin[15]:inonevariant,namelyMINCSP,theob-jectiveistondanassignmenttominimizethenum-berofunsatisedconstraints,whileintheother,namelyMINONES,thegoalistondasatisfyingassign-mentwithminimumnumberofones.Thesetwoclassestogethercaptureanentirespectrumofimportantmini-mizationproblemsincluding-MinCut,vertexcover,hittingsetwithboundedsizesets,integerprogramswithtwovariablesperinequality,graphbipartization,clausedeletioninCNFformulae,andnearestcodeword.Ourmainresultisthatthereexistsanitepartitionofthespaceofallconstraintsetssuchthatforanygiven,theapproximabilityofMINCSPandMINONESiscompletelydeterminedbythepartitioncontainingit.Moreover,wepresentacompactsetofrulesthatdeter-mineswhichpartitioncontainsagivenfamily.Ourclassicationidentiesthecentralelementsgoverningtheapproximabilityofproblemsintheseclasses,byuni-fyingalargecollectionalgorithmicandhardnessofap-proximationresults.Whencontrastedwiththeworkof[15],ourresultsalsoservetoformallyhighlightinher-entdifferencesbetweenmaximizationandminimizationproblems. FundamentalMathematicsResearchDepartment,BellLabs,700MountainAvenue,NJ07974.sanjeev@research.bell-labs.com.IBMThomasJ.WatsonResearchCenter,P.O.Box218,YorktownHeights,NY10598.madhu@watson.ibm.com.CentreUniversitaired'Informatique,Universit´edeGeneve,RueG´en´eral-Dufour24,CH-1211,Geneve,Switzerland.WorkdoneattheUniversityofRome“LaSapienza”.trevisan@cui.unige.ch.1.IntroductionInthispaperwepresentacompleteclassicationoftheapproximabilityofminimizationproblemsderivedfrom“Booleanconstraintsatisfaction”.Ourworkfol-lowstheworkofCreignou[5]andKhanna,SudanandWilliamson[15]whoobtainedsuchaclassicationformaximizationproblems.Thislineofresearchismotivatedbyanattempttounifythemanyknownpositiveandnegativeresultsontheapproximabilityofcombinatorialoptimizationprob-lems.Inthecaseofpositiveresults,manyparadigmshavebeenobtainedandtheseservetounifytheresultsnicely.Incontrast,thereisalackofsimilarunicationamongnegativeresults.Partofthereasonforthisisthathardnessresultstipicallytendtoexploiteveryfeatureoftheproblemwhosehardnessisbeingshown,ratherthanisolatingthe“minimal”featuresthatwouldsufcetoobtainthehardnesresult.Asaresultmanyinterest-ingquestionsabouthardproblemstendtoremainunre-solved.Khannaetal.[15]describeanumberofsuchinterestingquestions:(1)ArethereanyNP-hardprob-lemsinMAXSNPwhicharenotMAXSNP-hard?(2)Arethereany“natural”maximizationproblemswhichareapproximabletowithinpolylogarithmicfactors,butnobetter?(3)Istheresomeinherentdifferencebetweenmaximizationandminimizationproblemsamongcom-binatorialoptimizationproblems?Inordertostudysuchquestions,oreventoplacethemunderaformalsetting,oneneedstorstspecifytheoptimizationproblemsonewishestostudyinsomeuniformframework.Furthermore,onehastobecare-fultoensurethatitispossibleto“decide”whethertheoptimizationproblemstudiediseasyorhard(to,say,computeexactly).Unfortunately,barrierssuchasRice'stheorem(whichsaysthisquestionmaynotingeneralbedecidable)orLadner'stheorem(whichsaysproblemsmaynotbejusteasyorhard[20])forceustoseverelyrestricttheclassofproblemswhichcanbestudied.AworkofSchaefer[25]from1978isolatesoneclassofdecisionproblemswhichcanactuallybeclassiedcompletely.Heobtainsthisclassicationbyrestrictinghisattentionto“Booleanconstraintproblems”.Atypi- calprobleminthisclassisdenedbyanitesetof-niteBooleanconstraints(speciedby,say,atruthtable).Aninstanceofsuchaproblemspecies“constraintapplications”onBooleanvariableswhereeachcon-straintapplicationistheapplicationofoneofthecon-straintsfromtosomesubset(actually,orderedtuplewouldbemoreexact)ofthevariables.ThelanguageSATconsistsofallinstanceswhichhaveanassign-mentsatisfyingallconstraints.Schaeferdescribessixclassesoffunctionfamilies,suchthatifisasubsetofoneoftheseclasses,thenthedecisionproblemisinP,elseheshowsthatthedecisionproblemisNP-hard.Creignou[5]andKhannaetal.[15]extendthestudyabove,inanaturalway,tooptimizationprob-lems.Theydenetwoclassesofoptimizationprob-lems:MAXCSPandMAXONES(Actu-allytheworkofCreignou'sstudiesonlytheclassMAXCSP.).TheinstancesinbothcasesareconstraintsappliedonBooleanvariables,wheretheconstraintscomefrom.Intheformercase,theobjec-tiveistondanassignmentwhichmaximizesthenum-berofconstraintsthataresatised.Inthelattercase,theobjectiveistondanassignmenttotheBooleanvariableswhichsatisesalltheconstraintswhilemaxi-mizingtheweightoftheassignment(i.e.,thenumberofvariablessetto).InaresultsimilartothatofSchaefer'stheyshowthatthereexistsanitepartitionofthespaceofallfunctionfamiliessuchthattheapproximabilityofagivenproblemiscompletelydeterminedbasedonwhichpartitionthefamilybelongsto.TheinterestingaspectofthisclassicationresultisthatitmanagestocapturediverseproblemssuchasMAXFLOW,MAXCUTandMAXCLIQUE(whichareallapproximabletoverydif-ferentfactors)andyetuniesthe(non)-approximabilityresultsforallsuchproblems.Withintheframeworkofconstraintsatisfactionproblems,Khannaetal.settlethequestions(1)and(2)raisedabove.Ourworkisdirectedtowardsquestion(3).Weconsiderthetwocorrespondingclassesofmin-imizationproblemswhichwecallMINCSPandMINONES.Again,instancesofbothproblemsconsistofconstraintsfromappliedtoBooleanvariables.TheobjectiveinMINCSPistondtheassignmentwhichminimizesthenumberofunsat-isedconstraints.TheobjectiveforMINONESistondtheassignmentwhichsatisesallconstraintswhileminimizingthenumberofthevariablessetto.Foreachclassofoptimizationproblemsourmainthe-oremisinformallystatedasfollows:Thereexistsa-nitepartitionofthespaceofallfunctionfamilies,suchthattheapproximabilityoftheproblemMINCSP(resp.MINONES)isdeterminedcompletelybywhichpartitionitliesin.Westresshoweverthatthereisoneimportantrespectinwhichourclassicationisdif-ferentfrompreviousones.Ourpartitionsincludeseveralclasseswhoseapproximabilityisstillnotcompletelyun-derstood.Thuswhileourresultshowsthatthenum-berof“distinct”levelsofapproximability(amongmini-mizationproblemsderivedfromconstraintsatisfaction)isnite—itonlyplacesanupperboundonthenumberoflevels—itisunabletopinitdownexactly.Bypin-ningdownacompleteproblemforeachpartition,we,howeverturnthisseemingweaknessintoastrengthbyhighlightingsomeimportantproblemswhoseapprox-imabilitydeservesfurtherattention.Eventhoughthetransitionfrommaximizationprob-lemstominimizationproblemsisanobviousnextstep,successinthistransitionisnotimmediate.Forstarters—thetransitionfromSATtoMAXCSPiscompletelyanalogoustothetransitionfromSNPtoMAXSNP.Yet,thereisnominimizationanalogofMAXSNP.Theobviousdifcultyseemstobethatweareimmediatelyconfrontedbyahostofproblemsforwhichdistinguish-ingthecasewheretheoptimumiszero,fromthecaseforwhichtheoptimumisnon-zeroisNP-hard.Thetradi-tionalapproachtodealwithzero/oneproblemhasbeentorestrictthesyntaxusingwhichthepredicatewithintheSNPconstructisused-therebyrulingoutthehard-nessofthezero/oneproblem(seee.g.[18,19]).Ourapproach,viaconstraintsatisfaction,howeverdoesnotplaceanysuchrestrictions.Wesimplycharacterizealltheproblemsforwhichthe0/1problemishard,andthenhavingdoneso,movetotherestoftheproblems.Allthedifferentlevelsofapproximabilitythatareseenemergenaturally.Despitethiscompletelyobliviousapproachtoden-ingtheclassesMINCSPandMINONEStheclassesendupcapturingnumerousnaturaloptimizationprob-lems—withverydistinctlevelsofapproximability.Forstarters,the-MINCUTproblemisoneoftheprob-lemscapturedbyMINCSPwhichiswellknowntobecomputableexactlyinP.(ThiswasalreadyshownandusedbyKhannaetal.[15].)AttheconstantlevelofapproximabilityweseeproblemssuchasVERTEXCOVER[11,22],HittingSetwithboundedsizesets[13],Integerprogramswithtwovariablesperinequality[12].(Thereferencescitedaftertheproblemsshowthattheproblemisapproximabletowithinconstantfactors.)Thenwecometotwoopenproblems:MINUNCUT[10]andMIN2CNFDELETION[17]bothofwhichareknowntobeapproximabletowithinpolylogarithmicfactorsandknowntobehardtoapproximatetowithinsomeconstantfactor.Theexactapproximabilityofbothproblemsremainsopen.Atahigherlevelofapproxima-bilityistheNEARESTCODEWORDproblem[1]whichisknowntobeapproximabletowithinpolynomialfac-torsbutishardtoapproximatetowithinfac-tors.Foreachoftheseproblemsweshowthatthere isaconstraintfamilysuchthateitherMINCSPorMINONESisisomorphictotheproblem.Theabilitytostudyallthesedifferentproblemsinauniformframeworkandextractthefeaturesthatmaketheprob-lemseasier/harderthantheothersshowstheadvantageofstudyingoptimizationproblemsundertheconstraintsatisfactionframework.Lastly,wepointoutthatitisnotonlythenega-tiveresultsthatareuniedbyourframeworkbutalsothepositiveresults.Ourpositiveresultshighlightoncemoretheutilityofthelinearprogramming(LP)relax-ationfollowedbyroundingapproachtodevisingap-proximationalgorithms.Thisapproach,whichplaysasignicantroleinalltheabovementionedresultsof[22,13,12,10,17],alsoplaysacrucialroleinobtainingconstantfactorapproximationalgorithmsforoneofthepartitionsoftheMINCSPproblemsandoneparti-tionoftheMINONESproblems.Onelimitationofourresultsisthattheyfocusonproblemsinwhichtheinputinstanceshavenorestric-tionsinthemannerinwhichconstraintsmaybeimposedontheinputvariables.Thisisthereasonwhymanyoftheproblemsturnouttobeashardasshown.Some-timessignicantinsightmaybegleanedfromrestrict-ingtheprobleminstances.Awidelyprescribedcondi-tionisthattheincidencegraphonthevariablesandtheconstraintsshouldformaplanargraph.ThisrestrictionhasbeenrecentlystudiedbyKhannaandMotwani[14]andtheyshowthatitleadstopolynomialtimeapprox-imationschemesforageneralclassofconstraintsatis-factionproblems.Anotherinputrestrictionofinterestcouldbethatvariablesareallowedtoparticipateonlyinaboundednumberofconstraints.Weareunawareofanyworkonthisfront.Animportantextensionofourworkwouldbetoconsiderconstraintfamilieswhichcontainconstraintsofunboundedarity(suchasthoseconsideredin).SuchanextensionwouldallowustocaptureproblemssuchasSETCOVER.Insummary,ourworkreectsyetanothersmallsteptowardsthebiggoalofunderstandingthestructureofoptimizationproblems.2.PreliminariesThenotionofconstraintsandconstraintapplicationsandourclassesofproblemsofinteresthavealreadybeendenedinformallyabove.Weformalizetheminthenexttwosubsections.Wenextreviewsomebasicconceptsanddenitionsinapproximability,reductionsandcom-pleteness.Finally,wepresentourclassicationtheo-remsandgiveanoverviewofhowtheremainderofthispaperisorganized.2.1Constraints,ConstraintApplicationsandConstraintFamiliesAconstraintisafunction.Aconstraintapplicationisapairf;;:::;i,wheretheen]indicatetowhichoftheBooleanvari-ablestheconstraintisapplied.Werequirethatfor.Acontraintfamilyisanitecollectionofconstraints;:::;f.Constraintsandconstraintfamiliesaretheingredientsthatspecifyanoptimizationproblem.Thusitisnecessarythattheirdescriptionbenite.Constraintapplicationsareusedtospecifyin-stancesofoptimizationproblemsandthefactthattheirdescriptionlengthsgrowwiththeinstancesizeiscru-ciallyexploitedhere.Whilethisdistinctionbetweenconstraintsandconstraintapplicationsisimportant,wewilloftenblurthisdistinctionintherestofthispaper.Inparticularwemayoftenlettheconstraintapplicationf;;:::;ireferjusttotheconstraint.Inparticular,wewilloftenusetheexpression“”whenwemean“,whereistherstpartof”.Wenowdescribetheoptimizationproblemsconsideredinthispaper.Denition1(MINCSP)INPUT:Acollectionofconstraintapplicationsoftheform;:::;i,onBooleanvari-ables;x;:::;xwhereandisthearityof.OBJECTIVE:FindaBooleanassignmentto'ssoastominimizethenumberofunsatisedconstraints.IntheweightedproblemMINWEIGHTCSPtheinputincludesnon-negativeweights::::;wandtheobjectiveistondanassignmentwhichminimizesthesumoftheweightsoftheunsatisedconstraints.Denition2(MINONES)INPUT:Acollectionofconstraintapplicationsoftheform;:::;i,onBooleanvari-ables;x;:::;xwhereandisthearityof.OBJECTIVE:FindaBooleanassignmentto'swhichsatisesalltheconstraintsandminimizestheto-talnumberofvariablesassignedtrue.IntheweightedproblemMINWEIGHTONEStheinputincludesnon-negativeweights::::;wandtheobjectiveistondanassignmentwhichsatisesallconstraintsandminimizesthesumoftheweightsofvariablesassignedto.RepresentationoffunctionsWewilloftenworkwiththemaxtermrepresentationoffunctions:Denition3[Maxterm]Givenafunc-tion;x;:::;x,asubsetofliteralsdenedoverthevariables'siscalledamaxtermifsettingeachof theliteralsfalsedeterminesthefunctiontobefalseandifitisaminimalsuchcollection.Weexpressamaxterm;l;:::;lwhereeachisorforsome,as.Thusif;m;:::;mareallthemaxtermsofafunction,thenmayberepresentedas.Thisiscalledamaxtermrepresentationofafunction.PropertiesoffunctionfamiliesWenowdescribethemainpropertiesthatareusedtoclassifytheapproxima-bilityoftheoptimizationproblems.Theapproximabil-ityofafunctionfamilyisdeterminedbywhichofthepropertiesthefamilysatises.WestartwiththesixpropertiesdenedbySchaefer: Aconstraintis0-valid(resp.1-valid)if;:::;(resp.;:::;). Aconstraintisweaklypositive(resp.weaklynega-tive)ifitcanbeexpressedasaCNF-formulahav-ingatmostonenegatedvariable(resp.atmostoneunnegatedvariable)ineachclause. Aconstraintisafneifitcanbeexpressedasacon-junctionoflinearequalitiesover. Aconstraintis2cnfifitisexpressibleasa2CNF-formula(thatis,aCNFformulawithatmosttwoliteralsperclause).Theabovedenitionsextendtoconstraintfamiliesnat-urally.Forinstance,aconstraintfamilyis0-validifeveryconstraintis0-valid.Usingtheabovede-nitionsSchaefer'stheoremmaybestatedasfollows:Foranyconstraintfamily,SATisinPifis-validor-validorweaklypositiveorweaklynegativeorafneor2cnf;elsedecidingSATisNP-hard.SomemorepropertiesweredenedbyKhannaetal.[15]todescribetheapproximabilityoftheproblemstheyconsidered.Wewillneedthemforourresultsaswell. if2-monotoneif;:::;xisexpressibleas





(i.e.,isexpressibleasaDNF-formulawithatmosttwoterms-onecontainingonlypositiveliteralsandtheothercontainingonlynegativeliterals). Aconstraintiswidth-2afneifitisexpressibleasaconjunctionoflinearequationsoversuchthateachequationhasatmost2variables. Aconstraintis-closedifforallassignments,(.Theaboveproperties,alongwithSchaefer'sorigi-nalsetofpropertiessufcefor[5]and[15]toclas-sifytheapproximabilityofthemaximizationproblems SuchclausesareusuallycalledHornclauses.MAXCSPandMAXONES.AstatementoftheirresultsisincludedinAppendixA.Lastlyweneedonedenitionofourown,beforewecanstateourresults. AconstraintisIHS-(forImplicativeHittingSet-Bounded+)ifitisexpressibleasaCNFfor-mulawheretheclausesareofoneofthefollowingtypes:


forsomepositiveinteger,or,or.IHS-constraintsandcon-straintfamiliesaredenedanalogously(witheveryliteralbeingreplacedbyitscomplement).AfamilyisaIHS-familyifthefamilyisaIHS-familyoraIHS-family.ProblemscapturedbyMINCSPandMINONESWeenumerateheresomeinterestingminimizationprob-lemswhichare“captured”by(i.e.,areequivalenttosomeproblemin)MINCSPandMINONES.Thefol-lowinglistisinterestingforseveralreasons.First,ithighlightstheimportanceoftheclassesMINCSPandMINONESasclassesthatcontaininterestingminimiza-tionproblems.Furthermore,theseproblemsturnouttobe“complete”problemsforthepartitionstheybelongto-thustheyarenecessaryforafullstatementofourresults.Last,forseveraloftheproblemslistedbelow,theirapproximabilityisfarfrombeingwell-understood.Wefeelthattheseproblemsaresomehowrepresentativeofthelackofourunderstandingoftheapproximabilityofminimizationproblems. Thewell-knownHittingSetproblem,whenre-strictedtosetsofboundedsizescanbecapturedasMINONESfor


.Also,ofinteresttoourpaperisaslightgener-alizationofthisproblemwhichwecalltheImplica-tiveHittingSet-BProblem(MINIHS-)whichisMINCSPfor


g[f:g[f:.TheMINONESversionofthisproblemwillbeofinteresttousaswell.TheHittingSet-problemiswell-knowntobeapprox-imabletowithinafactorof.Weshowthat,infactMINIHS-isapproximabletowithinafactorof. MINUNCUTMINCSP.ThisproblemhasbeenstudiedpreviouslybyKleinetal.[16]andGargetal.[10].TheproblemisknowntobeMAXSNP-hardandhencenotapproximabletowithinaconstantfactor.Ontheotherhand,theproblemisknowntobeapproximabletowithinafactorof[10]. MIN2CNFDELETIONMINCSPy;.ThisproblemhasbeenstudiedbyKleinetal.[17].Theyshowthat theproblemisMAXSNP-hardandthatitisap-proximabletowithinafactorofloglog. NEARESTCODEWORDMINCSP;x.ThisisaclassicalproblemforwhichhardnessofapproximationresultshavebeenshownbyAroraetal.[1].TheMINONESversionofthisproblemisessentiallyidenticaltothisproblem.Forbothproblems,thehardnessre-sultofAroraetal.[1]saysthatapproximatingthisproblemtowithinafactorofishard,unlessNPQP.Nonon-trivialapproximationguaran-teesareknownforthisproblem(thetrivialboundbeingafactorof,whichiseasilyachievedsincedecidingifallequationsaresatisableamountstosolvingalinearsystem). Lastlywealsomentiononemoreproblemwhichisrequiredtopresentourmaintheorem.MINHORNDELETIONMINCSP.ThisproblemisessentiallyashardastheNEARESTCODEWORD.2.2Approximability,ReductionsandCom­pletenessFinally,beforepresentingourresults,wementionsomebasicnotionsonapproximability.Acombinatorialoptimizationproblemisdenedoverasetofinstances(admissibleinputdata);anitesetsoloffeasibleso-lutionsisassociatedtoanyinstance.Anobjectivefunc-tionattributesanintegervaluetoanysolution.Thegoalofanoptimizationproblemis,givenaninstance,ndasolutionsolofoptimumvalue.Theoptimumvalueisthelargestoneformaximizationproblemsandthesmallestoneforminimizationproblems.Acombina-torialoptimizationproblemissaidtobeanNPOprob-lemifinstancesandsolutionsareeasytorecognize,so-lutionsareshort,andtheobjectivefunctioniseasytocompute.Seee.g.[4]forformaldenitions.Denition4(PerformanceRatio)Anapproxima-tionalgorithmforanNPOproblemhasperformanceratioif,givenanyinstanceofwithjIj,itcomputesasolutionofvaluewhichsatises optopt Asolutionsatisfyingtheaboveinequalityisreferredtoasbeing-approximate.WesaythataNPOprob-lemisapproximabletowithinafactorifithasapolynomial-timeapproximationalgorithmwithperfor-manceratio.Denition5(ApproximationClasses)AnNPOprob-lemisintheclassPOifitissolvabletooptimalityinpolynomialtime.isintheclassAPX(resp.log-APX/poly-APX)ifthereexistsapolynomial-timealgorithmforwhoseperformanceratioisboundedbyacon-stant(resp.logarithmic/polynomialfactorinthesizeoftheinput).Completenessinapproximationclassescanbede-nedusingappropriateapproximationpreservingre-ducibilities.Thesereducibilitiestendtobeabitsub-tleandwewillbecarefultospecifythereducibilitiesusedinthispaper.Inthispaper,weheavilyusetwono-tionsofreducibilitesdenedbelow.(1)A-reducibilitywhichensuresthatifisA-reducibletoandisapproximableforsomefunction,thenis-approximable,forsomeconstantsand.Inparticularifisapproximabletowithinsomeconstantfactor(resp.,factor),thenisalsoapproximabletowithinsomeconstantfactor(resp.,factor).(2)AP-reducibilitywhichisamorestringentnotionofreducibility,inthateveryAP-reductionisalsoanA-reductionThisreducibilityhasthefeaturethatifAP-reducestoandhasaPTAS,thenhasaPTAS.Unfortunatelyneitheroneofthesereducibilitiesalonesufcesforourpurposes—weneedtousethemorestringentreducibilitytoshowAPX-hardnessofproblemsandweneedtheexibilityoftheweakerreducibilitytoprovidetheotherhardnessresults.Fortunately,resultsshowingAPX-hardnessfol-lowdirectlyfrom[15]andsothenewreductionsofthispaperareallA-reductions.Denition6(AP-reducibility[6])ForaconstantandtwoNPOproblemsand,wesaythatisAP-reducibletoiftwopolynomial-timecomputablefunctionsandexistsuchthatthefollowingholds:(1)Foranyinstanceof,isaninstanceof.(2)Foranyinstanceof,andanyfeasiblesolutionfor,isafeasiblesolutionfor.(3)Foranyinstanceofandany,ifisan-approximatesolutionfor,thenisan(1+(-approximatesolutionfor,wherethenotationiswithrespecttojIj.WesaythatisAP-reducibletoifaconstantexistssuchthatis-AP-reducibleto.Denition7(A-reducibility[7])AnNPOproblemissaidtobeA-reducibletoanNPOproblemiftwopolynomialtimecomputablefunctionsandandaconstantexistsuchthat:(1)Foranyinstanceof,isaninstanceof.(2)Foranyinstanceofandanyfeasiblesolutionfor,isafeasiblesolutionfor.(3)Foranyinstanceofandany,ifisa-approximatesolutionforthenisan-approximatesolutionfor. Remark8TheoriginaldenitionsofAP-reducibilityandA-reducibilityweremoregeneral.Undertheorig-inaldenitions,theA-reducibilitydoesnotpreservemembershipinlog-APX,anditisnotclearwhetherev-eryAP-reductionisalsoanA-reduction.Therestrictedversionsdenedherearemoresuitableforourpur-poses.Inparticular,itistruethattheVertexCoverproblemisAPX-completeunderourdenitionofAP-reducibility.Denition9(APXandpoly-APX-completeness)AnAPXproblemisAPX-completeifanyAPXprob-lemisAP-reducibleto.Apoly-APXproblemispoly-APX-completeifanypoly-APXproblemisA-reducibleto.ItiseasytoprovethatifisAPX-complete(resp.poly-APX-complete)thenaconstantexistssuchthatitisNP-hardtoapproximatewithin(1+(resp.).OneofourhardnessresultwillbeprovedbymeansofareductionfromtheMINTOTALLABEL-COVERprob-lem,denedasfollows.Denition10(MINTOTALLABEL-COVER)Anin-stanceoftheMINTOTALLABEL-COVERproblemcon-tainsintegerparameters,,,,and;andfunctionssR]!Q1;q2:[R]!Q2;V:[R][A1][A2]!f0;1gAfeasiblesolutionisapairoffunctions;p,whereeQ1]!2[A1]anddQ2]!2[A2],suchthatforeveryyR],thereexistsandsuchthat;a;a.Theob-jectivefunctiontobeminimizedis.Thisisavariation,introducedbyAmaldiandKann,oftheMINLABEL-COVERproblem[21,1](intheMINLABEL-COVERproblemtheobjectivefunctiontobeminimizedis).Areductionfromthemulti-proverproof-systemsof[24,2]showsthat,forany,itisNP-hardtoapproximateMINTOTALLABEL-COVERwithin.There-ductioninquestionissimilartothestandardonefrommulti-proverproofsystemstoMINLABEL-COVER[21,1]andomittedfromthisextendedabstract.2.3MainResultsWenowpresentthemainresultsofthispaper.Thetheoremusestheshorthandis-completetoindicatethattheproblemisequivalent(underA-reductions)totheproblem.Theorem11(MINCSPClassication)Foreverycon-straintset,MINCSPiseitherinPOorAPX-completeorMINUNCUT-completeorMIN2CNFDELETION-completeorNEARESTCODEWORD-completeorMINHORNDELETION-completeorthedecisionprob-lemisNP-hard.Furthermore,(1)Ifis-validor-validor2-monotone,thenMINCSPisinPO.(2)ElseifisIHS-thenMINCSPisAPX-complete.(3)Elseifiswidth-2afnethenMINCSPisMINUNCUT-complete.(4)Elseifis2CNFthenMINCSPisMIN2CNFDELETION-complete.(5)ElseifisafnethenMINCSPisNEARESTCODEWORD-complete.(6)ElseifisweaklypositiveorweaklynegativethenMINCSPisMINHORNDELETION-complete.(7)ElsedecidingiftheoptimumvalueofaninstanceofMINCSPiszeroisNP-complete.Theorem12(MINONESClassication)Foreveryconstraintset,MINONESiseitherinPOorAPX-completeorNEARESTCODEWORD-completeorMINHORNDELETION-completeorpoly-APX-completeorthedecisionproblemisNP-hard.Further-more,(1)Ifis-validorweaklynegativeorafnewithwidth,thenMINONESisinPO.(2)Elseifis2CNForIHS-thenMINONESisAPX-complete.(3)ElseifisafnethenMINONESisNEARESTCODEWORD-complete.(4)ElseifisweaklypositivethenMINONESisMINHORNDELETION-complete.(5)Elseifis-validthenMINONESispoly-APXcomplete(6)ElsendinganyfeasiblesolutiontoMINONESisNP-hard.TechniquesAsintheworkofKhannaetal.[15]twosimpleideasplayanimportantroleinthispaper.(1)Thenotionofimplementationsfrom[15](alsoknownasgadgets[3,26])whichshowshowtousetheconstraintsofafamilytoenforceconstraintsofadifferentfam-ily,therebylayingthegroundworkofareductionfromMINCSPtoMINCSP.(2)Theideaofworkingwithweightedversionsofminimizationprob-lems.Eventhoughourtheoremsonlymakestatementsaboutunweightedversionsofproblems,allourresultsuseasintermediatestepstheweightedversionsoftheseproblems.Theweightsallowustomanipulateprob-lemsmorelocally.However,simpleandwell-knownideaseventuallyallowustogetridoftheweightsand therebyyieldinghardnessoftheunweightedproblemaswell.Asaside-effectwealsoshow(inSection3.2)thattheunweightedandweightedproblemsareequallyhardtoapproximateinalltherelevantcasesofMINCSPandMINONESproblems.ThisextendstominimizationproblemstheresultsofCrescenzietal.[8].AmoredetailedlookatimplementationsandweightedproblemsfollowsinSection3.InSection4weshowthecontainmentresultsfortheMINCSPre-sult.Thenewelementhereistheconstantfactorapprox-imationalgorithmforIHS-families.InSection5weshowthehardnessresults.ThenewelementhereisthecharacterizationoffunctionswhicharenotexpressibleasIHS-andtheMINHORNDELETION-completenessresultsforweaklypositiveandnegativefamilies.WeshowaclosecorrespondencebetweenMINCSPandMINONESproblemsinSection6.Finally,inSections7and8,wegiveourpositiveandnegativeresultsforMINONESproblems.3.Warm-up3.1ImplementationsSupposewewanttoshowthatforsomeconstraintset,theproblemMINONESisAPX-hard.WewillstartwithaproblemthatisknowntobeAPX-hard,suchasVERTEXCOVER,whichisthesameasMINONES.WewillthenhavetoreducethisproblemtoMINONES.Themaintechniqueweusetodothisisto“implement”theconstraintusingconstraintsfromtheconstraintset.Thefollowingdenitionshowshowtoformalizethisnotion.(Thedef-initionispartofamoregeneraldenitionofKhannaetal[15].Infact,theirdenitionisneededforAP-reductions,butsincewedon'tprovideanynewAP-reductions,wedon'tneedtheirfulldenitionhere.)Denition13(PerfectImplementation[15])Acollectionofconstraintapplications;:::;Coverasetofvariables;x;:::;xand;y;:::;yiscalledaperfect-implementationofaconstraintiffthefollowingconditionsaresatis-ed:(1)Foranyassignmentofvaluestosuchthatistrue,thereexistsanassignmentofvaluestosuchthatalltheconstraintsaresatised,(2)Foranyassignmentofvaluestosuchthatisfalse,noassignmentofvaluestocansatisfyalltheconstraints.Aconstraintsetperfectlyimplementsaconstraintifthereexistsaperfect-implementationofusingconstraintsofforsome.Werefertothesetastheconstraintvariablesandthesetastheauxiliaryvariables.Aconstraint1-implementsitselfperfectly.Itiseas-ilyseenthatperfectimplementationscomposetogether,i.e.,ifperfectlyimplements,andperfectlyim-plements,thennf[Fperfectlyimple-ments.Inordertoseetheutilityofimplementations,itisbettertoworkwithweightedproblems.3.2WeightedProblemsForafunctionfamily,theproblemMINWEIGHTCSPhasasinstancesweightedconstraints;:::;Cwithnon-negativeweights;:::;wonBooleanvariables;:::;x.Theobjectiveistondanassignmenttowhichminimizestheweightofunsatisedconstraints.AninstanceoftheproblemMINWEIGHTONEShasasinstancesconstraints;:::;ConweightedBooleanvariables;:::;xwithnon-negativeweights;:::;w.Theobjectiveistondtheassignmentwhichminimizesthesumofweightsofvariablessettoamongallassignmentsthatsatisfyallconstraints.Thefollowingpropositionshowshowimplementationsareusefulforreductionsamongweightedproblems.Proposition14Ifaconstraintfamilyperfectlyim-plementseveryfunction,thenMINCSP(resp.MINWEIGHTCSP,MINWEIGHTONES)isA-reducibletoMINCSP(resp.MINWEIGHTCSP,MINWEIGHTONES).Proof:Letbelargeenoughsothatanyconstraintfromhasaperfect-implementationusingconstraintsfrom.LetbeaninstanceofMINWEIGHTCSPandletbetheinstanceofMINWEIGHTCSPobtainedbyreplacingeachconstraintofwiththere-spective-implementation.ItiseasytocheckthatanyassignmentforofcostyieldsanassignmentforwhosecostisbetweenV=kand.Itisimmediatetocheckthatiftheformersolutionis-approximate,thenthelatteriskr-approximate.Whileweightedproblemsallowfortheconvenientuseofimplementations,thereisreallynotmuchofadifferencebetweenweightedandunweightedprob-lems.ItiseasytoshowthatMINWEIGHTONESA-reducestoMINONES.Itisalsoeasytoseethatifweareallowedtorepeatthesameconstraintmanytimes,thenMINWEIGHTCSPA-reducestoMINCSP.Finally,itturnsoutthattheequiva-lenceholdsevenwhenwearenotallowedtorepeatcon-straints.ThisissummarizedinthefollowingTheorem.Theorem15(Weight-removingTheorem)Foranyconstraintfamily,MINWEIGHTONESA-reducestoMINONES.Ifperfectlyimplements,thenMINWEIGHTCSPA-reducestoMINCSP. Asarststeptowardsestablishingthisre-sult,werecallthatfromtheresultsof[8],itfollowsthatwheneverMINWEIGHTCSP(resp.MINWEIGHTONES)isinpoly-APX,thenitisAP-reducible(andhenceA-reducible)totherestrictionwhereweightsarepolynomiallybounded(inparticular,theycanbeassumedtobeboundedby;m,whereisthenumberofconstraintsandthenumberofvariables).Forthisreason,fromnowon,weightedproblemswillalwaysbeassumedtohavepolynomiallyboundedweights.Moreover,inaMINWEIGHTCSPinstance,wewillsometimesseeaweightedconstraintofweightasacollectionofidenticalconstraints.InaMINWEIGHTCSPinstancewecanassumethatnoconstrainthasweightzero(otherwisewecanremovetheconstraintwithoutchangingtheproblem).WealsoassumethatinaMINWEIGHTONESinstancenovari-ablehasweightzero.Otherwise,wemultiplyalltheweightsby(numberofvariables)andthenwechangethezero-weightsto1.ThisnegligiblyperturbstheproblemandgivesanAP-reduction.Thisisformal-izedbelow.ProofofTheorem15:Webeginbyshowingthatforanyfamily,MINWEIGHTCSPAP-reducestoMINCSPPf(x=y)g).Forthis,weuseanargu-mentsimilartothereductionfromMAX3SATtoMAX3SAT(see[23]),howeverwedon'tneedtouseex-panders.LetbeaninstanceofMINWEIGHTCSPovervariableset;:::;x.Foranyyn],letbethenumberoftheconstraintswhereappears.Wemake“copies”of,andcallthem;:::;y.Wesubstitutethe-thoccurrenceofby.Werepeatthissubstitutionforanyvari-able.Additionally,forrn],weaddallthepossi-ble“consistency”constraintsoftheformforj;hhocci],.Callthenewinstance;observethatcontainsnorepetitionofconstraints.Moreover,anyassignmentforcanbeconvertedintoanassignmentthatsatisesallthecon-sistencyconstraintswithoutincreasingthecost.Indeed,if,forsome,notallthehavethesamevalueunder,thenwegivevalue0toallofthem.Thiscan,atmost,contradictalltheconstraintscontaininganoccurrenceofaswitchedvariable,butthissatisesmanymoreconsis-tencyconstraintsthanthosethatgotcontradicted.Wenextshowthatforanyfamily,MINWEIGHTONESAP-reducestoMINONES.Tobeginwith,notethatifMINWEIGHTONESisinPO,thenitistriv-iallyAP-reducibletoanyNPOproblem(including,inparticular,MINONES).Theinterestingcasethusariseswhenisnot0-validnorwidth-2afnenorweaklynegative.AscanbeseenfromtheproofofLemma48below,insuchcaseeitherperfectlyim-plementsorallthebasicconstraintsofareoftheform


forsome.Ifperfectlyimplements,thenforanyvari-ableofweightweintroducenewvari-ables;:::;yandtheimplementationsofthecon-straints,,...,.Eachvari-ablehasnowcost1.Anysolutionsatisfyingtheoriginalsetofconstraintscanbeconvertedintoasolutionforthenewsetofconstraintsbylettingforallln],,wi1].Thecostremainsthesame.Anysolutionforthenewsetofconstraintsclearlysatisestheoriginalone(andwiththesamecost).Ifallthebasicconstraintsofareoftheform


(i.e.ifallconstraintsaremonotonefunctions)thenweproceedasfollows.Foranyvari-ableofweightweintroducenewvariables;:::;y.Anyconstraint;:::;xissubstitutedbythe


constraints;:::;yyw1];:::;jjwk]g:Itisnotdifculttoverifythatifwehaveafeasibleas-signmentforthenewproblemsuchthat,forsomei;j,,thenwecansetforalllwi]withoutcontradictinganyconstraint.Sinceno0ischangedtoa1,asolutionforthenon-weightedinstancecanbecon-vertedintoasolutionfortheweightedinstancewithoutincreasingthecost.3.3BasesandFirstReductionsInthissubsectionwesetupsomepreliminaryresultsthatwillplayaroleinthepresentationofourresults.First,wedevelopsomeshorthandnotationforthecon-straintfamilies:(1)(respectively,)isthefamilyof0-valid(respectively,1-valid)functions;(2)isthefamilyof2-monotonefunctions;(3)isthefam-ilyofIHS-functions;(4)isthefamilyofwidth-2afnefunctions;(5)isthefamilyof2CNFfunc-tions;(6)isthefamilyofafnefunctions;(7)isthefamilyofweaklypositivefunctions;(8)isthefamilyofweaklynegativefunctions.Denition16(Basis)Aconstraintfamilyisabasisforaconstraintfamilyifanyconstraintofcanbeexpressedasaconjunctionofconstraintsdrawnfrom.Thus,forexample,abasisforanafneconstraintistheset[Fwhereand,abasisforawidth-2afneconstraintistheset;x;x;x,andabasisfora2CNFconstraintisthesety;y;y;x;. Theabovedenitionismotivatedbythefactthatifisabasisfor,thenanapproximationalgo-rithmsforMINCSP()(resp.MINONES)yieldsanapproximationalgorithmforMINCSP()(resp.MINONES).Thisisassertedbelow.Theorem17Ifisabasisfor,thenMINWEIGHTCSP(resp.MINWEIGHTONES)isA-reducibletoMINWEIGHTCSP(resp.MINWEIGHTONES).TheabovetheoremfollowsfromProposition14andthenexttwopropositions.Proposition18If


,thenthefamily;:::;fperfectly-implements.Proof:Thecollection;:::;fisaperfect-implementationof.Proposition19Ifaconstraintfamilyperfectlyim-plementseveryfunction,thenMINWEIGHTONESisAP-reducibletoMINWEIGHTONES.Proof:ConsideraninstanceofMINWEIGHTONESandsubstituteeachconstraintbyaperfectimplementation,thusobtaininganinstanceofMINWEIGHTONES.Giveweight0totheauxiliaryvariables.Eachfeasiblesolutionforcanbeextendedtoafeasiblesolutionforwiththesamecost.Conversely,anyfeasiblesolutionfor,whenrestrictedtothevariablesofisfeasibleforandhasthesamecost.ThisisanAP-reduction.Tosimplifythepresentationofalgorithms,itwillbeusefultoobservethat,forafamily,ndinganap-proximationalgorithmforMINCSPisequivalenttondinganapproximationalgorithmforarelatedfamilythatwecall.Denition20Fora-aryconstraintfunction,wedene;:::;x;:::;.Forafamily;:::;fwedene;:::;fProposition21Forevery,MINWEIGHTCSPisA-reducibletoMINWEIGHTCSP.Proof:Thereductionsubstituteseveryconstraintfromwiththeconstraintfrom.Asolu-tionforthelatterproblemisconvertedintoasolutionfortheformeronebycomplementingthevalueofeachvariable.Thetransformationpreservesthecostofthesolution.AtechnicalresultbyKhannaetal.[15]willbeusedextensively.Lemma22([15])Letbeafamilythatcontainsanot0-validandanot1-validfunction.Then(1)IfcontainsafunctionthatisnotC-closed,thenperfectlyimplementstheunaryconstraintsand.(2)Otherwise,perfectlyimplementsthebinaryconstraintsand.Onerelevantconsequence(thatalsousesanideafrom[3])isthefollowing.Lemma23Letbeafamilythatcontainsanot0-validandanot1-validfunction.ThenMINWEIGHTCSPPfx;(:x)g)isA-reducibletoMINWEIGHTCSP.Proof:IfcontainsafunctionthatisnotC-closed,thenandcanbeperfectlyimplementedusingcon-straintsfrom,andsowearedone.Otherwise,givenaninstanceofMINWEIGHTCSPPfx;(:x)g)onvariables;:::;xandconstraints;:::;C,wedeneaninstanceofMINWEIGHTCSPwhosevariablesare;:::;xandadditionallyonenewauxil-iaryvariable.Eachconstraintoftheform(resp.)inisreplacedbyaconstraint(resp.).Alltheotherconstraintsarenotchanged.Thusalsohasconstraints.Givenasolution;:::;a;aforwhichsatisesofthesecon-straints,noticethattheassignment;:::;alsosatisesthesamecollectionofconstraints(sinceev-eryfunctioninis-closed).Inoneofthesecasestheassignmenttoisfalseandthenwenoticethatacon-straintofissatisedifandonlyifthecorrespondingconstraintinissatised.Thuseverysolutiontocanbemappedtoasolutionofwiththesameobjec-tivefunction.4.ContainmentResults(Algorithms)forMINCSPInthissectionweshowthecontainmentresultsde-scribedinTheorem11.Mostresultsdescribedherearesimplecontainmentresultswhichfolloweasilyfromthenotionofa“basis”.ThemoreinterestingresulthereisaconstantfactorapproximationalgorithmforIHS-whichispresentedinLemma25.Lemma24If,forsome,thenMINWEIGHTCSPissolvableexactlyinP.Proof:Creignou[5]andKhannaetal.[15]showthatthecorrespondingmaximizationproblemissolvableexactlyinP.Ourlemmafollowsimmediately(sincetheexactproblemsareinterreducible).Lemma25If,thenMINWEIGHTCSPAPX.Proof:ByTheorem17andProposition21itsufcestoprovethelemmafortheproblemsMINWEIGHTCSPIHS-.Wewillshowthatforevery,MINWEIGHTCSPIHS-is-approximable.GivenaninstanceofMINWEIGHTCSPIHS-onvariables;:::;xwithconstraints;:::;C withweights;:::;w,wecreatealinearprogramonvariables;:::;y(correspondingtotheBooleanvari-ables;:::;x)andvariables;:::;z(correspond-ingtotheconstraints;:::;C).ForeveryconstraintintheinstancewecreateanLPconstraintasfol-lows:1.If


,for,wecreatetheconstraint


2.If,wecreatetheconstraint3.IfwecreatetheconstraintInadditionweaddtheconstraints;yforeveryi;j.ItmaybeveriedthatanyintegersolutiontotheabovedescribedLPcorrespondstoanassignmenttotheMINCSPproblemwiththevariablesettoiftheconstraintisnotsatised.ThustheobjectivefunctionfortheLPistominimize.Givenanyfeasiblesolutionvector;:::;y;z;:::;ztotheLPabove,weshowhowtoobtainavector;:::;y;z;:::;zthatisalsofeasiblesuchthat+1).Firstwesetand.Observethatthevector;:::;y;:::;zisalsofeasibleandgivesasolutionofvalueatmost.Wenowhowtogetaninte-gralsolutionwhosevalueisatmost.Forthispartwerstsetifandif.NowweremoveeveryconstraintintheLPthatismaderedundant.Noticeinparticularthateveryconstraintoftype(1)isnowredundant(eitheroroneofthe'shasalreadybeensettoandhencethecon-straintwillbesatisedbyanyassignmenttotheremain-ingvariables).Wenowobservethat,ontheremainingvariables,theLPconstructedabovereducestoan-MINCUTLPrelaxation,andthereforehasanoptimalintegralsolution.Weset'sandtosuchanintegralandoptimalsolution.Noticethatthesoobtainedsolu-tionisintegralandsatises.Lemma26Foranyfamily,;xperfectlyimplementsthefamily.Proof:ByProposition18itsufcestoimplementthebasicwidth-2afnefunctions:namely,thefunctions,,and.Therstandthethirdfunctionsareinthetargetfamily.Thefunctionisperfectly2-implementedbytheconstraintsAUXandAUX.ThefunctionisimplementedbytheconstraintsAUXandAUX.AsaconsequenceoftheabovelemmaandLemma23,weget:Lemma27Foranyfamily,MINWEIGHTCSPA-reducestoMINWEIGHTCSP.ThefollowinglemmasshowreducibilitytoMIN2CNFDELETION,NEARESTCODEWORDandMINHORNDELETION.Lemma28Foranyfamily,thefamily2CNFperfectlyimplementseveryfunctionin.Proof:Againitsufcestoconsiderthebasicconstraintsofandthisissomesubsetofy;y;y;x;Thefamily2CNFcontainsalltheabovefunctionsex-ceptthefunctionwhichisimplementedbytheconstraintsAUXandAUX.Lemma29Foranyfamily,thefamily;xperfectlyimplementseveryfunctionin.Proof:Itsufcestoshowimplementationofthebasicafneconstraints,namely,constraintsoftheformandforsomep;q.Wefocusontheformertypeastheimplementationofthelatterisanalogous.First,weobservethattheconstraintisimplementedbyNowtheconstraintcanbeimplementedbyThewidth-2constraintsintheabovecanbeexpandedasbefore.Finally,theconstraintforanycanbeimplementedasfollows.Weintroducethefol-lowingsetofconstraintsusingtheauxiliaryvariables;z;:::;z. .........Lemma30Foranyfamily,thefamilyperfectlyimplementseveryfunc-tionin.Proof:A-aryweaklypositiveconstraint(for)iseitheroftheform:::oroftheform:::.For,theimplementationofis;a,andtheimplementationofis.For,theim-plementationofis(theconstrainthasinturntobeimplementedwiththealreadyshownmethod).For,weusethetextbookreductionfromSATto3SAT(seee.g.[9,Page49])andweobservethatwhenappliedto-aryweaklypositiveconstraintsityieldsaperfectimplemen-tationusingonly3-aryweaklypositiveconstraints.5.HardnessResults(Reductions)forMINCSPLemma31(TheAPX-hardCase)If,for,andthenMINWEIGHTCSPisAPX-hard.Proof:Followsimmediatelyfromtheresultsof[15].Lemma32(TheMINUNCUT-hardCase)If,for,andthenMINWEIGHTCSPisMINUNCUT-hard.Proof:Itsufcestoshowthatwecanperfectlyimple-menttheconstraint.Consideraconstraintsuchthat.Weknowthatcanbeex-pressedasaconjunctionofconstraintsdrawnfromthefamily;x;x;x.Noticefur-therthatalloftheseconstraintsexceptfortheconstraintarealsoin.Thusmustcontain,asoneofitsbasicprimitives,theconstraint.Nowanexistentialquanticationoveralltheremainingvariablesingivesusaperfectimplementationof.FortheMIN2CNFDELETION-hardnessproof,weneedthefollowingtwosimplelemmas.Lemma33Letbea2CNFfunctionwhichisnotwidth-2afne.Thencanperfectlyimplementsomefunctioninthefamily.Proof:Letbea2CNFfunctiononthevariables;:::;x.isaconjunctionofconstraintsoftheform,and.Consideradirectedgraphonvertices(onecorrespondingtoeverylit-eralor)whichhasadirectededgefromaliteraltoaliteral,ifthisisaconstraintimposedby.Weclaimthatthegraphmusthaveverticesandsuchthatthereisadirectedpathfromtobutnottheotherwayaround.(Ifnot,thencanbeexpressedasacon-junctionofequalityandinequalityconstraints.)Existen-tiallyquantifyingoverallothervariables(exceptthoseinvolvedinand)wendthatimplementsthecon-straint,whichisoneoftheconstraintsfrom.Lemma34Givenanyfunctionandthefunction,wecanperfectlyimplementallthefunctionsin.Lemma35(TheMIN2CNFDELETION-hardCase)If,for,andthenMINWEIGHTCSPisMIN2CNFDELETION-hard.Proof:Weneedtoshowthatwecanperfectlyimple-menttheconstraintsand.Since,itmustcontainaconstraintwhichisnotaIHS-constraintandaconstraintwhichisnotaIHS-constraint.Sincebothandare2CNFcon-straints,itmeansthatmusthaveasaba-sicconstraintandmusthaveasabasiccon-straintintheirrespectivemaxtermrepresentations.Ob-servethatthemaxtermrepresentationsofneithernorcanhavethebasicconstraintsand.Usingthisobservationwemayconcludethatanexis-tentialquanticationoverallvariablesbesidesx;yinwilleitherperfectlyimplementtheconstraintortheconstraint.Similarly,canperfectlyimplementeithertheconstraintor.Ifwegetbothand,wearedone.Oth-erwise,wehaveaperfectimplementationofthefunc-tion.Since,theremustexistaconstraintwhichisnotwidth-2afne.UsingLemmas33and34,wecannowconcludeaperfectim-plementationofthedesiredconstraints.Lemma36Ifbutforany,thenMINWEIGHTCSPisNEARESTCODEWORD-hard.Proof:Khannaetal.[15]showthatinthiscaseper-fectlyimplementstheconstraint


forsomeandsome.ThusthefamilyyfT;Fimplementsthefunctions;x.ThusNEARESTCODEWORDMINCSP;xisA-reducibletoMINWEIGHTCSPPfF;T.Sinceisneither 0-validnor1-valid,wecanuseLemma23toconcludethatMINWEIGHTCSPisNEARESTCODEWORD-hard.Lemma37([1])NEARESTCODEWORDishardtoap-proximatetowithinafactorof.Proof:TherequiredhardnessofthenearestcodewordproblemisshownbyAroraetal.[1].Thenearestcode-wordproblem,asdenedinAroraetal.,workswiththefollowingproblem:Givenanmatrixandan-dimensionalvector,ndan-dimensionalvec-torwhichminimizestheHammingdistancebetweenand.ThusthisproblemcanbeexpressedasaMINCSPproblemwithafneconstraintsover-variables.Theonlytechnicalpointtobenotedisthattheseconstraintshaveunboundedarity.Inordertogetridofsuchlongconstraints,wereplaceaconstraintoftheform


intoconstraints,,etc.onauxil-iaryvariables;:::;z.(ThesameimplementationwasusedinLemma29.)Thisincreasesthenumberofconstraintsbyafactorofatmost,butdoesnotchangetheobjectivefunction.ItremainstoseetheMINHORNDELETION-hardcase.Wewillhavetodrawsomenon-trivialconse-quencesfromthefactthatafamilyisnotIHS-.Lemma38Assumeandeitheror.ThencontainsanonC-closedfunction.Proof:Followsfromthefactthata-closedweaklypositivefunctionisalsoweaklynegative.Lemma39Ifisaweaklypositivefunctionnotex-pressibleasIHS-,thenf;x;canperfectlyimplementthefunction.Proof:SinceisnotIHS-,anymaxtermrep-resentationofmusthaveeitheramaxtermoramaxterm.Butsinceisweaklypositive,wemusthavethefor-merscenario.Werstshowthatcanperfectlyimple-mentthefunctionsand.Togetthefor-mer,wesetallliteralsin,besidesand,tofalseandexistentiallyquantifyovertherest.Sinceisamaxterm,thenewfunctionthusobtainedmusteitherbeorjust.Intheformercase,wearedone,otherwise,x;y;fy;xper-fectlyimplementstheconstraint.Toobtainaperfectimplementationof,asimilarargumentcanbeusedbysettingallliteralsinbesidesandtofalse.Wenextshowhowthesamefunctioncanalsobeusedtoobtainaperfectimplementationsofand.Todoso,wenowsetalltheliteralsinbesides,andtofalse.Existentiallyquantifyingoveranyothervariables,wegetafunctionwithatruthtableasgiveninFigure1. 110 Figure1.Truth­tableoftheconstraintIfthenrestrictinggivestheconstraint.Thiscontradictstheweaklypositiveassump-tionandhence.Ifor,wegetafunction.Elseand.Nowif,weagaingetbyexistentiallyquanti-fyingover,andif,wegetthecomplementof1-in-3sat.Thecomplementof1-in-3satfunctionalongwithcanonceagainimplement—sim-plyset.Thuswehaveaperfectimplementationof.Nowusingthefactthatwehavethefunction,wecanimplementbythefol-lowingcollectionofconstraints:x;a;bThiscompletestheproof.Lemma40(TheMINHORNDELETION-hardCase)If,for,andeitheror,thenMINWEIGHTCSPisMINHORNDELETION-hard.Proof:FromtheabovelemmasandfromLemma22wehavethatMINWEIGHTCSPisA-reducibletoMINWEIGHTCSP.Lemma41MINHORNDELETIONishardtoapproxi-matetowithin.Proof:ReductionfromtheMINTOTALLABEL-COVERproblem.Let;q;VbeaninstanceofMINTOTALLABEL-COVER,whereeR]![Q1],,R]![Q2]anddR][A1][A2]!f0;1g.ForanyyR],wedene;a;a;a.Wenowdescribethereduction.Forany,,A1],anddA2]wehaveavariablewhoseintendedmeaningisthevalueof;a;a.Moreover,forany(respectively,)andany(resp.)wehaveavariableq;a(resp.q;a),withtheintendedmeaningthatitsvalueis1ifandonlyif(respectively,).Foranyq;a(resp.q;a)variablewehavetheweight-oneconstraintq;a(resp.q;a.)Thefollowingcon-straints(eachwithweight)enforcethevariablestohavetheirintendedmeaning.Duetotheirweight,itisneverconvenienttocontradictthem. r]:W(a1;a2)2Acc(r)vr;a1;a28r2[r];aaA1];aaA2]:vr;a1;a2)wq1(r);a18r2[r];aaA1];aaA2]:vr;a1;a2)xq2(r);a2Theconstraintsoftherstkindcanbeperfectlyimplementedwithand(seeLemma30).ItcanbecheckedthatthisisanA-reductionfromMINTOTALLABEL-COVERtoMINHORNDELETION.6.MINONESvs.MINCSPWebeginthissectionwiththefollowingeasyrelationbetweenMINCSPandMINONESproblems.Proposition42Foranyconstraintfamily,MINWEIGHTONESisA-reducibletoMINWEIGHTCSPPf:xg).Proof:LetbeaninstanceofMINWEIGHTONESovervariables;:::;xwithweights;:::;w.Letbethelargestweight.WeconstructaninstanceofMINWEIGHTCSPPf:xg)byleavingthecon-straintsof(eachwithweight),andaddingaconstraintofweightforany;:::;n.When-evertheconstraintsofaresatisable,itwillbealwaysconvenienttosatisfythemin.ReducingaMINCSPproblemtoaMINONESproblemisslightlylessobvious.Proposition43(1)If,forany,perfectlyimple-ments,thenMINWEIGHTCSPA-reducestoMINWEIGHTONES.(2)If,forany,perfectlyimplements,thenMINWEIGHTCSPA-reducestoMINWEIGHTONES.Proof:Inbothcases,weuseanauxiliaryvariableforanyconstraint.Thevariabletakesthesameweightastheconstraint.Theoriginalvariableshaveweightzero.Intherstcase,aconstraintisreplacedby(theimplementationof);inthesecondcaseby(theimplementationof).Givenanassign-mentfortherstcase,wemayassumeaswellthatthessatisfy,sinceifissatisedbytheas-signmentthereisnopointinhaving.Thus,wenotethatthetotalweightofnon-zerovariablesintheMINONESinstanceequalsthetotalweightofnon-satisifedconstraintsintheMINCSPinstance.7.ContainmentResultsforMINONESLemma44(Poly-timeSolvableCases)Iffor,thenMINWEIGHTONESissolvableexactlyinpolynomialtimeProof:FollowsfromtheresultsofKhannaetal.[15]andfromtheobservationthatforafamily,solvingtooptimalityMINWEIGHTONESreducestosolvingtooptimalityMAXWEIGHTONES.Lemma45Iffor,thenMINWEIGHTONESisinAPX.Proof:Forthecase,a2-approximatealgo-rithmisgivenbyHochbaumetal.[12].Considernowthecase.FromTheorem17itissufcienttoconsideronlybasicIHS-constraints.SinceIHS-constraintsareweaklynegative,wewillrestricttobasicIHS-constraints.Weuselinear-programmingrelaxationsanddeterministicrounding.Letbethemaximumarityofafunctionin,wewillgivea-approximatealgorithm.Let;:::;CbeaninstanceofMINWEIGHTONESovervari-ableset;:::;xwithweights;:::;w.Thefollowingisanintegerlinearprogrammingformu-lationofndingtheminimumweightsatisfyingassign-mentfor.Subjectto::::::;:::;n(SCB)Considernowthelinearprogrammingrelaxationob-tainedbyrelaxingtheconstrainsinto.Werstndanoptimumsolutionfortherelaxation,andwethendenea0/1solutionbysettingif,andif.Itiseasytoseethatthisroundingincreasesthecostofthesolutionatmosttimesandthattheobtainedsolutionisfeasiblefor(SCB).Lemma46Forany,MINWEIGHTONESisA-reducibletoNEARESTCODEWORD.Proof:FromLemma29andProposition19,wehavethatMINWEIGHTONESAP-reducestoMINWEIGHTONES;x.FromProposition42,wehavethatMINWEIGHTONESA-reducestoNEARESTCODEWORD.Lemma47Forany,MINWEIGHTONESisA-reducibletoMINHORNDELETION.Proof:FollowsfromLemma30,Proposition19,andProposition42. 8.HardnessResultsforMINONESLemma48(APX-hardCases)IfdoesnotsatisfythehypothesisofLemma44,thenMINWEIGHTONESisAPX-hard.Proof:Thispartessentiallyfollowsfromtheproofof[15].Themajorstepsareasfollows:Werstarguethateitherimplementssomefunctionoftheform


,orthefunctionsorthefunction.Intherstcase,wegetaproblemthatisashardasVertexCover.Inthesec-ondcasewegetamuchharderproblem(NCP).Inthenalcaseweneedtoworksomemore.Inthiscaseagainweshowthatwithf;x;wecanimplementthefunction.Furthermore,weshowthatforanyfunction,MINWEIGHTONESf;x;AP-reducestoMINWEIGHTONESf;x.ThusonceagainwearedowntoafunctionwhichisatleastashardasVERTEXCOVER.Fromnowonwewillassumethatisnot0-valid,norweaklynegative,norwidth-2afne.Lemma49Ifisafnebutnotwidth-2afnenor0-validthenMINWEIGHTONES;xisAP-reducibletoMINWEIGHTONES.Proof:From[15]wehavethatimplementsthefunc-tion


forsomeandsome.Alsotheexistenceofnon-validfunctionimplieswecaneither(essentially)implementthefunc-tionorthefunction.Intheformercasewecansetthevariables;:::;xtoandthusim-plementeithertheconstraintsandortheconstraintsand.Inthelattercase,wecangetridofthevariablesin


inpairsandthusei-therimplementsthefunctionsoritimplementsthefunctions.Intherstandthirdcaseslistedaboveweimmedi-atelyimplementthefamily;xandsowearedone.Inthesecondandfourthcasesthiswillnotbepossible(inthesecondcasewealwayshave1-validconstraintandinthelastcasewealwayshaveconstraintsofevenwidth).Sowewillshowhowtore-ducetheproblemMINWEIGHTONES;xtotheseproblems.Thebasicideabe-hindthereductionsisthatifwehaveavailableavariablewhichweknowiszero,thenwecanimplementtheconstraint.Inthesecondcaseabove,weonlyneedtoimplementtheconstraintandthisisdoneusingtheconstraintsAUXandAUX.Inthefourthcaseabove,thecon-straintisimplementedusingtheconstraint.Tocreatesuchavariablewesimplyintroduceineveryinstanceofthereducedproblemanauxiliaryvariableandplaceaverylargeweightonit,sothatanysmallweightassignmenttothevariablesisforcedtomakeazero.Lemma50MINWEIGHTONES;xisNEARESTCODEWORD-hardandhardtoapproximatetowithinafactorof.Proof:TheNEARESTCODEWORD-hardnessfollowsfromLemma29andProposition43.ThehardnessofapproximationisduetoLemma37.Lemma51MINWEIGHTONESz;xz;xishardtoapproximatewithinforany.Proof:FollowsfromLemma41andProposition43.Lemma52IfisweaklypositiveandnotIHS-(nor0-valid)thenMINWEIGHTONESisMINHORNDELETION-hard.Proof:SimilartotheproofofLemma39.Lemma53Ifisnot2CNF,norIHS-,norafne,norweaklypositive(nor0-validnorweaklyneg-ative),thenMINONESispoly-APX-hardandMINWEIGHTONESishardtoapproximatetowithinanyfactor.Proof:Werstshowhowtohandletheweightedcase.Thehardnessfortheunweightedcasewillfolloweas-ily.Considerafunctionwhichisnotweaklypositive.Forsuchan,thereexistsassignmentsandsuchthatandandiszeroineverycoordinatewhereiszero.(Suchainputpairex-istsforeverynon-monotonefunctionandeverymono-tonefunctionisalsoweaklypositive.)Nowletbetheconstraintobtainedfrombyrestrictingittoin-putswhereisone,andsettingallotherinputstozero.Thenisasatisablefunctionwhichisnot-valid.WecannowapplySchaefer'stheorem[25]toconcludethatSAT((ff0g)ishardtodecide.WenowreduceaninstanceofdecidingSAT((ff0g)toapproximat-ingMINWEIGHTCSP.GivenaninstanceofSAT((ff0g)wecreateaninstancewhichhassomeauxiliaryvariables;:::;Wwhichareallsupposedtobezero.Thisinenforcedbygivingthemverylargeweights.Wenowreplaceeveryoccurenceofthecon-straintinbytheconstraintonthecorrespondingvariableswiththe'sinplacewhichweresettozerointoobtain.Itisclearthatifa“small”weightsolu-tionexiststotheresultingMINWEIGHTCSPproblem,thenissatisable,elseitisnot.ThusweconcludeitisNP-hardtoapproximateMINWEIGHTCSPtowithinanyboundedfactors.Fortheunweightedcase,itsufcestoobservethatbyusingpolynomiallyboundedweightsabove,wegetapoly-APXhardness.Furtheronecangetridofweightsentirelybyreplicatingvariables. Lemma54([25])Letbeaconstraintfamilythatisnot0-valid,nor1-valid,norweaklypositive,norweaklynegative,norafne,nor2CNF.Then,givenasetofcon-straintsfromitisNP-hardtodecideiftheyaresatsi-able.AcknowledgementsWethanktheanonymousrefereesandJean-PierreSeifertforseveralhelpfulcomments.References[1]S.Arora,L.Babai,J.Stern,andZ.Sweedyk.Thehard-nessofapproximateoptimainlattices,codes,andsys-temsoflinearequations.InProceedingsofthe34thIEEESymposiumonFoundationsofComputerScience,pages724–733,1993.[2]S.AroraandM.Sudan.Improvedlowdegreetestinganditsapplications.InProceedingsofthe29thACMSymposiumonTheoryofComputing,1997.Toappear.[3]M.Bellare,O.Goldreich,andM.Sudan.Freebits,PCP'sandnon-approximability–towardstightresults(3rdversion).TechnicalReportTR95-24,ElectronicColloquiumonComputationalComplexity,1995.Pre-liminaryversioninProc.ofFOCS'95.[4]D.BovetandP.Crescenzi.IntroductiontotheTheoryofComplexity.PrenticeHall,1993.[5]N.Creignou.Adichotomytheoremformaximumgener-alizedsatisabilityproblems.JournalofComputerandSystemSciences,51(3):511–522,1995.[6]P.Crescenzi,V.Kann,R.Silvestri,andL.Trevisan.Structureinapproximationclasses.InProceedingsofthe1stCombinatoricsandComputingConference,pages539–548.LNCS959,Springer-Verlag,1995.[7]P.CrescenziandA.Panconesi.Completenessinapprox-imationclasses.InformationandComputation,93:241–262,1991.PreliminaryversioninProc.ofFCT'89.[8]P.Crescenzi,R.Silvestri,andL.Trevisan.Toweightornottoweight:Whereisthequestion?InProceedingsofthe4thIEEEIsraelSymposiumonTheoryofComputingandSystems,pages68–77,1996.[9]M.GareyandD.Johnson.ComputersandIntractability:aGuidetotheTheoryofNP-Completeness.Freeman,1979.[10]N.Garg,V.Vazirani,andM.Yannakakis.Approxi-matemax-owmin-(multi)cuttheoremsandtheirappli-cations.SIAMJournalonComputing,25(2):235–251,1996.PreliminaryversioninProc.ofSTOC'93.[11]F.Gavril.Manuscriptcitedin[9],1974.[12]D.Hochbaum,N.Megiddo,J.Naor,andA.Tamir.Tightboundsand2-approximationalgorithmsforintegerpro-gramswithtwovariablesperinequality.MathematicalProgramming,62:69–83,1993.[13]D.Johnson.Approximationalgorithmsforcombina-torialproblems.JournalofComputerandSystemSci-ences,9:256–278,1974.[14]S.KhannaandR.Motwani.Towardsasyntacticchar-acterizationofPTAS.InProceedingsofthe28thACMSymposiumonTheoryofComputing,1996.[15]S.Khanna,M.Sudan,andD.Williamson.Acom-pleteclassicationoftheapproximabilityofmaximiza-tionproblemsderivedfrombooleanconstraintsatisfac-tion.InProceedingsofthe29thACMSymposiumonTheoryofComputing,1997.Toappear.[16]P.Klein,A.Agarwal,R.Ravi,andS.Rao.Aprroxima-tionthroughmulticommodityow.InProceedingsofthe31stIEEESymposiumonFoundationsofComputerScience,pages726–737,1990.[17]P.Klein,S.Plotkin,S.Rao,and´E.Tardos.Approxi-mationalgorithmsforsteineranddirectedmulticuts.ToappearonJournalofAlgorithms.,1996.[18]P.KolaitisandM.Thakur.LogicaldenabilityofNPoptimizationproblems.InformationandComputation,115(2):321–353,1994.[19]P.KolaitisandM.Thakur.ApproximationpropertiesofNPminimizationclasses.JournalofComputerandSystemSciences,50:391–411,1995.PreliminaryversioninProc.ofStructures91.[20]R.Ladner.Onthestructureofpolynomialtimere-ducibility.JournaloftheACM,22(1):155–171,1975.[21]C.LundandM.Yannakakis.Onthehardnessofap-proximatingminimizationproblems.JournaloftheACM,41:960–981,1994.PreliminaryversioninProc.ofSTOC'93.[22]G.NemhauserandL.Trotter.Vertexpacking:structuralpropertiesandalgorithms.MathematicalProgramming,8:232–248,1975.[23]C.H.PapadimitriouandM.Yannakakis.Optimization,approximation,andcomplexityclasses.JournalofCom-puterandSystemSciences,43:425–440,1991.Prelimi-naryversioninProc.ofSTOC'88.[24]R.RazandS.Safra.Asub-constanterror-probabilitylow-degreetest,andasub-constanterror-probabilityPCPcharacterizationofNP.InProceedingsofthe29thACMSymposiumonTheoryofComputing,1997.Toap-pear.[25]T.Schaefer.Thecomplexityofsatisabilityproblems.InProceedingsofthe10thACMSymposiumonTheoryofComputing,pages216–226,1978.[26]L.Trevisan,G.Sorkin,M.Sudan,andD.Williamson.Gadgets,approximation,andlinearprogramming.InProceedingsofthe37thIEEESymposiumonFounda-tionsofComputerScience,pages617–626,1996.AppendixA.ClassicationTheoremsofCreignou[5]andKhannaet.al.[15]Theorem55(MAXCSPClassi-cationTheorem)[5,15]Foreveryconstraintset,theproblemMAXCSPisalwayseitherinPorisAPX-complete.Furthermore,itisinPifandonlyifis0-validor1-validor2-monotone. Theorem56(MAXONESClassicationTheorem)[15]Foreveryconstraintset,MAXONESiseithersolvableex-actlyinPorAPX-completeorpoly-APX-completeordecidablebutnotapproximabletowithinanyfactorornotdecidable.Furthermore,(1)Ifis-validorweaklypositiveorafnewithwidth,thenMAXONESisinP.(2)ElseifisafnethenMAXONESisAPX-complete.(3)Elseifisstrongly-validorweaklynegativeor2CNFthenMAXONESispoly-APXcom-plete.(4)Elseifis0-validthenSATisinPbutnd-ingasolutionofpositivevalueisNP-hard.(5)ElsendinganyfeasiblesolutiontoMAXONESisNP-hard.