Generating Hard Satisable Formulas by Hiding Solutions - PDF document

GeneratingHardSatisableFormulasbyHidingSolutionsDeceptivelyHaixiaJiaandCristopherMooreandDougStrainComputerScienceDepartmentUniversityofNewMexicofhjia,mooreg@cs.unm.edu,doug.strain@gmail.comAbstractTotestincompletesearchalgorithmsforconstraintsatisfac-tionproblemssuchas3-SAT,weneedasourceofhard,butsatisable,benchmarkinstances.AsimplewaytodothisistochoosearandomtruthassignmentA,andthenchooseclausesrandomlyfromamongthosesatisedbyA.How-ever,thismethodtendstoproduceeasyproblems,sincethemajorityofliteralspointtowardthe“hidden”assignmentA.Lastyear,(Achlioptas,Jia,&Moore2004)proposedaprob-lemgeneratorthatcancelsthiseffectbyhidingbothAanditscomplement A.WhiletheresultingformulasappeartobejustashardforDPLLalgorithmsasrandom3-SATformulaswithnohiddenassignment,theycanbesolvedbyWalkSATinonlypolynomialtime.HereweproposeanewmethodtocanceltheattractiontoA,bychoosingaclausewitht�0literalssatisedbyAwithprobabilityproportionaltoqtforsomeq1.Byvaryingq,wecangenerateformulaswhosevariableshavenobias,i.e.,whichareequallylikelytobetrueorfalse;wecanevencausetheformulato“deceptively”pointawayfromA.Wepresenttheoreticalandexperimentalresultssuggestingthatthesefor-mulasareexponentiallyhardbothforDPLLalgorithmsandforincompletealgorithmssuchasWalkSAT.IntroductionToevaluatesearchalgorithmsforconstraintsatisfactionproblems,weneedgoodsourcesofbenchmarkinstances.Real-worldproblemsarethebestbenchmarksbydenition,buteachsuchproblemhasstructuresspecictoitsappli-cationdomain;inaddition,ifwewishtogathergooddataonhowtherunningtimesofouralgorithmsscale,weneedentirefamiliesofbenchmarkswithvaryingsizeanddensity.Onewaytollthisneedistogeneraterandominstances.Forinstance,for3-SATwecangenerateinstanceswithnvariablesandmclausesbychoosingeachclauseuniformlyfromamongthe8n3
possibilities.Wecanthenvarytheseformulasaccordingtotheirsizeandtheirdensityr=m=n.Whilesuchformulaslackmuchofthestructureofreal-worldinstances,theyhavebeeninstrumentalinthedevelop-mentandstudyofnewsearchmethodssuchassimulatedan-nealing(Johnsonetal.1989),thebreakoutprocedure(Mor-ris1993),WalkSAT(Selman,Kautz,&Cohen1996),andSurveyPropagation(M´ezard&Zecchina2002). Copyrightc\r2005,AmericanAssociationforArticialIntelli-gence(www.aaai.org).Allrightsreserved.However,ifwewishtotestincompletealgorithmssuchasWalkSATandSurveyPropagation(SP),weneedasourceofhard,butsatisableproblems.Incontrast,aboveacriticaldensityr4:27,therandomformulasdenedabovearealmostcertainlyunsatisable.Randomformu-lasatthisthresholdappeartobequitehardforcompletesolvers(Cheeseman,Kanefsky,&Taylor1991;Hogg,Hu-berman,&Williams1996;Mitchell,Selman,&Levesque1992);butforpreciselythisreason,itisnotfeasibletogen-eratelargeproblemsatthethresholdandthenlterouttheunsatisableones.WhileotherclassesofsatisableCSPshavebeenproposed,suchasthequasigroupcompletionproblem(Shaw,Stergiou,&Walsh1998;Kautzetal.2001;Achlioptasetal.2000),wewouldliketohaveproblemsgeneratorsthatare“native”to3-SAT.Anaturalwaytogeneraterandomsatisable3-SATfor-mulasistochoosearandomtruthassignmentA2f0;1gn,andthenchoosemclausesuniformlyandindependentlyfromamongthe7n3
clausessatisedbyA.TheproblemwiththisisthatsimplyrejectingclausesthatconictwithAcausesanunbalanceddistributionofliterals;inparticular,onaveragealiteralwillagreewithitsvalueinthehiddenassignment4=7ofthetime.Thus,especiallywhentherearemanyclauses,asimplemajorityheuristicorlocalsearchwillquicklyndA.Moresophisticatedversionsofthis“hid-denassignment”scheme(Asahiro,Iwama,&Miyano1996;VanGelder1993)improvematterssomewhatbutstillleadtobiasedsamples.Thusthequestionishowtoavoidthis“attraction”tothehiddenassignment,Oneapproach(Achlioptas,Jia,&Moore2004)istochooseclausesuniformlyfromamongthosethataresat-isedbybothAanditscomplement A.Thisisin-spiredbyrecentworkonrandomk-SATandNot-All-EqualSAT(Achlioptas&Moore2002b),inwhichsymmetrywithrespecttocomplementationreducesthevarianceofthenum-berofsolutions;theideaisthatAand Acanceleachothers'attractionsout,makingeitheronehardtond.Indeed,theresultingformulasappeartotakeDPLLsolversexponentialtimeand,ingeneral,tobejustashardasrandom3-SATformulaswithnohiddenassignment.Ontheotherhand,WalkSATsolvestheseformulasinpolynomialtime,sinceafterafewvariablesaresetinawaythatagreeswithoneofthehiddenassignments,neighboringvariablesdevelopcor-relationsconsistentwiththese(Bartheletal.2002). Inthispaper,wepursueanalternateapproach,inspiredby(Achlioptas&Peres2003),whoreweightedthesatisfy-ingassignmentsinanaturalway.Wehidejustoneassign-ment,butwebiasthedistributionofclausesasfollows:foreachclause,wechoosearandom3-tuple(ormoregenerally,ak-tuple)ofvariables,andconstructaclausewitht�0lit-eralssatisedbyAwithprobabilityproportionaltoqtforsomeconstantq1.(Notethatthenaiveformulasdis-cussedaboveamounttothecaseq=1.)Thispenalizestheclauseswhichare“moresatised”byA,andreducestheextenttowhichvariableoccurrencesaremorelikelytoagreewithA.Aswewillseebelow,bychoosingqappro-priatelywecanrebalancethedistributionofliterals,sothateachvariableisaslikelytoappearpositivelyasoftenasneg-ativelyandnolongerpointstowarditsvalueinA.Byreduc-ingqfurther,wecanevenmakeitmorelikelythatavariableoccurrencedisagreeswithA,sothattheformulabecomes“deceptive”andpointsawayfromthehiddenassignment.Wecalltheseformulas“q-hidden,”todistinguishthemfromthenaive“1-hidden”formulasdiscussedabove,the“2-hidden”formulasstudiedin(Achlioptas,Jia,&Moore2004),andthe“0-hidden”formulasconsistingofrandom3-SATformulaswithnohiddenassignment.Liketheseotherfamilies,ourq-hiddenformulasarereadilyamenabletoallthemathematicaltoolsthathavebeendevelopedforstudy-ingrandomk-SATformulas,includingmomentcalculationsandthemethodofdifferentialequations.Belowwecalcu-latetheexpecteddensityofsatisfyingassignmentsasafunc-tionoftheirdistancefromA,andanalyzethebehavioroftheUnitClause(UC)algorithmonq-hiddenformulas.Wethenpresentexperimentsonseveralcompleteandincom-pletesolvers.Wendthatourq-hiddenformulasarejustashardforDPLLalgorithmsas0-hiddenformulas,andaremuchharderthannaive1-hiddenformulas.Inaddition,wendthatlocalsearchalgorithmslikeWalkSATndourfor-mulasmuchharderthananyoftheseotherfamilies,takingexponentialasopposedtopolynomialtime.Moreover,therunningtimeofWalkSATincreasessharplyasourformulasbecomemoredeceptive.TheexpecteddensityofsolutionsFor2[0;1],letXbethenumberofsatisfyingtruthas-signmentsinarandomq-hiddenk-SATformulathatagreeonafractionofthevariableswiththehiddenassignmentA;thatis,theirHammingdistancefromAis(1)n.WewishtocalculatetheexpectationE[X].Bysymmetry,wecantakeAtobetheall-trueassignment.Inthatcase,aclausewitht&#x-5.1;䝀0positiveliteralsischosenwithprobabilityqt (1+q)k1:LetBbeatruthassignmentwherenofthevariablesaretrueand(1)narefalse.Then,analogousto(Achliop-tas,Jia,&Moore2004),weuselinearityofexpectation,independencebetweenclauses,theselectionoftheliteralsineachclausewithreplacement,andStirling'sapproxima-tionforthefactorialtoobtain(wheresuppressestermspolynomialinn): 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Density of solutions with r=6a q=0.5q=0.618 q=1 0 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 aDensity of solutions with q=0.5r=3 r=4 r=5 r=5.6 r=7 Figure1:Thenthrootf()oftheexpectednumberofsolu-tionswhichagreewiththehiddenassignmentonafractionofthevariables.Herek=3.Theupperpartofthegureshowsf()forq=1,q=0:618andq=0:5atr=6.Thelowerpartshowsf()forq=0:5andvaryingr.Notethatatr=5:6,wehavef()1forall1=2.E[X]=nnPr[Bsatisesarandomclause]m=nn 1kXt=1ktqt(1)tkt (1+q)k1!mfk;r;q()nwheref()=1 (1)11(q(1)+)kk (1+q)k1r:LookingatFigure1,weseethatthebehavioroffnear=1=2changesdramaticallyaswevaryq.Forq=1(i.e.,naive1-hiddenformulas),f0(1=2)ispositive,givinglocalsearchalgorithmsa“push”towardsthehiddenassignment.Ontheotherhand,ifqisthepositiverootqof(1q)(1+q)k11=0 thenf0(1=2)=0.Analogousto(Achlioptas&Peres2003),thisisalsothevalueofqatwhichliteralsareequallylikelytoagreeordisagreewithA.Intuitively,then,ifq=qwewouldexpectalocalsearchalgorithmstartingfromarandomassignment—forwhichistightlyconcentratedaround1=2—tohavenolocalinformationtellingitinwhichdirectionthehiddenassignmentlies.Wecalltheseq-hiddenformulasbalanced;fork=3,qisthegoldenratio(p 51)=2=0:618:::Forsmallervaluesofqsuchasq=0:5showninFig-ure1,f0(1=2)becomesnegative,andweexpectalocalsearchalgorithmstartingatarandomassignmenttomoveawayfromA.Indeed,f()hasalocalmaximumatsome1=2,andforsmallrtherearesolutionswith1=2.Whenrissufcientlylarge,however,f()1forall1=2,andtheprobabilityanyofthese“alternate”so-lutionsexistisexponentiallysmall.Weconjecturethatforeachqqthereisathresholdrc(q)atwhichwithhighprobabilitytheonlysolutionsarethoseclosetoA.Set-tingmaxff()j1=2g=1yieldsanupperboundonrc(q),whichweshowinFigure4below.Forinstance,rc(0:5)5:6asshowninFigure1.Wecallsuchformulasdeceptive,sincelocalsearchal-gorithmssuchasWalkSAT,DPLLalgorithmssuchaszChaffthatuseamajorityheuristicintheirsplittingrule,andmessage-passingalgorithmssuchasSPwillpresumablysearchinthewrongdirection,andtakeexponentialtimetocrossthelocalminimuminf()tondthehiddenassign-ment.Ourexperimentsbelowappeartoconrmthisintu-ition.Inaddition,allthreetypesofalgorithmsappeartoen-counterthemostdifcultyatroughlythesamedensityrc(q),whereweconjecturethe“alternate”solutionsdisappear.UnitClauseheuristicandDPLLalgorithmsUnitClause(UC)isalinear-timeheuristicwhichperma-nentlysetsonevariableineachstepasfollows:ifthereareanyunitclauses,satisfythem;otherwise,pickarandomlit-eralandsatisfyit.Forrandom3-SATformulas,UCsucceedswithconstantprobabilityforr8=3,andfailswithhighprobabilityforr&#x-5.1;䝀8=3(Chao&Franco1986).UCcanbethoughtastherstbranchofasimpleDPLLalgorithmS,whosesplittingruletakesarandomunsetvariableandtriesitstruthvaluesinrandomorder;thusUCsucceedsifSsuc-ceedswithoutbacktracking.Ontheotherhand,(Cocco&Monasson2004;Coccoetal.2005)showedthatS'sex-pectedrunningtimeisexponentialinnforanyr&#x-5.1;䝀8=3;seealso(Achlioptas,Beame,&Molloy2001),whousedlowerboundsonresolutioncomplexitytoshowthatStakesexpo-nentialtimewithhighprobabilityifr&#x-5.1;䝀3:81.Ingeneral,itappearsthatsimpleDPLLalgorithmsbegintotakeexpo-nentialtimeatexactlythedensitywherethecorrespondinglinear-timeheuristicfails.Inthissection,weanalyzetheperformanceofUConourq-hiddenformulas.Specically,weshowthatinthebal-ancedcasewhereq=q,UCfailsforr&#x-5.1;䝀8=3justasfor0-hiddenformulas.Basedonthis,weconjecturethattherunningtimeofS,andothersimpleDPLLalgorithms,isexponentiallylargeforourformulasatthesamedensityasfor0-hiddenones.Asin(Achlioptas,Jia,&Moore2004),weanalyzethebehaviorofUConarbitraryinitialdistributionsof3-SATclausesusingthemethodofdifferentialequations.Forsim-plicityweassumethatAistheall-trueassignment.AroundofUCconsistsofa“freestep,”inwhichwesatisfyarandomliteral,andtheensuingchainofunit-clausepropagations.For0i3and0ji,letSi;j=si;jnbethenum-berofclausesoflengthiwithjpositiveliteralsandijnegativeones,andsi=Pjsi;j.LetX=xnbethenumberofvariablessetsofar,andletmTandmFbetheexpectednumberofvariablessettrueandfalseinaround.ThenwecanmodelthediscretestochasticprocessoftheSi;jwiththefollowingdifferentialequationsforthesi;j:ds3;j dx=3s3;j 1x(1)ds2;j dx=2s2;j 1x+mF(j+1)s3;j+1+mT(3j)s3;j (mT+mF)(1x)Theunitclausesaregovernedbyatwo-typebranchingpro-cess,withtransitionmatrixM=1 1xs2;12s2;02s2;2s2;1:Asin(Achlioptas&Moore2002a),aslongasthelargesteigenvalueofMislessthan1,thebranchingprocessissub-critical,andsummingovertheroundgivesmFmT=(IM)1
1=21=2:Wethensolvetheequation(1)withtheinitialconditionss3;0=0ands3;j=3jqj (1+q)31for0j3.Inthebalancedcaseq=q,wendthatUCsucceedsonq-formulaswithconstantprobabilityifandonlyifr8=3,justasfor0-hiddenformulas.Therea-sonisthat,asfor2-hiddenformulas,theexpectednumberofpositiveandnegativeliteralsarethesamethroughouttheprocess.ThissymmetrycausesUCtobehavejustasitwouldonrandom3-SATformulaswithoutahiddenassignment.Wenotethatforqq,UCsucceedsatslightlyhigherdensities,atwhichitcanndoneofthe“alternate”solutionswith1=2.Athigherdensitieswherethesealternateso-lutionsdisappear,ourexperimentalresultsbelowshowthatthese“deceptive”formulastakeDPLLalgorithmsexponen-tialtime,andforr&#x-5.1;ä¡£rc(q)theyareharderthan0-hiddenformulasofthesamedensity.ExperimentalresultsDPLLInthissectionwediscussthebehaviorofDPLLsolversonourq-hiddenformulas.WefocusonzChaff(Zhang);ourresultsfromOKsolver(Kullmann2002)arequalitativelysimilar.Figure2showszChaff'smedianrunningtimeon 4 4.5 5 5.5 6 6.5 7 7.5 8 101 102 103 104 105 zChaff performance with n=200rMedian number of Decisions over 49 trials q=0.2 50 100 150 200 250 300 101 102 103 104 105 106 zChaff performance with r=5.5NMedian number of Decisions over 49 trials q=0.3 Figure2:TheupperpartofthegureshowszChaff'sme-dianrunningtimeover49trialson0-hidden,1-hiddenandq-hiddenformulaswithn=200andrrangingfrom4.0to8.0.Thelowerpartshowsthemedianrunningtimewithr=5:5andnrangingfrom50to300.0-hidden,1-hidden,andq-hiddenformulasforvariousval-uesofq.Weseethefollowingphenomena:Ourq-hiddenformulaswithq=q=0:618:::areaboutashardas0-hiddenones,andpeakincomplexitynearthesatisabilitythreshold.Thisisconsistentwiththepicturegivenintheprevioustwosections:namely,thatthese“bal-anced”formulasmakeitimpossibleforalgorithmstofeeltheattractionofthehiddenassignment.Incontrast,naive1-hiddenformulasarefareasier,sincetheattractiontothehiddenassignmentisstrong.Theq-hiddenformulaswithqqarethemostinterest-ingones.Thehardnessoftheseformulasshowstwophases:atlowdensitytheyarerelativelyeasy,andtheirhardnesspeaksatadensityrc(q).Aboverc(q)theytakeexponen-tialtime;asfor0-hiddenformulas,althoughasrincreasesfurtherthecoefcientoftheexponentialdecreasesastheclausesgeneratecontradictionsmorequickly.Webelievethatthispeakrc(q)isthesamethresholdden-sitydenedearlier(seeFigure4below)abovewhichtheonlysolutionsarethoseclosetothehiddenassignment.Thesituationseemstobethefollowing:belowrc(q),thereare“alternate”solutionswith1=2,andzChaffisledtothesebyitssplittingrule.Aboverc(q),thesealternatesolu-tionsdisappear,andzChafftakesexponentialtimetondthevicinityofthehiddenassignment,sincetheformulade-ceptivelypointsintheotherdirection.Moreover,foraxedraboverc(q)theseformulasbecomeharderasqdecreasesandtheybecomemoredeceptive.Toillustratethisfurther,thelowerpartofFigure2showszChaff'smedianrunningtimeon0-hiddenformulas,1-hiddenformulas,andq-hiddenformulasforq=q(bal-anced)andq=0:3(deceptive).Wealsocomparewiththe2-hiddenformulasof(Achlioptas,Jia,&Moore2004).Wexr=5:5,whichappearstobeaboverc(q)forboththesevaluesofq.Atthisdensity,the0-hidden,2-hidden,andbalancedq-hiddenformulasareallcomparableindifculty,while1-hiddenformulasaremucheasierandthedeceptiveformulasappeartobesomewhatharder.SPSurveyPropagationorSP(M´ezard&Zecchina2002)isarecentlyintroducedincompletesolverbasedoninsightsfromthereplicamethodofstatisticalphysicsandageneral-izationofbeliefpropagation.WetestedSPon0-hiddenfor-mulasandq-hiddenformulasfordifferentvaluesofq,usingn=104andvaryingr.For0-hiddenformulas,SPsucceedsuptor=4:25,quiteclosetothesatisabilitythreshold.Forq-hiddenformulaswithq=q,SPfailsat4:25justasitdoesfor0-hiddenformulas,suggestingthatitndsthesefor-mulasexactlyashardas0-hiddenoneseventhoughtheyareguaranteedtobesatisable.Fornaive1-hiddenformulas,SPsucceedsatasignicantlyhigherdensity,uptor=5:6.Presumablythenaive1-hiddenformulasareeasierforSPsincethe“messages”fromclausestovariables,likethema-jorityheuristic,tendtopushthealgorithmtowardsthehid-denassignment.Inthebalancedcaseq=q,thisattrac-tionissuccessfullysuppressed,causingSPtofailatessen-tiallythesamedensityasfor0-hiddenformulas,closetothesatisabilitythreshold,eventhoughourq-hiddenformulascontinuetobesatisableatalldensities.Incontrast,the2-hiddenformulasof(Achlioptas,Jia,&Moore2004)aresolvedbySPuptoasomewhathigherdensityr4:8.Thusitseemsthatthereweightingapproachofq-hiddenformulasdoesabetterjobofconfusingSPthanhidingtwocomple-mentaryassignmentsdoes.Forqq,SPsucceedsuptosomewhathigherdensities,eachofwhichmatchesquitecloselythevaluerc(q)atwhichzChaff'srunningtimepeaks(seeFigure4below).Build-ingonourconjecturethatthisisthedensityabovewhichtheonlysolutionsarethoseclosetothehiddenassignment,weguessthatSPsucceedsforrrc(q)preciselybecausethelocalgradientinthedensityofsolutionspushesittowardsthe“alternate”solutionswith1=2.Aboverc(q),thesesolutionsnolongerexist,andSPfailsbecausetheclausessenddeceptivemessages,demandingthatvariablesbesetoppositetothehiddenassignment. WalkSATWeconcludewithalocalsearchalgorithm,WalkSAT.Foreachformula,wedidupto104restarts,with104stepsperattempt,whereeachstepdoesarandomorgreedyipwithequalprobability.IntheupperpartofFigure3wemeasureWalkSAT'sperformanceonq-hiddenformulaswitharangeofvaluesofq,includingq=1,q=q,anddeceptiveval-uesofqrangingfrom0:2to0:5.Weusedn=200andletrrangefrom4to8.Evenfortheserelativelysmallfor-mulas,weseethatforthethreemostdeceptivevaluesofq,thereisadensityatwhichthemedianrunningtimejumpsto108,indicatingthatWalkSATfailstosolvetheseformulas.Forinstance,q-hiddenformulaswithq=0:4appeartobeunfeasibleforWalkSATfor,say,r�5.Webelievethat,consistentwiththediscussionabove,lo-calsearchalgorithmslikeWalkSATgreedilyfollowthegradientinthedensityofsolutionsf().Forqq,thisgradientisdeceptive,andluresWalkSATawayfromthehiddenassignment.Atdensitiesbelowrc(q),therearemanyalternatesolutionswith1=2andWalkSATndsoneofthemveryeasily;butfordensitiesaboverc(q),theonlysolutionsarethosenearthehiddenassignment,andWalkSAT'sgreedcausesittowanderforanexponentiallylongtimeinthewrongregion.Thispictureissupportedbythefactthat,asFigure4showsbelow,thedensityatwhichWalkSAT'srunningtimejumpsupwardcloselymatchesthethresholdsrc(q)thatweobservedforzChaffandSP.ThelowerpartofFigure3looksatWalkSAT'smedianrunningtimeataxeddensityasafunctionofn.Wecom-pare1-hiddenand2-hiddenformulaswithq-hiddenoneswithq=qandtwodeceptivevalues,0:5and0:3.Wechooser=5:5,whichisaboverc(q)forallthreeval-uesofq.Therunningtimeof1-hiddenand2-hiddenfor-mulasisonlypolynomial(Achlioptas,Jia,&Moore2004;Bartheletal.2002).Incontrast,eveninthebalancedcaseq=q,therunningtimeisexponential,andtheslopeofthisexponentialincreasesdramaticallyaswedecreaseqandmaketheformulasmoredeceptive.WenotethatitmightbepossibletodevelopaheuristicanalysisofWalkSAT'srunningtimeinthedeceptivecaseusingthemethodsof(Se-merjian&Monasson2004;Coccoetal.2005).ThethresholddensityAswehaveseen,thereappearstobeacharacteristicdensityrc(q)foreachvalueofqqatwhichtherunningtimeofDPLLalgorithmslikezChaffpeaks,atwhichWalkSAT'srunningtimebecomesexponential,andatwhichSPceasestowork.Weconjecturethatinallthreecases,thekeyphe-nomenonatthisdensityisthatthesolutionswith1=2disappear,leavingonlythoseclosetothehiddenassign-ment.Figure4showsourmeasuredvaluesofrc(q),andindeedtheyarequitecloseforthethreealgorithms.Wealsoshowtheanalyticupperboundonrc(q)resultingfromset-tingmaxff()j1=2g=1,abovewhichtheexpectednumberofsolutionswith1=2isexponentiallysmall. 4 4.5 5 5.5 6 6.5 7 7.5 8 102 103 104 105 106 107 108 WalkSAT performance with n=200rMedian number of flips over 49 trials q=0.2 100 200 300 400 500 600 102 104 106 108 WalkSAT performance with r=5.5Median number of flips over 49 trialsN q=0.3 Figure3:TheupperpartshowsWalkSAT'smedianrun-ningtimeover49trialsonq-hiddenformulaswithn=200andrrangingfrom4to8;thelowerpartshowsthemedianrunningtimewithr=5:5andnrangingfrom50to600.ConclusionsWehaveintroducedasimplenewwaytohidesolutionsin3-SATproblemsthatproducesinstancesthatarebothhardandsatisable.Unlikethe2-hiddenformulasof(Achliop-tas,Jia,&Moore2004)wheretheattractionofthehiddenassignmentiscancelledbyalsohidingitscomplement,hereweeliminatethisattractionbyreweightingthedistributionofclausesasin(Achlioptas&Peres2003).Indeed,bygo-ingbeyondthevalueoftheparameterqthatmakesourq-hiddenformulasbalanced,wecancreatedeceptiveformulasthatleadalgorithmsinthewrongdirection.Experimentally,ourformulasareashardorharderforDPLLalgorithmsas0-hiddenformulas,i.e.,random3-SATformulaswithoutahid-denassignment;forlocalsearchalgorithmslikeWalkSAT,theyaremuchharderthan0-hiddenor2-hiddenformulas,takingexponentialratherthanpolynomialtime.Ourformu-lasarealsoamenabletoallthemathematicaltoolsdevelopedforthestudyofrandom3-SAT;herewehavecalculatedtheir 0.2 0.3 0.4 0.5 0.6 4 5 6 7 8 9 10 11 12 rc(q)q Upper bound Figure4:Thedensityrc(q)atwhichtherunningtimeofzChaffpeaks,WalkSATpeaksorexceeds108ips,andSPstopsworking.Weconjecturealloftheseeventsoccurbecauseatthisdensitythealternatesolutionswith1=2disappear,leavingonlythoseclosetothehiddenassignment.Shownalsoistheanalyticupperbounddescribedinthetext.expecteddensityofsolutionsasafunctionofdistancefromthehiddenassignment,andusedthemethodofdifferentialequationstoshowthatUCfailsforthematthesamedensityasitdoesfor0-hiddenformulas.Weclosewithseveralexcitingdirectionsforfuturework:1.Conrmthatthereisasinglethresholddensityrc(q)atwhicha)thealternatesolutionsfarfromthehiddenas-signmentdisappear,b)therunningtimeofDPLLalgo-rithmsismaximized,c)SPstopsworking,andd)therun-ningtimeofWalkSATbecomesexponential.2.ProvethatsimpleDPLLalgorithmstakeexponentialtimeforr&#x-5.1;ä¡£rc(q),inexpectationorwithhighprobability.3.Calculatethevarianceofthenumberofsolutionsasafunctionof,andgivingimprovedupperandlowerboundsonthedistributionofsolutionsandrc(q).AcknowledgmentsH.J.issupportedbyanNSFGraduateFellowship.C.M.andD.S.aresupportedbyNSFgrantsCCR-0220070,EIA-0218563,andPHY-0200909.C.M.thanksTracyConradandSheWhoIsNotYetNamedfortheirsupport.ReferencesAchlioptas,D.,andMoore,C.2002a.Almostallgraphswithaveragedegree4are3-colorable.STOC199–208.Achlioptas,D.,andMoore,C.2002b.Theasymptoticorderoftherandomk-SATthreshold.FOCS779–788.Achlioptas,D.,andPeres,Y.2003.Thethresholdforran-domk-SATis2k(ln2o(k)).STOC223–231.Achlioptas,D.;Beame,P.;andMolloy,M.2001.Asharpthresholdinproofcomplexity.STOC337–346.Achlioptas,D.;Gomes,C.;Kautz,H.;andSelman,B.2000.Generatingsatisableprobleminstances.AAAI256.Achlioptas,D.;Jia,H.;andMoore,C.2004.Hidingsatis-fyingassignments:twoarebetterthanone.AAAI131–136.Asahiro,Y.;Iwama,K.;andMiyano,E.1996.Randomgenerationoftestinstanceswithcontrolledattributes.DI-MACSSeriesinDisc.Math.andTheor.Comp.Sci.26.Barthel,W.;Hartmann,A.;Leone,M.;Ricci-Tersenghi,F.;Weigt,M.;andZecchina,R.2002.Hidingsolutionsinrandomsatisabilityproblems:Astatisticalmechanicsapproach.Phys.Rev.Lett.88(188701).Chao,M.,andFranco,J.1986.Probabilisticanalysisoftwoheuristicsforthe3-satisabilityproblem.SIAMJ.Comput.15(4):1106–1118.Cheeseman,P.;Kanefsky,R.;andTaylor,W.1991.Wherethereallyhardproblemsare.IJCAI163–169.Cocco,S.,andMonasson,R.2004.Heuristicaverage-caseanalysisofthebacktrackresolutionofrandom3-satisabilityinstances.Theor.Comp.Sci.320:345–.Cocco,S.;Monasson,R.;Montanari,A.;andSemerjian,G.2005.Approximateanalysisofsearchalgorithmswith“physical”methods.InPercus,A.;Istrate,G.;andMoore,C.,eds.,ComputationalComplexityandStatisticalPhysics.OxfordUniversityPress.Hogg,T.;Huberman,B.;andWilliams,C.1996.Phasetransitionsandcomplexity.ArticialIntelligence81.Johnson,D.;Aragon,C.;McGeoch,L.;andShevon,C.1989.Optimizationbysimulatedannealing:anexperimen-talevaluation.OperationsResearch37(6):865–892.Kautz,H.;Ruan,Y.;Achlioptas,D.;Gomes,C.;Selman,B.;andStickel,.2001.Balanceandlteringinstructuredsatisableproblems.IJCAI351–358.Kullmann,O.2002.InvestigatingthebehaviourofaSATsolveronrandomformulas.TechnicalReportCSR23-2002,UniversityofWalesSwansea.M´ezard,M.,andZecchina,R.2002.Randomk-satisability:fromananalyticsolutiontoanewefcientalgorithm.Phys.Rev.E66:056126.Mitchell,D.;Selman,B.;andLevesque,H.1992.HardandeasydistributionsofSATproblems.AAAI459–465.Morris,P.1993.Thebreakoutmethodforescapingfromlocalminima.AAAI40–45.Selman,B.;Kautz,H.;andCohen,B.1996.Localsearchstrategiesforsatisabilitytesting.2ndDIMACSChallangeonCliques,Coloring,andSatisability.Semerjian,G.,andMonasson,R.2004.Astudyofpurerandomwalkonrandomsatisabilityproblemswith“phys-ical”methods.LNCS2919:120–.Shaw,P.;Stergiou,K.;andWalsh,T.1998.Arccon-sistencyandquasigroupcompletion.ECAIworkshoponnon-binaryconstraints.VanGelder,A.1993.Problemgeneratormkcnf.c.DI-MACSchallengearchive.Zhang,L.zChaff.ee.princeton.edu/˜chaff/zchaff.php.