Table Of Content

6 Renormalizationgroupapproach tosatisfiability 0 0 S.N. Coppersmith 2 Department of Physics, University of Wisconsin, n 1150 University Avenue, Madison, WI 53706 USA a (Dated:February2,2008) J Satisfiability is a classic problem in computational complexity theory, in which one wishes to determine 7 1 whether an assignment of values to a collection of Boolean variables exists in which all of a collection of clausescomposed oflogical OR’softhesevariablesistrue. Here, arenormalizationgrouptransformation is ] constructedandusedtorelatethepropertiesofsatisfiabilityproblemswithdifferentnumbersofvariablesineach h clause.Thetransformationyieldsnewinsightintophasetransitionsdelineating“hard”and“easy”satisfiability c problems. e m PACSnumbers:05.10.Cc,89.20Ff,75.10.Nr - t a t Computational complexity theory addresses the (1 -2 4), (-1 3 -4), (2 3 4), (-1 -3 4). s . question of how fast the resourcesrequiredto solve a 2-SAT can be solved in polynomialtime [1], while t a givenproblemgrow with the size of the inputneeded K-SATwithK 3isknowntobeNP-complete[4]: m to specify the problem.[1] P is the class of problems ≥ if a polynomialalgorithm for 3-SAT exists, then P is - thatcanbesolvedinpolynomialtime,whichmeansa equal to NP. The complexity of SAT is intimately re- d timethatgrowsasapolynomialofthesizeoftheprob- n latedtothepresenceofphasetransitions[5,6,7,8,9, lem specification, while NP is the class of problems o 10, 11]. For random problems with N variables, M c forwhichasolutioncanbeverifiedinpolynomialtime. clauses, and K literals per clause, as M is increased [ Whether or notP is distinct from NP has been a cen- there is a phase transition from a satisfiable phase, in tralunansweredquestionincomputationalcomplexity 2 which almost all random instances are satisfiable, to v theoryfordecades.[2] anunsatisfiablephase,inwhichalmostallrandomin- 0 Satisfiability (SAT) is a classic problem in compu- stances are unsatisfiable. The most difficult instances 6 tational complexity. An often-studied type of SAT is are near this SAT-unSAT transition. It has also been 1 K-SAT, in which one attempts to find assignment of 9 shown that there is a transition as the parameter K is N variables such that the conjunction (AND) of M 0 changed between 2 and 3, at Kc 2.4, at which the 5 constraints,orclauses,eachofwhichisthedisjunction natureoftheSAT-unSATtransition∼changes[5]. 0 (OR) of K literals, each literal being either a negated Here, we investigatethe relationshipbetweensatis- / orun-negatedvariable,istrue.(Thiswayofwritingthe t fiability problems with different values of K by con- a problem, as a conjunctionof clauses that are disjunc- m structingarenormalizationgrouptransformation,sim- tions,iscalledconjunctivenormalform.)Forexample, ilartothoseusedforphasetransitionproblems[12,13, d- the3-SATinstancewiththe4variablesx1,x2,x3,and 14,15],thatreducesthenumberofdegreesoffreedom, n x4 andthefourclauses whilepossiblyincreasingthenumberandrangeofin- o teractions[16](whichinthiscontextisthenumberof c (x1 =1ORx2 =0ORx4 =1) literalsperclause). Todothis,wenotethattheexpres- : v AND (x1 =0ORx3 =1ORx4 =0) sion i X AND (x2 =1ORx3 =1ORx4 =1) (( 1x), ( 2x), ...( P x), r AND (x1 =0ORx3 =0ORx4 =1), (1) A A A a ( 1 x), ( 2 x))...,( Q x)) B − B − B − is satisfiable because it it is true for the assignments issatisfiableifandonlyif x1 =1,x2 =1,x3 =1,x4 =1. Below,wewillwrite satisfiability problemsin conjunctivenormalformus- (( 1 1), ( 1 2),...( 1 Q), ing the notation of [3], where the ANDs and ORs are A B A B A B impliedandtheliteralshavepositiveornegativesigns (A2B1), (A2 B2),...(A2 BQ), dependingonwhetherornottheyarenegated. Forex- ..., ample,theexpressionofEq.(1)iswritten ( P 1), ( P 2), ..., ( P Q)) A B A B A B 2 is satisfiable. Here, the ’s and ’s are arbitrary framework for understanding a phase transitions be- i i A B clausesandxisavariable. (Theeasiestwaytoseethe tween “easy” and “hard” satisfiability problems iden- equivalenceistonotethatbothexpressionsaresatisfi- tifiedin[5]. able if andonly ( 1 AND 2 AND ... P) OR ( 1 Wepresentevidencethatthechangeinthenatureof A A A B AND 2 AND... Q)is.) Thefirststepoftherenor- the SAT-unSAT phase transition critical value K B B c malization procedure is to use this identity to elimi- ∼ 2.4[5]isintimatelyrelatedtowhetherornotthenum- nate a givenvariable. In thisstep, P clauses in which berofclausesproliferatesexponentiallyuponrepeated agivenvariablecomesinun-negatedandQclausesin application of the renormalization group (RG) trans- which the same variable comes in negated are elimi- formation. Note that when K = 2 the clause length natedandreplacedwith PQ “resolution”[17] clauses. decreasesuponrenormalization,sincetheresolutionof Thus,eliminatinga“frustrated”[18]variable(onethat two 2-clauses is a 2-clause, so no clause gets longer, enters into different clauses negated and un-negated) andsomeof theresultingclauseshavea duplicatelit- increases the numberof clauses if PQ-(P+Q)>0. The eral and so get shorter. Having a large number of 2- resolution of two clauses of length K and K has i j clauseslimitsthegrowthinthenumberoflongclauses lengthK +K 2.Notethatresolvingtwo2-clauses i j because of subsumption, so there is a qualitative dif- − yieldsa2-clause,resolvinga2-clausewithaclauseof ferenceinthebehaviordependingonwhethertheratio length 3 yieldsa clause length , andresolving ofthe numberof 2-clausestothe numberofvariables K ≥ K twoclausesofwithlengths 1 3and 2 3yields growsorshrinksuponrenormalization. K ≥ K ≥ aclausewithlengthgreaterthanboth 1and 2. K K We show numerical data for an RG implementa- One then simplifies the resulting satisfiability ex- tioninwhichsuccessivevariablesarechosenrandomly pressionbynotingthat and eliminated if they occur in a clause of minimum 1. Duplicateclausesareredundant, length. This procedure is used because it focuses on short clauses, which are much more restrictive than 2. Duplicateliteralsinagivenclauseareredundant, long clauses. Figure 1 shows α , the ratio of M , K K the number of clauses of length K to N, the number 3. Ifavariableentersintooneclausebothnegated of variables remaining in the problem, as a function andun-negated,thentheclausemustbetrueand of K, as the RG proceeds. The average and standard canberemoved, deviation of numerical data from 5 realizations at the SAT-unSAT transition with p = 0.2 and p = 0.6 are 4. If a clause has one literal, then the value of the shown(usingparametervaluesforthetransitionloca- correspondingvariableisdetermined,and tionsfrom[5]).Largenumbersoflongclausesaregen- eratedwhenp=0.6>p andnotwhenp=0.2<p . 5. If a subset of the literals in a clause comprisea c c differentclause,thentheclausewithmoreliter- alsisredundant. IthasbeenproventhattheSAT-unSATtransitionfor 2-SAToccursatα=1[25],andfor2 K <2.4,the ≤ This last point means, for example, that if an expres- SAT-unSATtransitionisbelievedtooccurwhenα2 = sioncontainsboth(13-45)and(13),then(13-45) 1 [5]. Figure 2 (left) shows that when K = 2.2, α2 canberemoved,becauseitissatisfiedautomaticallyif increasesuponrenormalizationwhen α2 > 1 and de- (13)issatisfied. creasesuponrenormalizationwhen α2 < 1. Because Thisprocedureisknownincomputerscienceas“the addingadditionalthree-clausesdoesnotaffectthebe- Davis-Putnam procedure of 1960 [19] with subsump- haviorofthetwo-clauses,andbecausetwo-clausesare tion[20],”andwasoriginallyproposedasamethodfor much more restrictive than longer clauses, the two- solvingsatisfiabilityinstances.Itdoesnotperformwell clausesdominatetheproblemwheneverα2 > 1. One in practice[21], andhasbeenproventorequireexpo- wouldexpectthat3-clausesbythemselveswouldpro- nentialtimeonsomeinstances[22,23].However,here liferatewhenN , the initialnumberofvariables, initial the aim is not to solve a given instance, but rather to wasequalto3M3/2. (Thisestimate,theanalogofthe investigatethe“flow”oftheproblemitselfasvariables resultfortwo-clauses,followsfromsettingthenumber are eliminated [12, 24]. In particular, this renormal- ofliteralsinall3-clausestotwicethenumberofvari- ization group (RG) transformation provides a natural ables,whichmeansthatonaverageeachvariableenters 3 αK2431 α =N1in.NNNNNN2iti5//////aNNNNNNl,= piiiiii1nnnnnn=iiiiii0ttttttiiiiii0aaaaaa0llllll.======2100000.....987662468 αK2467835 NNNNNN//////NNNNNNiiiiiinnnnnniiiiiittttttiiiiiiaaaaaallllll======100000.....987662468 N αpin==it0i2a.l.6=275 α201..2551 α=1.0p=0.2 ber of clauses)/(number of variables))105 S23--AccllTaa-uuussneeS ppArrooTll iitffreearrnaasttiiiootinno n (est) α=1.25 m 1 α=1.67 nu 00K (clause lengt5h) 00 K (cl5ause length) 10 00Frac0ti.o1n of0 v.2ariab0l.e3s eli0m.4inate0d.5 α (( 02 K (c2la.5use length) 3 FIG.1:PlotofαK,theratioofMK,thenumberofclausesof FIG.2:Left:plotofα2,theratioofthenumberof2-clauses tothenumberofvariables,versusN,thenumberofundeci- lengthK,toN,thenumberofvariables,atdifferentstages matedvariables,forinstanceswithp=0.2anddifferentini- of the renormalization process. The points plotted are the tialvaluesofα.Thenumericaldataareaveragesandstandard meanandstandarddeviationoftheresultsfromfiveindepen- deviationsoffiverealizationsofsystemsofsize500whenthe dentsystemrealizations. Theparametersarechosentobeat the SAT-unSAT transition with p = 0.2 < pc (left panel) initial α = 1, size 400 when the initial α = αc = 1.2, and size 300 when the initial α = 1.67. The numerical and p = 0.6 > pc (right panel). Whenp > pc theclause lengthincreasesmarkedlyandthenumber ofclausesgrows dataareconsistent withthehypothesisthatwhenK < Kc enormously. theSAT-unSATtransitionoccurswhenα2 neitherdecreases norincreasesuponrenormalization. Right: Schematicphase diagram showing the SAT-unSAT transition (using data of Ref. [5]), the region in which the number of 2-clauses in- creasesuponrenormalization (theredhatchedregioninthe intoone3-clausenegatedandone3-clauseun-negated. leftofthefigure)andanestimateoftheregioninwhichthe Moreover, eliminating one variable on average yields numberofclauseswithK ≥ 3increasesuponrenormaliza- twolessliteralsandonelessvariable,sothat,ignoring tion(thebluehatchedregionintherightofthefigure). The fluctuations, the relationship remains true.) However, SAT-unSATtransition line crosses into the region in which long clauses proliferate exponentially at the intersection of becausethe2-clausespreventthe3-clausesfromprolif- thethreelines. Theestimatefor3-clauseproliferationgiven erating, adding3-clauses to the 2-clauseschangesthe inthetextyieldsanintersectionat(K =2.4,α=1.25). natureoftheSAT-unSATtransitiononlywhenenough 3-clauseshavebeenaddedsothattheSAT-unSATtran- sitionoccurswithα2 <1. Inthisregime,underrenor- malization the 2-clauses disappear and so the clauses unSATtransition,fortheonsetofincreaseinthenum- allbecomelonger. SinceverylongclausesareORsof berof2-clauses,andourestimatefortheonsetofpro- manyliteralsandhenceeasyto satisfy, thenumberof liferation of clauses of length greater than or equalto clausesmustgoupsufficientlyfastfortheproblemto three. bedifficulttosolve—atlargeK,theSAT-unSATtran- sitionoccurswhentheratioofthenumberofclausesto WhenK > K , theSAT-unSATtransition[5, 6, 7, c thenumberofvariablesis 2K [11]. WhenK >K , 8,9,10,26]occursinaregimeinwhichtherenormal- c ∝ near the SAT-unSAT transition we expect the number ization transformation causes both the typical clause ofclausestogrowgeometricallywithiterationnumber, length and the total number of clauses to grow. We and the numerical data are consistent with the maxi- conjecture that the SAT-UNSAT transition occurs at mumnumberofclausesobtainedduringtherenormal- the value of M/N at which the rate of exponential izationprocessincreasingexponentiallywiththeinitial growthisacriticalvalue. A“replica-symmetrybreak- number of variables, with an exponent that increases ing” transition [7, 9, 10, 26] at a somewhat smaller monotonically with (M/N) , the initial ratio of valueofαcanbeinterpretedintermsofpropagationof initial thenumberofclausestothenumberofvariables. Fig- constraintson eliminatedliterals, as will be discussed ure2(right)showsaphaseboundarylinesfortheSAT- elsewhere.[27]Becauseoftheexponentialclausepro- 4 liferation, numerical investigation of these transitions [3] SATLIB - The Satisfiability Library; using this renormalization group is limited to small http://www.satlib.org. sizes. However, the renormalization group may still [4] S.Cook,inProceedingsofthethirdannualACMsym- posium on Theory of computing (ACM, New York, be useful for investigating these transitions using an- 1971),pp.151–158. alytic techniques appropriatefor large K [10, 11] for [5] R.Monasson, R.Zecchina, S.Kirkpatrick, B.Selman, anyK >K ,thoughitwillbenecessarytounderstand c andL.Troyansky,Nature400,133(1999). how to account for possible RG-induced correlations [6] S. Kirkpatrick and B. Selman, Science 264, 1297 betweenclauses. (1994). WhenK > Kc,thenumberofclausescontinuesto [7] M. Me´zard, G. Parisi, and R. Zecchina, Science 297, increaseunderrenormalizationuntilitisnolongerun- 812(2002). likelythatagivencompoundclausecontainsarepeated [8] G.Biroli,R.Monasson, andM.Weigt,Eur.Phys.J.B 14,551(2000). variable (in additionto the decimated one), which we [9] M.Me`zardandR.Zecchina, Phys.Rev.E66, 056126 expecttooccurwhentherenormalizedclauselengthis (2002). oforder√N [28]. Because itappearsthatthereisno [10] T.Mora,M.Me´zard,andR.Zecchina(2005),preprint impedimentto the growth in the effective value of K cond-mat/0506053. until it is of order √N, where the problem specifica- [11] D.Achlioptas,A.Naor,andY.Peres,Nature435, 759 tion itself is exponentially large in N, it appears that (2005). therenormalizationgroupprocedurecantransformthe [12] K.Wilson,Phys.Rev.B4,3174(1970). [13] K.Wilson,ScientificAmerican241,158(1979). problemoutofNPandevenPSPACEaltogether. This [14] H.MarisandL.Kadanoff,Am.J.Phys.46,652(1978). propertymayindicatethatwhetherornotagivencom- [15] S.White,Phys.Rev.Lett.69,2863(1992). putational problem has a solution that can be verified [16] T. Niemeijer and J. V. Leeuwen, in Phase transitions using polynomially-bounded resources has no funda- andcriticalphenomena, vol.6 (Academic, NewYork, mentaleffectonthedifficultyofsolvingtheproblem. 1976). Insummary,atransformationinspiredbytherenor- [17] J.Robinson,J.Assoc.Comput.Machin.12,23(1965). malization groupis constructedand used to relate the [18] G.Toulouse,Phys.Rep.49,267(1979). [19] M.DavisandH.Putnam,J.oftheACM7,201(1960). behavior of satisfiability problems with different val- [20] S. Cook and R. Reckhow, J. Symbolic Logic 44, 36 uesofK,thenumberofliteralsperclause. Thetrans- (1979). formationprovidesusefulinsightintopreviouslyiden- [21] L. Zhang andS. Malik, inLecture NotesinComputer tified phase transitions of satisfiability problems and Science,Vol.2404(Springer-Verlag,London,2002),pp. mayyieldnew insightintothe questionof whetheror 17–36. notPisequaltoNP. [22] G.Tseitin,inStudiesinConstructiveMathematicsand The author gratefully acknowledges finan- Mathematical Logic, Part II, edited by A. Slisenko (ConsultantsBureau,1968),pp.115–125. cial support from grants nsf-dmr0209630 and [23] A.Haken,Theor.Comp.Sci.39,297(1985). nsf-emt0523680, and the hospitality of the Aspen [24] L.Kadanoff,Physics2,263(1966). CenterforPhysics,wheresomeofthisworkwasdone. [25] B.Bollobas, C.Borgs,J.Chayes, J.Kim,andD.Wil- son,RandomStructuresandAlgorithms18,201(2001). [26] S. Mertens, M. Me´zard, and R. Zecchina (2005), http://arxiv.org/abs/cs.CC/0309020. [27] S.Coppersmith,unpublished. [1] C. Papadimitriou, Computational Complexity [28] P.DiaconisandF.Mosteller,J.Amer.Statist.Assoc.84, (Addison-Wesley,1994). 853(1989). [2] Clay Mathematics Institute Millenium Problems; http://www.claymath.org/millennium/.