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ON THE SCALING WINDOW OF MODEL RB CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 8 0 Abstract. This paper analyzes the scalingwindow ofarandom CSP model 0 (i.e. modelRB)forwhichwecanidentifythethresholdpointsexactly,denoted 2 by rcr or pcr. For this model, we establish the scaling window W(n,δ) = (r−(n,δ), r+(n,δ)) such that the probability of a random instance being n satisfiable is greater than 1−δ for r < r−(n,δ) and is less than δ for r > a r+(n,δ). Specifically,weobtainthefollowingresult J 5 W(n,δ)=(rcr−Θ(n1−ε1lnn), rcr+Θ(nl1nn)), 2 where 0 ≤ ε < 1 is a constant. A similar result with respect to the other parameter p is also obtained. Since the instances generated by model RB ] C have been shown to be hardat the threshold, this is the first attempt, as far asweknow,toanalyzethescalingwindowofsuchamodelwithhardinstances. C . s c [ 1. Introduction 1 v The Constraint Satisfaction Problem(CSP), originated from artificial intelli- 1 gence, has become an important and active field of statistical physics, informa- 7 tion theory and computer science. The CSP area is very interdisciplinary, since 8 3 it embeds ideas from many research fields, like artificial intelligence, databases, . programming languages and operation research. A constraint satisfaction problem 1 0 consists of a finite set U = u1,u2, ,un of n variables, each ui associated with { ··· } 08 aardeolmataioinn,odfevfianluedesoDnis,oamnedsaubsesettofucon,ustra,ints,.uEachofonf tvhaericaobnlestsr,acianltlesdCiit1si2s·c··oikpeis, v: denoting their legal tuples of valu{esi.1 Ai2so·lu··tionikt}o a CSP is an assignment of a i valueto eachvariablefromits domainsuchthatallthe constraintsofthis CSPare X satisfied. A constraint is said to be satisfied if the tuple of values assigned to the r variables in this constraint is a legal one. A CSP is called satisfiable if and only if a ithasatleastonesolution. ThetaskofaCSP isto finda solutionortoprovethat no solution exists. Given a CSP, we are interested in polynomial-time algorithms, that is, algorithms whoserunning time is bounded by a polynomialin the number of variables. Cook’s Theorem[2] asserts that satisfiability is NP-complete and at least as hard as any problem whose solutions can be verified in polynomial time. Most of the interesting CSPs are NP-complete problems. We know that k-SAT problem is a canonical version of the CSPs, in which variables can be assigned the value True or False(called Boolean variables). A lot of efforts have been devoted to k-SAT and it is widely believed that no efficient algorithm exists for k-SAT. However, Key words and phrases. Constraint Satisfaction Problem; Model RB; Satisfiability; Phase Transition;ScalingWindow. Supported in part by National 973 Program of China(Grant No. 200532CB1902) and NSFC(GrantNos. 60473109 and60403003). 1 2 CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 it is shown that most instances of k-SAT can be solved efficiently, so perhaps genuine hardness is only present in a tiny fraction of all instances. In 1990s, a remarkable progress[3, 13, 11, 12] was made that the the really difficult instances is related to phase transition phenomenon, as suggested in the pioneering work of Fu and Anderson[6]. The study of phase transitions has attracted much interest subsequently[9, 12]. In recent years, random k-SAT has been well studied both from theoretical and algorithmic point of views. If k = 2 then it is known that there is a satisfiability threshold at α = 1 (here α represents the ratio of clauses m to variables n), c below which the probability of a random instance being satisfiable tends to 1 and abovewhichit tends to 0 as n approachesinfinity[4]. This wassharpenedin[8,14]. Random 2-SAT is now pretty much understood. However,for k 3, the existence ≥ of the phase transition phenomenon has not been established, not even the exact value of the threshold point[1, 10]. To gain a better understanding of how the phase transition scales with problem size,thefinite-sizescalingmethodhasbeenintroducedfromstatisticalmechanics[11, 7]. We use finite-size scaling, a method from statistical physics in which observing how the width of a transition narrows with increasing sample size gives direct ev- idence for critical behavior at a phase transition. Finite-size scaling is the study of changes in the transition behavior due to finite-size effects, in particular, broad- ening of the transition region for finite n. More precisely, for 0 < δ < 1, let r (n,δ) be the supremum over r such that the probability of a random CSP in- − stance being satisfiable is at least 1 δ, and similarly, let r (n,δ) be the infimum + − over r such that the probability of a random CSP instance being satisfiable is at most δ. Then, for r within the scaling window W(n,δ) = (r (n,δ), r (n,δ)) + − the probability is between δ and 1 δ. And for all δ, r (n,δ) r (n,δ) 0 as + − | − − | → n . For random 2-SAT, it has been determined that the scaling window is → ∞ W(n,δ)=(1 Θ(n 1/3), 1+Θ(n 1/3))[2]. − − − Model RB is a random CSP model proposed by Xu and Li to overcome the trivial insolubility of standard CSP models[16]. For this model, we can not only establish the existence of phase transitions, but also pinpoint the threshold points exactly, denoted by r or p . Moreover, it has been proved that almost all in- cr cr stancesofmodelRBhavenotree-likeresolutionproofsoflessthanexponentialsize [16]. This implies that unlike random 2-SAT, model RB can be used to generate hardinstances,whichhasalsobeenconfirmedbyexperiments[7]. Motivatedbythe work on the scaling window of random 2-SAT, in this paper, we study the scaling windowofmodelRBandobtainthatW(n,δ)=(r Θ( 1 ), r +Θ( 1 )). cr− n1−εlnn cr nlnn And we also obtain similar results about the other control parameter p. Themaincontributionofthispaperisnottopresentnewmethodsforcomputing the scaling window, but to show that for an interesting model with hard instances (i.e. model RB), not only can the threshold points be located exactly, but also the scaling window can be deteremined using standard methods. This means that hopefully, more mathematical properties about the threshold behavior of model RB can be obtained in a relatively easy way, which will help to shed light on the phase transition phenomenon in NP-complete problems. The rest of the paper is organized as follows. In the next section, we will give a brief introduction about model RB. The main results of this paper and their proofs will be given in Section 3 and Section 4 respectively. Finally, we will conclude in Section 5. ON THE SCALING WINDOW OF MODEL RB 3 2. Model RB WecanpinpointthethresholdlocationformodelRBproposedbyXuandLi[16]. The way of generating random instances for model RB is: (1). Given a set U of n variables, select with repetition m = rnlnn random con- straints. Each random constraint is formed by selecting without repetition k of n variables, where k 2 is an integer. ≥ (2). Next, for each constraint we select uniformly at random without repetition q = p dk illegal tuples of values, i.e., each constraint contains exactly (1 p) dk · − · legalones,whered=nα isthedomain sizeofeachvariableandα>0isaconstant. In this paper, the probability of a random CSP instance being satisfiable is denoted by Pr(Sat). It is proved that for model RB the phase transition phenomenon occurs at rcr =−ln(1α p) or pcr =1−e−αr as n approaches infinity[16]. More precisely, we have the follo−wing two theorems. Theorem 2.1[16] Let r = α . If α > 1, 0 r . cr n →∞ sTahtiesofyretmhe i2n.e2q[u1a6l]itLyetkep−crαr=11−, teh−enαr. If α> k1, r >0 are two constants and k, α ≥ lim Pr(Sat)=1 when pp . cr n →∞ 3. Main results Our main results are the following two theorems. Theorem 3.1 For all sufficiently small δ > 0, there exist r (n,δ) and r (n,δ) + − such that the following holds: Pr(Sat)>1 δ, when r<r (n,δ); − − Pr(Sat) < δ, when r>r (n,δ), + where r (n,δ) = r Θ( 1 ), r (n,δ) = r +Θ( 1 ). So that the scaling − cr − n1−εlnn + cr nlnn window of model RB is 1 1 W(n,δ)=(r Θ( ), r +Θ( )). cr− n1 εlnn cr nlnn − It is easy to see that r (n,δ) r (n,δ) 0, as n . + | − − |→ →∞ Theorem 3.2 For all sufficiently small δ > 0, there exist p (n,δ) and p (n,δ) + − 4 CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 such that the following holds: Pr(Sat)>1 δ, when pp (n,δ), + where p (n,δ)=p Θ( 1 ), p (n,δ)=p +Θ( 1 ). So that the scaling − cr− n1−εlnn + cr nlnn window of Model RB is 1 1 W(n,δ)=(p Θ( ), p +Θ( )). cr− n1 εlnn cr nlnn − It is not difficult to see that p (n,δ) p (n,δ) 0, as n . + | − − |→ →∞ Remark 3.1 If n , then r (n,δ),r (n,δ) r , p (n,δ),p (n,δ) p . + cr + cr → ∞ − → − → For every sufficiently small δ, Theorem 3.1 and Theorem 3.2 hold. So we can obtain lim Pr(Sat)=1 when r <r or pr or p>p . cr cr n →∞ This is the result of Xu and Li[16]. 4. Proof of the results To prove the main results, we need the following lemmas. Lemma 4.1 Let c=α+1 r kp, then c<1. cr − Proof We know that r = α , then cr −ln(1 p) − αkp c = α+1+ ln(1 p) − α[kp+ln(1 p)] = 1+ − ln(1 p) − Assume that f(p)=kp+ln(1 p), hence we have f (p)= 1 +k. − ′ −1 p By the condition of Theorem 2.1, we have k 1 , hence f−(p) 0. That is ≥ 1 p ′ ≥ f(p) is a monotone increasing function. − So f(p) > f(0), that is kp+ln(1 p) > 0. It is obvious that ln(1 p) < 0 − − because of 0<p<1. And α> 1 is a constant. k Hence α[kp+ln(1−p)] <0. ln(1 p) − Therefore, it is proved that c=1+ α[kp+ln(1−p)] <1. ln(1 p) − Lemma 4.2 Let c=α+1 rkp , then c<1. cr − Proof We know that pcr =1 e−αr, so − α c = α+1 rk(1 e−r) − − α α = 1 r[ +k(1 e−r)] − −r − Let α = x, then x ( ,0). Suppose h(x) = x+k(1 ex), then h(x) = −r ∈ −∞ − ′ 1 kex. − ON THE SCALING WINDOW OF MODEL RB 5 By the condition of Theorem 2.2, kex = ke−αr 1, hence h′(x) 0. That is ≥ ≤ h(x) is a monotone decreasing function. So h(x) > h(0), that is h(x) > 0. And r > 0 is a constant, hence it is proved that c=1−r[−αr +k(1−e−αr)]<1. Proof of Theorem 3.1 Let N denote the number of satisfying assignments for a random CSP instance, we can obtain that E(N) = dn(1 p)rnlnn − (4.1) = nαn(1 p)rnlnn − Assume that E(N)<δ, by (1) we get (4.2) [α+rln(1 p)]nlnn<lnδ − lnδ (4.3) α+rln(1 p)< − nlnn α lnδ lnδ (4.4) r > + =r + cr −ln(1 p) nlnnln(1 p) nlnnln(1 p) − − − Using the Markov inequality Pr(Sat) E(N), we get Pr(Sat)<δ for ≤ 1 (4.5) r >r +Θ( ). cr nlnn Here note that f = Θ(g) represents there exist two finite constants c > 0 and 1 c >0 such that c <f/g<c . 2 1 2 In the following, we use Cauchy inequality Pr(Sat) E2(N) to prove when ≥ E(N2) r<r +Θ( 1 ), we have Pr(Sat)>1 δ. cr nlnn − In the remaining partof the paper,the expressionof E(N2) will play an important role in the proof of the main results. The derivation of this expression can be found in [16]. For the convenienceof the reader,we give anoutline of it as follows. Definition 4.1Let t ,t representsanorderedassignmentpairtothenvariables i j h i inU,whichsatisfiesaCSPinstanceifandonlyifbotht andt satisfytheCSPin- i j stance. And P( t ,t ) denotes the probabilityof t ,t satisfying a CSP instance. i j i j Definition 4.2hTheisimilarity number S of an ashsignmient pair t ,t is the num- i j h i ber of variables t and t take the identical values. It is obvious that 0 S n, i j ≤ ≤ and let s= S. Let A be the set of assignments whose similarity number is equal n S to S. We can get the expression of E(N2) is n A P( t ,t ) = A P( t ,t ) S i j S i j | | h i | | h i SX=0 n dk−1 S dk−2 S = dn (d 1)n S[ q k + q (1 k )]rnlnn (cid:18)S(cid:19) − − (cid:0) dk (cid:1) · (cid:0)n(cid:1) (cid:0) dk (cid:1) · − (cid:0)n(cid:1) q k q k (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) First we need to estimate E(N2). We can rewrite the above equation as the following one A P( t ,t )= E2(N)[1+ p (sk+ g(s))]rnlnn (4.6) | S| h i ji 1 p n · (1 1 )n ns−( 1 )ns n (1+O(1)) − nα − nα ns n (cid:0) (cid:1) 6 CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 where g(s)= k(k−1)(sk−sk−1). 2 When n is sufficiently large, except E2(N), the dominant contribution to (4.6) comes from p 1 f(s) = (1+ sk)rnlnn( )ns 1 p nα − (4.7) = e[rln(1+1−ppsk)−αs]nlnn Weputh(s)=rln(1+ p sk) αsandfocusonthefunctionh(s),differentiating 1 p − h(s) twice with respect to−s we get rkpsk 2[(k 1)(1 p) psk] − (4.8) h′′(s)= − − − (1 p+psk)2 − Applying the condition k 1 , we get (k 1)(1 p) psk 0 on the interval ≥ 1 p − − − ≥ [0,1], then h (s) 0. So h(s) is−a convex function. It is easy to see that h(0)=0 ′′ ≥ andh(1)= rln(1 p) α. Sowhenr <r Θ(1/(n1 εlnn)), wehaveh(1) 0. cr − − − − − ≤ Ontheinterval0<s<1,wegeth(s)<0. Sothereexist0<δ <1and0<δ <1 1 2 such that when r < r Θ(1/(n1 εlnn)), h(s) is mainly decided by the values cr − − s [0,δ ] [1 δ ,1]. Soweonlyneedtoconsiderthosetermss [0,δ ] [1 δ ,1] 1 2 1 2 ∈ ∪ − ∈ ∪ − to estimate (4.6). This is different from the proof in Xu and Li[16] for establishing theexistenceofphasetransitions,whereonlythosetermss [0,δ ]wereconsidered. 1 ∈ (i) s [0,δ ] 1 ∈ We can learn from Xu and Li[16] that 1 (4.9) A P( t ,t ) E2(N)(1+O( )) S i j | | h i ≤ n X s∈[0,δ1] (ii) s [1 δ ,1] 2 ∈ − It is easily known that if s [1 δ ,1], we can obtain sk sk 1 < 0, thus 2 − g(s)= k(k−1)(sk−sk−1) <0. So w∈e can−get the following inequalit−y 2 p A P( t ,t ) E2(N)(1+ sk)rnlnn S i j | | h i ≤ 1 p − 1 1 n 1 (1 )(n ns)( )ns (1+O( )) − · − nα nα (cid:18)ns(cid:19) n n 1 (4.10) = E(N)(1 p+psk)rnlnn(nα 1)n ns (1+O( )) − − − (cid:18)ns(cid:19) n When s=1(S =n), we obtain 1 (4.11) A P( t ,t )=E(N)(1+O( )); S i j | | h i n ON THE SCALING WINDOW OF MODEL RB 7 When s= n t(S =n t), where 1 t n. We can get that n− − ≤ ≪ n t n 1 A P( t ,t ) E(N) [1 p+p( − )k](nα 1)t (1+O( )) S i j | | h i ≤ · − n − (cid:18)t(cid:19)· n E(N)e−p[1−(nn−t)k]rnlnn(nα 1)t n (1+O(1)) ≤ − (cid:18)t(cid:19)· n n(α+1)t 1 E(N) (1+O( )) ≤ nrnpt(nk−O(n12)) n n(α+1)t 1 = E(N) (1+O( )) nrkpt−O(n1) n (4.12) E(N)(nα+1+O(n1))t(1+O(1)) ≤ nrkp n When n is sufficiently large, let c= α+1 r kp =α+1+ αkp . Thus it is − cr ln(1 p) divided into two cases to discuss the value of c. − Case 1: c<0. When s= n 1, by (4.12) we can obtain −n 1 (4.13) A P( t ,t ) E(N) nc (1+O( )) S i j | | h i ≤ · · n When s= n 2, by (4.12) we have −n 1 (4.14) A P( t ,t ) E(N) n2c (1+O( )) S i j | | h i ≤ · · n ··· ··· ··· So we can get 1 A P( t ,t ) E(N)(1+n2+n2c+ ) (1+O( )) S i j | | h i ≤ ··· · n X s∈[1−δ2,1] (4.15) = E(N)(1+O(nc)) It is shown from (i) and (ii) that n E(N2) = A P( t ,t ) S i j | | h i SX=0 = A P( t ,t )+ A P( t ,t ) S i j S i j | | h i | | h i X X s∈[0,δ1] s∈[1−δ2,1] 1 (4.16) E2(N)(1+O( ))+E(N)(1+O(nc)) ≤ n Consequently, by the Cauchy inequality, we have E2(N) E2(N) Pr(Sat) ≥ E(N2) ≥ E2(N)(1+O(1))+E(N)(1+O(nc)) n (4.17) > 1 δ − 1 δ+O(nc) (4.18) E(N)> − δ O(1) − n Putting 1−δδ+OO((1n)c) =ϑ, hence we have − n (4.19) αnlnn+rnlnnln(1 p)>lnϑ − 8 CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 lnϑ αnlnn α lnϑ (4.20) r < − = + nlnnln(1 p) −ln(1 p) nlnnln(1 p) − − − So we obtain that lnϑ (4.21) r <r + cr nlnnln(1 p) − Thus when r <r +Θ( 1 ) , we have the result Pr(Sat)>1 δ. Case 2: c 0. cr nlnn − ≥ When 1 t n, by the right side of (4.10), we can get ≤ ≪ t n [1 p+p(1 )]rnlnn(nα 1)t − − n − (cid:18)t(cid:19) 1 n 1 = n−rkpt+O(n1)nαt(1 )t√2πn( )t(1+O( )) − nα t n √2πn n(α+1+O(n1)−rkp)t 1 (4.22) · (1+O( )) ≤ tt n Now when n is sufficiently large, let u = n(α+1−rkp)t = nct. Then u = t tt tt t ectlnn tlnt. Ifweputω =ctlnn tlnt,wecangetω =clnn lnt 1,thenω =0 when−t= nc. Anditistknowntha−t 0 c<1 by Lemt′ma4.1. S−o A−P( t ,t t)′ has the maximeal value √2πn enec at the≤point of t= nc. So we can|haSv|e h i ji · e (4.23) AS P( ti,tj ) E(N)√2πn enecn(1+O(1)) | | h i ≤ · n X s∈[1−δ2,1] We use the Cauchy inequality E2(N) E2(N) Pr(Sat) ≥ E(N2) ≥ (E2(N)+E(N)√2πn enecn)(1+O(1)) · n (4.24) > 1 δ − (4.25) E(N)> 1−δ+O(n1)√2πn enecn δ O(1) · − n √2π(1 δ+O(1)) Let − n =λ, then we get δ O(1) − n nc 3 (4.26) αnlnn+rnlnnln(1 p)>lnλ+ + lnn − e 2 αnlnn+ nc + 3lnn+lnλ r < − e 2 nlnnln(1 p) − 1 3 lnλ (4.27) = r + + + − cr en1 clnnln(1 p) 2nln(1 p) nlnnln(1 p) − − − − So when r <r +O( 1 ), we have Pr(Sat)>1 δ. cr n1−clnn − Combining the above cases,it is provedthat the scaling window of model RB is 1 1 W(n,δ)=(r Θ( ), r +Θ( )), cr− n1 εlnn cr nlnn − whereε= c+2|c|,c<1anditisobviousthat|rcr+Θ(nl1nn)−(rcr−Θ(n1−ε1lnn))|→0 (n ). Thus, we finish the proof of Theorem 3.1. →∞ ON THE SCALING WINDOW OF MODEL RB 9 Remark 4.1 By Lemma 4.1, we claim that c increases with p and decreases with α. Therefore, when 0 c < 1, the convergence rate of r (n,δ) approaching r cr ≤ − decreases with p and increases with α. Proof of Theorem 3.2 Similarly, we can also use (4.3) to obtain that α lnδ (4.28) ln(1 p)< + − −r rnlnn p > 1 e−αr+rnlnlnδn − = 1 e−αr +e−αr(1 ernlnlnδn) − − lnδ α = pcr+e−r[1 (1+O( ))] − rnlnn 1 (4.29) = p +Θ( ) cr nlnn So when p>p +Θ( 1 ), we have Pr(Sat)<δ. cr nlnn Similar to the proofof Theorem 3.1,when n is sufficiently large,let c=α+1 − rkp . So by Lemma 4.2 we can also divide c into two cases, that is to say c < 0 cr and 0 c<1. Therefore, we have the followings. ≤ By (4.19), we can get lnϑ αnlnn α lnϑ (4.30) ln(1 p)> − = + − rnlnn −r rnlnn p < 1 e−αr+rnlnlnϑn − = 1 e−αr +e−αr(1 ernlnlnϑn) − − lnϑ α = pcr+e−r[1 (1+O( ))] − rnlnn 1 (4.31) = p Θ( ) cr − rnlnn By (4.26), we have nc 3 (4.32) αnlnn+rnlnnln(1 p)>lnλ+ + lnn − e 2 p < 1 e−αr+nec+r32nllnnnn+lnλ − = 1 e−αr +e−αr(1 enec+r32nllnnnn+lnλ) − − 1 α = pcr+e−r[1−(1+O(n1 clnn))] − 1 (4.33) = p Θ( ) cr− n1 clnn − Thus the results are as follows: 1 Pr(Sat)>1 δ, when pp +Θ( ), cr nlnn 10 CHUNYANZHAO1,KEXU2,ANDZHIMINGZHENG3 where ε= c+|c|, c<1 and 0 ε<1. 2 ≤ Therefore the scaling window of model RB with respect to parameter p is 1 1 W(n,δ)=(p Θ( ), p +Θ( )) cr− n1 εlnn cr nlnn − Remark4.2SimilartoRemark4.1,byLemma4.2,weobtainthattheconvergence rate of p (n,δ) approaching p increases with both r and α. cr − Note that especially, when n , we have →∞ Pr(Sat) 0,when r>r or p>p , cr cr → Pr(Sat) 1,when r<r or p<p . cr cr → This is the result of Xu and Li[16]. 5. 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