On the Scaling Window of Model RB

ON THE SCALING WINDO W OF MODEL RB CHUNY AN ZHAO 1 , KE XU 2 , AND ZHIMING ZHENG 3 Abstract. This pap er analyzes the scaling windo w of a random CSP mo del (i.e. model RB) for which we can iden tify the threshold points exactly , denoted b y r cr or p cr . F or this mo del, we establish the scaling wi ndow W ( n, δ ) = ( r − ( n, δ ) , r + ( n, δ )) s uch that the probability of a random instance b eing satisfiable is greater than 1 − δ for r < r − ( n, δ ) and is less than δ for r > r + ( n, δ ). Specifically , we obtain the following r esult W ( n, δ ) = ( r cr − Θ( 1 n 1 − ε ln n ) , r cr + Θ( 1 n ln n )) , where 0 ≤ ε < 1 is a constan t. A simil ar result with resp ect to the other parameter p is also obtained. Since the instances generated by mo del RB ha ve b een sho wn to b e hard at the threshold, this i s the first atte mpt, as far as w e kno w, to analyz e the scaling windo w of such a model with hard i nstances. 1. Introduction The Constraint Satis fa ction Problem(CSP), orig ina ted from a rtificial in telli- gence, has bec ome an imp orta nt and active field of s tatistical physics, informa - tion theo r y and co mputer sc ie nce. The CSP are a is very int erdisciplinar y , since it embeds ideas from many resea rch fields, like ar tificial int elligence, databas es, progra mming la nguages and op era tio n resear ch. A constraint satisfac tio n problem consists of a finite s et U = { u 1 , u 2 , · · · , u n } o f n v ariables, each u i asso ciated w ith a domain o f v alues D i , and a s et of c o nstraints. Each of the constr aints C i 1 i 2 ··· i k is a relation, defined on some subset { u i 1 , u i 2 , · · · , u i k } of n v ariables, called its scop e, denoting their lega l tuples of v alues. A solution to a CSP is an assignment o f a v alue to each v aria ble from its domain suc h that all the constraints of this CSP a re satisfied. A constraint is said to be satisfie d if the tuple of v alues assigned to the v ariables in this constraint is a lega l one. A CSP is called satisfiable if and only if it has at leas t one so lution. The task of a CSP is to find a solution o r to pr ov e that no solution ex ists. Given a CSP , we are interested in p olynomial-time alg orithms, that is, algo - rithms who se running time is b ounded by a polynomia l in the num ber of v aria ble s . Co ok’s Theo rem[2] asserts that sa tisfiability is NP-complete and a t least a s hard as any pro blem whose solutions can b e verified in p olynomial time. Mo st of the int eresting CSPs a re NP-complete pr oblems. W e k now that k -SA T problem is a canonical v ersion of the CSPs, in which v ar iables can b e a ssigned the v a lue T rue or F alse(called Bo ole an v ariables). A lot of efforts hav e b een devoted to k -SA T and it is widely believed tha t no efficie nt algor ithm exists for k -SA T. How ever, Key wor ds and phr ases. Constraint Satisfaction Problem; Model RB; Satisfiability; Phase T ransition; Scaling Windo w. Supported in part by National 973 Program of China(Grant No. 200532CB190 2) and NSFC( Grant Nos. 604731 09 and 60403 003). 1 2 CHUNY AN ZHA O 1 , KE XU 2 , AND ZHIMING ZHENG 3 it is shown that most instances of k -SA T can be so lved efficiently , so pe r haps genuine har dnes s is o nly present in a tiny fra ction of a ll instances . In 1990s, a remark a ble progress[3, 13, 11, 1 2] was made that the the really difficult ins tances is related to phase transition phenomenon, as suggested in the pioneering w ork of F u and Anderson[6]. The study of phase trans itions has attracted muc h in terest subsequently[9, 1 2]. In rece n t years, r andom k -SA T has b een well studied b o th from theoretica l and algorithmic p oint of views. If k = 2 then it is k nown that there is a satisfiabil- it y threshold a t α c = 1 (here α r epresents the ra tio of clauses m to v ariables n ), below whic h the pr o bability of a ra ndom instance b eing satisfiable tends to 1 a nd ab ov e whic h it tends to 0 as n appro a ches infinity[4]. This was sharp ened in[8 , 1 4]. Random 2-SA T is now pretty m uch understo o d. How ever, for k ≥ 3, the existence of the phase transition phenomenon has not b een established, not even the exact v alue of the threshold p oint[1, 10]. T o gain a better unders ta nding of how the phase tra ns ition scales with problem size, the finite-size scaling metho d has been in tro duced from statistical mechanics[11, 7]. W e use finite-siz e scaling, a metho d from statistical physics in which observing how the width of a transitio n nar r ows with increasing s ample size g ives direct ev- idence for critical behavior at a phase transition. Finite-size sca ling is the study of c hanges in the transition b ehavior due to finite-size effects, in par ticular, broad- ening o f the transition region for finite n . More precisely , for 0 < δ < 1, let r − ( n, δ ) be the supremum ov er r such that the probability of a rando m CSP in- stance b eing satisfiable is at lea st 1 − δ , and similar ly , let r + ( n, δ ) b e the infim um ov er r such that the probability of a rando m CSP instance b eing satisfiable is at most δ . Then, for r within the s caling w indow W ( n, δ ) = ( r − ( n, δ ) , r + ( n, δ )) the probability is betw een δ a nd 1 − δ . And for a ll δ , | r + ( n, δ ) − r − ( n, δ ) | → 0 as n → ∞ . F o r random 2-SA T, it has b een determined that the scaling window is W ( n, δ ) = (1 − Θ( n − 1 / 3 ) , 1 + Θ( n − 1 / 3 ))[2]. Mo del RB is a r andom C SP mo del pr op osed b y Xu a nd Li to ov ercome the trivial insolubility o f standard CSP models[1 6]. F or this mo del, we can not only establish the existence of phase transitions, but also pinp oint the threshold p oints exactly , denoted by r cr or p cr . Mor eov er, it has b een proved that almost all in- stances of model RB hav e no tree-like res olution pro o fs of less than exp onential size [16]. This implies that unlik e r andom 2- SA T, mo del RB can b e used to gener ate hard instances, which has also been confirmed b y exp eriments[7]. Motiv ated by the work on the scaling window of r andom 2 -SA T, in this pap er , we study the scaling window of mo del RB and obta in that W ( n, δ ) = ( r cr − Θ( 1 n 1 − ε ln n ) , r cr + Θ( 1 n ln n )). And we also obtain simila r results about the other control pa rameter p . The main con tribution of this pap er is not to present new metho ds for computing the scaling window, but to sho w that for an interesting mo del with hard insta nces (i.e. mo del RB), not o nly can the threshold p o in ts b e lo cated exa ctly , but als o the scaling windo w ca n be deteremined us ing standard metho ds. This means that hop efully , more mathematical pr op erties abo ut the threshold b ehavior of mo del RB can b e obtained in a relatively ea sy wa y , which will help to shed light on the phase tra ns ition phenomenon in NP-complete problems. The rest of the pape r is organize d as follows. In the next section, we will give a brief in tro duction ab out mo del RB. The main results of this pape r and their pro ofs will be giv en in Sectio n 3 and Section 4 resp ectively . Finally , we will conclude in Section 5. ON THE SCALING WINDOW OF MODEL RB 3 2. Model RB W e can pinpoint the threshold location for mo del RB propo sed by Xu and Li[16]. The wa y of genera ting random instances for model RB is: (1). Given a set U of n variables, s ele ct with r ep etition m = rn ln n r andom c on- str aints. Each r andom c onstr aint is forme d by sele cting without r ep et ition k of n variables, wher e k ≥ 2 is an inte ger. (2). Next, for e ach c onstr aint we sele ct uniformly at r andom without r ep etition q = p · d k il le gal tuples of values, i.e. , e ach c onstr aint c ont ains exactly (1 − p ) · d k le gal ones, wher e d = n α is the domain size of e ach variable and α > 0 is a c onst ant. In this pap er, the pr obability of a random C SP instance being satisfiable is denoted by P r(Sat). It is proved tha t fo r mo del RB the phase transition phenom- enon o ccurs at r cr = − α ln(1 − p ) or p cr = 1 − e − α r as n approa ches infinit y[16]. Mo re precisely , w e hav e the follo wing tw o theor ems. The or em 2.1 [16] L et r cr = − α ln(1 − p ) . If α > 1 k , 0 < p < 1 ar e two c onstants and k , p satisfy the ine quality k ≥ 1 1 − p , then lim n →∞ P r ( S at ) = 1 w hen r < r cr , lim n →∞ P r ( S at ) = 0 w hen r > r cr . The or em 2.2 [16] L et p cr = 1 − e − α r . If α > 1 k , r > 0 ar e two c onstants and k , α satisfy the ine quality k e − α r ≥ 1 , then lim n →∞ P r ( S at ) = 1 w hen p < p cr , lim n →∞ P r ( S at ) = 0 w hen p > p cr . 3. Main resul ts Our main results are the following tw o theorems. Theorem 3.1 F o r all sufficiently s mall δ > 0, there exist r − ( n, δ ) a nd r + ( n, δ ) such that the following holds: P r ( S at ) > 1 − δ, when r < r − ( n, δ ); P r ( S at ) < δ , whe n r > r + ( n, δ ) , where r − ( n, δ ) = r cr − Θ( 1 n 1 − ε ln n ), r + ( n, δ ) = r cr + Θ( 1 n ln n ). So that the scaling window o f mo del RB is W ( n, δ ) = ( r cr − Θ( 1 n 1 − ε ln n ) , r cr + Θ( 1 n ln n )) . It is ea sy to see that | r + ( n, δ ) − r − ( n, δ ) | → 0, as n → ∞ . Theorem 3.2 F or all s ufficiently small δ > 0, there exist p − ( n, δ ) a nd p + ( n, δ ) 4 CHUNY AN ZHA O 1 , KE XU 2 , AND ZHIMING ZHENG 3 such that the following holds: P r ( S at ) > 1 − δ , whe n p < p − ( n, δ ); P r ( S at ) < δ , whe n p > p + ( n, δ ) , where p − ( n, δ ) = p cr − Θ( 1 n 1 − ε ln n ), p + ( n, δ ) = p cr + Θ( 1 n ln n ). So tha t the scaling window o f Mo del RB is W ( n, δ ) = ( p cr − Θ( 1 n 1 − ε ln n ) , p cr + Θ( 1 n ln n )) . It is not difficult to s e e that | p + ( n, δ ) − p − ( n, δ ) | → 0, as n → ∞ . Remark 3. 1 If n → ∞ , then r + ( n, δ ) , r − ( n, δ ) → r cr , p + ( n, δ ) , p − ( n, δ ) → p cr . F o r every sufficiently small δ , Theorem 3.1 and Theor e m 3.2 ho ld. So we can obtain lim n →∞ P r ( S at ) = 1 w hen r < r cr or p < p cr , lim n →∞ P r ( S at ) = 0 w hen r > r cr or p > p cr . This is the result of Xu and Li[16]. 4. Proof of the resul ts T o pro ve the main results, we need the follo wing lemmas. Lemma 4.1 Let c = α + 1 − r cr k p , then c < 1 . Pro of W e know that r cr = − α ln(1 − p ) , then c = α + 1 + αk p ln(1 − p ) = 1 + α [ k p + ln(1 − p )] ln(1 − p ) Assume that f ( p ) = k p + ln(1 − p ), hence w e ha ve f ′ ( p ) = − 1 1 − p + k . By the co nditio n of Theorem 2.1, we hav e k ≥ 1 1 − p , hence f ′ ( p ) ≥ 0. That is f ( p ) is a mono to ne increasing function. So f ( p ) > f (0), that is k p + ln(1 − p ) > 0. It is obvious that ln(1 − p ) < 0 bec ause of 0 < p < 1. And α > 1 k is a constan t. Hence α [ kp +ln(1 − p )] ln(1 − p ) < 0. Therefore, it is prov ed that c = 1 + α [ kp +ln(1 − p )] ln(1 − p ) < 1. Lemma 4.2 Let c = α + 1 − rk p cr , then c < 1. Pro of W e know that p cr = 1 − e − α r , so c = α + 1 − rk (1 − e − α r ) = 1 − r [ − α r + k (1 − e − α r )] Let − α r = x , then x ∈ ( −∞ , 0). Supp ose h ( x ) = x + k (1 − e x ), then h ′ ( x ) = 1 − k e x . ON THE SCALING WINDOW OF MODEL RB 5 By the condition of Theorem 2.2, k e x = k e − α r ≥ 1 , hence h ′ ( x ) ≤ 0. Tha t is h ( x ) is a monotone decreas ing function. So h ( x ) > h (0 ), that is h ( x ) > 0. And r > 0 is a constant, he nc e it is proved that c = 1 − r [ − α r + k (1 − e − α r )] < 1. Pro of of Theorem 3 .1 Let N denote the n um b er of satisfying assignments for a random CSP instance, w e can o bta in that E ( N ) = d n (1 − p ) r n ln n = n αn (1 − p ) r n ln n (4.1) Assume that E ( N ) < δ , by (1) we get (4.2) [ α + r ln(1 − p )] n ln n < ln δ (4.3) α + r ln (1 − p ) < ln δ n ln n (4.4) r > − α ln(1 − p ) + ln δ n ln n ln(1 − p ) = r cr + ln δ n ln n ln(1 − p ) Using the Ma rko v ineq uality P r(Sat) ≤ E ( N ), we get Pr(Sat) < δ for (4.5) r > r cr + Θ( 1 n ln n ) . Here note that f = Θ( g ) represents there exis t t wo finite consta nt s c 1 > 0 and c 2 > 0 such that c 1 < f /g < c 2 . In the following, we use Cauch y inequality Pr(Sa t) ≥ E 2 ( N ) E ( N 2 ) to prov e when r < r cr + Θ( 1 n ln n ), we hav e Pr(Sat) > 1 − δ . In the remaining part of the pap er, the expressio n of E ( N 2 ) will play a n impor - tant r o le in the pr o of of the main results. The der iv ation of this expression can b e found in [1 6]. F or the convenience of the r eader, w e give an outline of it a s follows. Definition 4.1 Let h t i , t j i represe nts an ordered assignment pair to the n v ariables in U , which satis fies a C SP instance if and only if b oth t i and t j satisfy the CSP in- stance. And P ( h t i , t j i ) denotes the probability of h t i , t j i satisfying a CSP instance. Definition 4.2 The simila rity n umber S of an as signment pair h t i , t j i is the num- ber of v ar iables t i and t j take the identical v alues. It is obvious that 0 ≤ S ≤ n , and let s = S n . Let A S be the set of a ssignments whose simila rity n umber is equal to S . W e can get the expression of E ( N 2 ) is | A S | P ( h t i , t j i ) = n X S =0 | A S | P ( h t i , t j i ) = d n  n S  ( d − 1 ) n − S [  d k − 1 q   d k q  ·  S k   n k  +  d k − 2 q   d k q  · (1 −  S k   n k  )] r n ln n First we need to estimate E ( N 2 ). W e can rewrite the ab ov e equatio n a s the following o ne (4.6) | A S | P ( h t i , t j i ) = E 2 ( N )[1 + p 1 − p ( s k + g ( s ) n )] r n ln n · (1 − 1 n α ) n − ns ( 1 n α ) ns  n ns  (1 + O ( 1 n )) 6 CHUNY AN ZHA O 1 , KE XU 2 , AND ZHIMING ZHENG 3 where g ( s ) = k ( k − 1)( s k − s k − 1 ) 2 . When n is sufficient ly large , exc e pt E 2 ( N ), the dominant contribution to (4.6) comes from f ( s ) = (1 + p 1 − p s k ) r n ln n ( 1 n α ) ns = e [ r ln(1+ p 1 − p s k ) − αs ] n ln n (4.7) W e put h ( s ) = r ln (1 + p 1 − p s k ) − αs and fo cus on the function h ( s ), differentiating h ( s ) twice with resp ect to s we get (4.8) h ′′ ( s ) = rk ps k − 2 [( k − 1)(1 − p ) − ps k ] (1 − p + ps k ) 2 Applying the condition k ≥ 1 1 − p , we get ( k − 1)(1 − p ) − ps k ≥ 0 on the in terv al [0 , 1], then h ′′ ( s ) ≥ 0. So h ( s ) is a conv ex function. It is eas y to see that h (0) = 0 and h (1) = − r ln(1 − p ) − α . So when r < r cr − Θ(1 / ( n 1 − ε ln n )), we have h (1) ≤ 0. On the interv al 0 < s < 1, we get h ( s ) < 0. So there exist 0 < δ 1 < 1 and 0 < δ 2 < 1 such that when r < r cr − Θ(1 / ( n 1 − ε ln n )), h ( s ) is mainly decided b y the v a lues s ∈ [0 , δ 1 ] ∪ [1 − δ 2 , 1]. So we o nly need to consider those terms s ∈ [0 , δ 1 ] ∪ [1 − δ 2 , 1] to estimate (4.6). This is diff erent fro m the pro of in Xu and Li[16] for es tablishing the existence of phase transitions, where only t hose terms s ∈ [0 , δ 1 ] w ere considered. (i) s ∈ [0 , δ 1 ] W e can learn fro m Xu and Li[16] that (4.9) X s ∈ [0 ,δ 1 ] | A S | P ( h t i , t j i ) ≤ E 2 ( N )(1 + O ( 1 n )) (ii) s ∈ [1 − δ 2 , 1] It is easily k nown that if s ∈ [1 − δ 2 , 1], w e can o bta in s k − s k − 1 < 0, thus g ( s ) = k ( k − 1)( s k − s k − 1 ) 2 < 0. So we can g et the following inequality | A S | P ( h t i , t j i ) ≤ E 2 ( N )(1 + p 1 − p s k ) r n ln n · (1 − 1 n α ) ( n − ns ) ( 1 n α ) ns  n ns  (1 + O ( 1 n )) = E ( N )(1 − p + ps k ) r n ln n ( n α − 1) n − ns  n ns  (1 + O ( 1 n )) (4.10) When s = 1( S = n ), w e obtain (4.11) | A S | P ( h t i , t j i ) = E ( N )(1 + O ( 1 n )); ON THE SCALING WINDOW OF MODEL RB 7 When s = n − t n ( S = n − t ), where 1 ≤ t ≪ n . W e can get tha t | A S | P ( h t i , t j i ) ≤ E ( N ) · [1 − p + p ( n − t n ) k ]( n α − 1) t  n t  · (1 + O ( 1 n )) ≤ E ( N ) e − p [1 − ( n − t n ) k ] rn ln n ( n α − 1) t  n t  · (1 + O ( 1 n )) ≤ E ( N ) n ( α +1) t n r npt ( k n − O ( 1 n 2 )) (1 + O ( 1 n )) = E ( N ) n ( α +1) t n r kpt − O ( 1 n ) (1 + O ( 1 n )) ≤ E ( N )( n α +1+ O ( 1 n ) n r kp ) t (1 + O ( 1 n )) (4.12) When n is sufficien tly larg e , let c = α + 1 − r cr k p = α + 1 + αkp ln(1 − p ) . Thus it is divided into t wo cases to discuss the v alue of c . Case 1: c < 0 . When s = n − 1 n , by (4.12) we can obtain (4.13) | A S | P ( h t i , t j i ) ≤ E ( N ) · n c · (1 + O ( 1 n )) When s = n − 2 n , by (4.12) we hav e (4.14) | A S | P ( h t i , t j i ) ≤ E ( N ) · n 2 c · (1 + O ( 1 n )) · · · · · · · · · So we can get X s ∈ [1 − δ 2 , 1] | A S | P ( h t i , t j i ) ≤ E ( N )(1 + n 2 + n 2 c + · · · ) · (1 + O ( 1 n )) = E ( N )(1 + O ( n c )) (4.15) It is sho wn from (i) and (ii) that E ( N 2 ) = n X S =0 | A S | P ( h t i , t j i ) = X s ∈ [0 ,δ 1 ] | A S | P ( h t i , t j i ) + X s ∈ [1 − δ 2 , 1] | A S | P ( h t i , t j i ) ≤ E 2 ( N )(1 + O ( 1 n )) + E ( N )(1 + O ( n c )) (4.16) Consequently , by the Cauch y inequality , we hav e P r ( S at ) ≥ E 2 ( N ) E ( N 2 ) ≥ E 2 ( N ) E 2 ( N )(1 + O ( 1 n )) + E ( N )(1 + O ( n c )) > 1 − δ (4.17) (4.18) E ( N ) > 1 − δ + O ( n c ) δ − O ( 1 n ) Putting 1 − δ + O ( n c ) δ − O ( 1 n ) = ϑ , hence we have (4.19) αn ln n + r n ln n ln(1 − p ) > ln ϑ 8 CHUNY AN ZHA O 1 , KE XU 2 , AND ZHIMING ZHENG 3 (4.20) r < ln ϑ − αn ln n n ln n ln(1 − p ) = − α ln(1 − p ) + ln ϑ n ln n ln(1 − p ) So we obtain that (4.21) r < r cr + ln ϑ n ln n ln(1 − p ) Thu s when r < r cr + Θ( 1 n ln n ) , w e ha ve the r esult P r ( S at ) > 1 − δ . Case 2: c ≥ 0 . When 1 ≤ t ≪ n , by the rig ht side of (4.10), we can get [1 − p + p (1 − t n )] r n ln n ( n α − 1) t  n t  = n − r kpt + O ( 1 n ) n αt (1 − 1 n α ) t √ 2 π n ( n t ) t (1 + O ( 1 n )) ≤ √ 2 π n · n ( α +1+ O ( 1 n ) − r kp ) t t t (1 + O ( 1 n )) (4.22) Now when n is sufficien tly large, let u t = n ( α +1 − rkp ) t t t = n ct t t . Then u t = e ct ln n − t ln t . If we put ω t = ct ln n − t ln t , w e can ge t ω ′ t = c ln n − ln t − 1, then ω ′ t = 0 when t = n c e . And it is kno wn that 0 ≤ c < 1 by Lemma 4.1 . So | A S | P ( h t i , t j i ) has the maximal v alue √ 2 π n · e n c e at the p oint of t = n c e . So w e can hav e (4.23) X s ∈ [1 − δ 2 , 1] | A S | P ( h t i , t j i ) ≤ E ( N ) √ 2 π n · e n c e n (1 + O ( 1 n )) W e use the Ca uch y inequality P r ( S at ) ≥ E 2 ( N ) E ( N 2 ) ≥ E 2 ( N ) ( E 2 ( N ) + E ( N ) √ 2 π n · e n c e n )(1 + O ( 1 n )) > 1 − δ (4.24) (4.25) E ( N ) > 1 − δ + O ( 1 n ) δ − O ( 1 n ) √ 2 π n · e n c e n Let √ 2 π ( 1 − δ + O ( 1 n )) δ − O ( 1 n ) = λ , then we get (4.26) αn ln n + r n ln n ln(1 − p ) > ln λ + n c e + 3 2 ln n r < − αn ln n + n c e + 3 2 ln n + ln λ n ln n ln(1 − p ) = − r cr + 1 en 1 − c ln n ln(1 − p ) + 3 2 n ln(1 − p ) + ln λ n ln n ln(1 − p ) (4.27) So when r < r cr + O ( 1 n 1 − c ln n ), we hav e P r ( S at ) > 1 − δ . Combining the ab ov e ca ses, it is pro ved that the scaling window of mo del RB is W ( n, δ ) = ( r cr − Θ( 1 n 1 − ε ln n ) , r cr + Θ( 1 n ln n )) , where ε = c + | c | 2 , c < 1 and it is obvious that | r cr + Θ( 1 n ln n ) − ( r cr − Θ( 1 n 1 − ε ln n )) | → 0 ( n → ∞ ). Thus, w e finish the pro of of Theor em 3.1. ON THE SCALING WINDOW OF MODEL RB 9 Remark 4.1 B y Lemma 4.1, w e claim that c increases with p and decreases with α . Therefore, when 0 ≤ c < 1 , the conv er gence ra te of r − ( n, δ ) approaching r cr decreases with p and increas es with α . Pro of of Theorem 3.2 Similarly , we can a lso use (4 .3) to o btain that (4.28) ln(1 − p ) < − α r + ln δ rn ln n p > 1 − e − α r + ln δ rn ln n = 1 − e − α r + e − α r (1 − e ln δ rn ln n ) = p cr + e − α r [1 − (1 + O ( ln δ rn ln n ))] = p cr + Θ( 1 n ln n ) (4.29) So when p > p cr + Θ( 1 n ln n ), we hav e P r ( S at ) < δ . Similar to the pro of of Theo rem 3.1, when n is sufficie ntly large, let c = α + 1 − rk p cr . So by Lemma 4.2 we can also divide c into t wo cases, that is to say c < 0 and 0 ≤ c < 1. Therefore, w e have the follo wings. By (4.1 9), w e c an get (4.30) ln(1 − p ) > ln ϑ − αn ln n rn ln n = − α r + ln ϑ rn ln n p < 1 − e − α r + ln ϑ rn ln n = 1 − e − α r + e − α r (1 − e ln ϑ rn ln n ) = p cr + e − α r [1 − (1 + O ( ln ϑ rn ln n ))] = p cr − Θ( 1 rn ln n ) (4.31) By (4.2 6), w e have (4.32) αn ln n + r n ln n ln(1 − p ) > ln λ + n c e + 3 2 ln n p < 1 − e − α r + n c e + 3 2 ln n +ln λ rn ln n = 1 − e − α r + e − α r (1 − e n c e + 3 2 ln n +ln λ rn ln n ) = p cr + e − α r [1 − (1 + O ( 1 n 1 − c ln n ))] = p cr − Θ( 1 n 1 − c ln n ) (4.33) Thu s the results are as follo ws: P r ( S at ) > 1 − δ , whe n p < p cr − Θ( 1 n 1 − ε ln n ); P r ( S at ) < δ , whe n p > p cr + Θ( 1 n ln n ) , 10 CHUNY AN ZHA O 1 , KE XU 2 , AND ZHIMING ZHENG 3 where ε = c + | c | 2 , c < 1 and 0 ≤ ε < 1. Therefore the scaling window o f mo del RB with resp ect to parameter p is W ( n, δ ) = ( p cr − Θ( 1 n 1 − ε ln n ) , p cr + Θ( 1 n ln n )) Remark 4.2 Simila r to Remark 4.1, b y Lemma 4.2, we obtain tha t the co nvergence rate of p − ( n, δ ) approaching p cr increases with both r a nd α . Note that especia lly , when n → ∞ , we hav e P r ( S at ) → 0 , whe n r > r cr or p > p cr , P r ( S at ) → 1 , whe n r < r cr or p < p cr . This is the result of Xu and Li[16]. 5. Conclusions In this paper , we obta in the scaling window o f model RB for whic h the phase transition p oint is known exac tly . As men tioned before, the scaling windo w o f ran- dom 2-SA T has also been de ter mined. How ever, this mo del is ea s y to solve because 2-SA T is in P class. Recently , both theoretica l[17] and exp erimental results[15] suggest that mo del RB is abundant with har d instances whic h are useful both for ev aluating the perfo r mance of algorithms and for understanding the nature of har d problems. As far as we know, this pap er is the first study on the scaling window of such a mo del with hard insta nc e s. W e hop e that it can help us to gain a b etter understanding of the phase transition pheno menon in NP - complete problems. References 1. D. Ac hlioptas and G. Sorkin, Optimal m y opic algorithms for random 3-SA T, Proceedings of the 41st Annua l IEEE Symp osium on F oundations of Comput ing (2000) 590-600. 2. B. Boll ob´ as, Chri stian Borgs, Jennifer Chay es, J.H. Kim, and D.B. Wilson, The scaling wi ndo w of the 2-SA T transition, Random Structures and Algorithms , 201-256(2001). 3. P . Cheeseman, B. Kanefsky and W. T a ylor, Where the reall y hard problems are, Pro c. 12th In t. 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Lecout re, Random Constraint Satisfaction: Easy Generation of Hard (Sat isfiable) Instances, Artificial In telligence , 171(2007):514-534, Earlier v ersion app eared in Pro c. of 19th IJCAI, pp.337-342, Scotland , 2005. 16. K. Xu and W. Li, Exact phase transition in random const raint satisfaction problems, Journal of Artificial In telligence Research , 12:93-103(2000 ). 17. K. Xu and W. Li, Many hard examples in exact phase transitions, Theoretical Computer Science , 355:291-302(200 6). 1 School of Science, Beijing University of Aeronautics and Astronautic s, Beijing 100083, China, Key Labora tor y of Mat hema tics, Informa tics and Beha vioral S emantics, Ministr y of Educa tion E-mail add r ess : xiaoyanzi @ss.buaa.e du.cn 2 School of Computers, Beijing University of Aeronautics and Astr onautics, Beijing 100083, China, Na tional Labora tor y of S oftw are Developmen t Environment E-mail add r ess : kexu@nlsd e.buaa.edu .cn 3 School of Science, Beijing University of Aeronautics and Astronautic s, Beijing 100083, China, Key Labora tor y of Mat hema tics, Informa tics and Beha vioral S emantics, Ministr y of Educa tion E-mail add r ess : zzheng@pk u.edu.cn