locally maximal solutions for random CSPs 109 Why almost all satisfiable k-cnf formulas are easy

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1 locall maximal solutions for random CSPs 109 Wh almost all satisfiable k-cnf formulas are eas

2 2007 Conference on Analsis of Algorithms, AofA 07 DMTCS proc. AH, 2007, Expected number of locall maximal solutions for random Boolean CSPs Nadia Creignou 1 and Hervé Daudé 2 and Olivier Dubois 3 1 LIF, UMR CNRS 6166 & Université de la Méditerranée, 163 avenue de Lumin, Marseille, France 2 LATP, UMR CNRS 6632 & Université de Provence, 39 rue Joliot-Curie, Marseille, France 3 LIP6, UMR CNRS 7606 & Université de Paris 6, 4 Place Jussieu, Paris, France received 14 Ma 2007, revised 19 th Januar 2008, accepted tomorrow. For a large number of random Boolean constraint satisfaction problems, such as random k-sat, we stud how the number of locall maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivit of the allowed constraint functions onl. As a b-product we provide a general tool for computing an upper bound of the satisfiabilit threshold for an problem of a large class of random Boolean CSPs. Kewords: Random structures, Constraint satisfaction problems, Boolean functions, Sensitivit, Satisfiabilit, Phase transition, Threshold. Contents 1 Introduction Random smmetric Boolean CSPs Locall maximal solutions and Sensitivit Expected number of (locall maximal) solutions Solutions versus locall maximal solutions Threshold upper bounds Proof of Theorem Introduction Constraint satisfaction is recognized as a fundamental problem in computer science, since combinatorial problems from man different application areas (artificial intelligence, databases, automated design, etc.) c 2007 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nanc, France

3 locall maximal solutions for random CSPs 111 can be expressed in a natural wa b means of constraints. Informall, an instance of a constraint satisfaction problem (CSP for short) consists of a set of variables, a set of possible values for the variables, and a set of constraints that restrict the combinations of values that certain tuples of variables ma take; the question is whether there is an assignment of values to variables that satisfies the given constraints. In the earl seventies, man variants of CSPs have been proved to be NP-complete on a problem b problem basis. In contrast Schaefer proposed to stud the standard constraint satisfaction problem parameterized b restricting the set of functions F, thus defining the constraint satisfaction problem CSP(F), so-called Generalized satisfiabilit [Sch78]. He studied the complexit of the associated decision problem and proved that ever Boolean CSP(F) is either NP-complete or solvable in polnomial time, thus revealing the seed of NP-completeness in SAT-like problems. Since then, there has been a growing bod of classification results for related problems (see [CKS01] for a surve). Similarl random instances of Boolean CSPs have earl attracted a lot of attention (see [MSL92], [Ba05]). Average case behavior and experiments have provided evidence of the existence of a phase transition for the probabilit of a random instance of specific CSPs being satisfiable (see [DMSZ01]). While the nature of the phase transition sharp or coarse- for (Boolean) CSPs is now well-understood [Mol03, CD04b, CD04a], due in particular to a probabilistic adaptation of Schaefer s framework, the location of the threshold is still performed on a problem b problem basis. A long series of works have been devoted to get lower and upper bounds (see respectivel, [Ach00, CF90, CR92, FS96] and [FP83, MdlV95, DB97, KKKS98, JSV00]). For some problems these two bounds coincide, namel 2-SAT [CR92, Goe96] and 3-XOR-SAT [DM02]. In this paper we are interested in random smmetric Boolean CSPs, defined in Section 2. Our main contribution is to investigate the evolution of the expected number of distinguished satisfing assignments, so-called locall maximal solutions (see [DB97, KKKS98]), when the usual order parameter c ratio of constraints to variables increases. In Section 3 we introduce a new parameter for CSP(F): the sensitivit polnomial of F. This polnomial is based on the sensitivit of Boolean functions, it collects the sensitivit of all points in f 1 (1) for ever f in F. In Section 4 we express the evolution of the expected number of locall maximal solutions as a function of c. This gives a new picture of the solution-space geometr of random CSPs (see Figure 1). In particular it provides non-trivial upper bounds, c, for all these problems, which are better than the ones, c #, obtained in considering the number of all solutions (see Table 1). 2 Random smmetric Boolean CSPs A Boolean constraint satisfaction problem consists of a finite set of Boolean variables {x 1,..., x n }, and a finite set of Boolean constraints. Each such constraint denotes the allowed combinations of values for the variables it affects. A Boolean constraint satisfaction problem asks to determine if a satisfing truth assignment exists. In the smmetric framework a constraint C on the tuple of distinct variables (x i1,..., x ik ) is given b a non trivial Boolean function f : {0, 1} k {0, 1} and a tuple of literals (x τ1 i 1,..., x τ k ik ), where τ i 0, 1, i = 1,..., k are called the signs and are such that for an variable x, x 0 denotes the negative literal x, while x 1 denotes the positive one, x. Such a constraint is denoted b C = f(x τ1 i 1,..., x τ k ik ), and is referred to as an f-constraint. A truth assignment I: V = {x 1,..., x n } {0, 1} (which can be seen as an element of {0, 1} n and which extends to literals as usual b I(x 1 ) = I(x) and I(x 0 ) = 1 I(x)) satisfies such a constraint if f(i(x τ1 i 1 ),..., I(x τ k ik )) = 1, which is denoted b I(C) = 1. We are here interested in Boolean constraint satisfaction problem CSP(F) in which the tpes of allowed

4 112 Nadia Creignou and Hervé Daudé and Olivier Dubois constraints F, i.e., the Boolean functions, are fixed. Thus, throughout the paper F will denote a finite multi-set of non-trivial Boolean functions of fixed arit k. Example 2.1 we get the following well-known problems: 3-SAT = CSP({f 1 }) with f 1 1 (1) = {0, 1}3 \ {000}. 3-XOR-SAT = CSP({f 2 }) with f2 1 (1) = {001, 010, 100, 111}. 1-in-3-SAT = CSP({f 3 }) with f3 1 (1) = {001, 010, 100}. NAE-3-SAT = CSP({f 4 }) with f 1 4 (1) = {0, 1}3 \ {000, 111}. Let L and n be integers, CSP n,l (F) is a random model for the Boolean constraint satisfaction problem with n variables and L constraints. Each constraint C i of a random F-formula is chosen as follows: Select a tuple of k distinct variables (x i1,..., x ik ) uniforml at random from the set of all k-tuples of variables from {x 1,..., x n }. Select a sign vector (τ 1,..., τ k ) uniforml at random from {0, 1} k Select a function uniforml at random from F. Let C i be the constraint f(x τ1 i 1,..., x τ k ik ). The total number of all possible constraints is thus F 2 k (n) k. For an fixed F, we are interested in studing the probabilit that a random formula in CSP n,l (F) is satisfiable, we denote this probabilit b Pr n,l (SAT(F)). 3 Locall maximal solutions and Sensitivit In order to get more insight in the probabilit that a random formula in CSP n,l (F) is satisfiable, it is natural to stud the expected numbers of assignments that certif the satisfiabilit of such a formula. Therefore, let Sol(Φ) {0, 1} n denote the set of satisfing assignments, we will be interested in E n,l (Sol). Some other (less numerous) satisfiabilit certificates are also of interest, namel the locall maximal satisfing assignments which are local maxima in the lexicographic ordering of assignments, where the neighborhood of an assignment is the Hamming ball of radius 1. We denote b MaxSol(Φ) {0, 1} n the set of locall maximal assignments of Φ and we will also be interested in E n,l (MaxSol). Formall these assignments can be defined as follows. Given a truth assignment I: V = {x 1,..., x n } {0, 1}, for t = 1,..., n let Ît denote the truth assignment on V defined b Ît(x t ) = 1 I(x t ) and Ît(x s ) = I(x s ) for s t. Given a formula Φ, let I be a satisfing assignment for Φ, i.e., I(Φ) = 1. We sa that x t is a sensitive variable for I with respect to Φ if Ît(Φ) = 0 and we denote b P s (I, Φ) the set of all such variables: P s (I, Φ) = {x t /Ît(Φ) = 0}. Definition 3.1 A solution I of a formula Φ is a locall maximal solution of Φ if and onl if P s (I, Φ) {x t /I(x t ) = 0}.

5 locall maximal solutions for random CSPs 113 In order to estimate E n,l (Sol), one has to compute the number of random formulas satisfied b a given assignment. Thus, a natural parameter defined on an Boolean function f: {0, 1} k {0, 1} naturall emerges: its weight, namel f 1 (1). It turns out that this information can be refined in order to exhibit the parameter that fits exactl the computation of E n,l (MaxSol): the sensitivit. A finer description of f can be given in describing how the elements of f 1 (1) are distributed on the hpercube {0, 1} k. Let v {0, 1} k, the number of its neighbors on which the value of f differs is called the sensitivit of f at v and is denoted b s f (v) = {v : f(v ) f(v), dist(v, v ) = 1}. The set f 1 (1) can be partitioned into (k + 1) sets, each of which consisting in the points whose sensitivit is exactl j, 0 j k; let θ j (f) be the cardinalit of each of these sets, that is Observe that θ j (f) = {v : f(v) = 1 and s f (v) = j}. k θ j (f) = f 1 (1). Thus, we define the sensitivit polnomial of f as: j=0 S f () = k θ r (f) r. This notion is naturall extended to finite multi-sets F of Boolean functions of same arit k: S F () = k r=0 r=0 θ r (F) r, where θ j (F) = f F θ j (f) 4 Expected number of (locall maximal) solutions 4.1 Solutions versus locall maximal solutions We are interested in a comparative stud of E n,l (Sol) and E n,l (MaxSol) that are the expected number of solutions and the expected number of locall maximal solutions of a random formula. Specificall, the number L of constraints will be equal to c n where c is the usual order parameter. Let us notice that our results can be better expressed in introducing a change of variables. For each set F the usual order parameter will be replaced b where c = γ F (), where γ F is defined in the following proposition. Proposition 4.1 Let F be a finite multi-set of Boolean constraint functions of arit k, and S F be its sensitivit polnomial. Let γ F be defined on (1, 2] b ( ) γ F () = ln SF() 2( 1) S F (1) (). The function γ F is continuous and strictl decreasing on (1, 2]. Since γ F (2) = 0 and lim γ F = +, 1+ this defines a valid change of variables from (1, 2] to [0, + ). Proof: Observe that ln( 2( 1) ) < 1 ( 1) on (1, 2]. Moreover, for all polnomial S with positive coefficients and for all > 1, one can verif that [(S ) 2 SS SS ]() < 0. Finall γ () is negative since it has the same sign as ln( 2( 1) ) [(S ) 2 SS )]() S()S () ( 1).

6 114 Nadia Creignou and Hervé Daudé and Olivier Dubois ln (2) 0,6 0,4 0,2 c# c* 5 6-0,2 Fig. 1: Evolution of the exponential order of the expected number of solutions (linear curve) versus the exponential order of the expected number of locall maximal solutions. The following theorem, which is our main result, describes the evolution of the expected number of solutions and locall maximal solutions of random smmetric CSP as a function of the order parameter c. It appears that the entrop, 1 n log E n,cn(sol) is a linear function of c, while 1 n log E n,cn(maxsol) is a nonmonotone function. In using Maple it is possible to get a snthetic picture of this. Thus, Figure 1 illustrates this theorem for the most famous example, namel 3-SAT = CSP({f 1 }) with f 1 1 (1) = {0, 1}3 \ {000}. Theorem 4.2 Let F be a finite multi-set of Boolean constraint functions of arit k, and S F be its sensitivit polnomial. Let γ F, χ F and Ψ F be the functions defined on (1, 2] b ( ) γ F () = ln SF() 2( 1) S F () χ F () = γ F () ln( S F(1) 2 k F ) + ln 2 Ψ F () = ( 1) ln( 1) ln( ( ) 2 ) + γ SF () F() ln 2 k F

7 locall maximal solutions for random CSPs 115 then for an c, and as n tends to infinit, the expected number of solutions and locall maximal solutions of a random F-formula with n variables and c n constraints satisf : 1 n log E n,cn(sol) χ F (γ 1 F (c)) (2) 1 n log E n,cn(maxsol) Ψ F (γ 1 F (c)) (3) Sketch of proof: The classical result that the entrop of an CSP is a linear function of the order parameter c is well-known and the first part of the theorem is an eas consequence of the following combinatorial observation: in our smmetric model, for an truth assignment I #{Φ CSP n,l (F) s.t. I Sol(Φ)} = ((n) k L f (1) ) 1 (4) f F : Since f F f 1 (1) = S F (1), the expected number of solutions is controlled b the parameter S F (1) ( ) cn E n,cn (Sol) = 2 n SF (1) 2 k (5) F The analsis is more cumbersome for locall maximal solutions. The first step relies on an analogue of (4) for locall maximal solutions. This combinatorial task involves the sensitivit polnomial of the set of Boolean functions F and is detailed in the last section (see Proposition 5.1). Then we derive the exponential order of the expected number of locall maximal solutions of an CSP. This second analtical task is based on asmptotical results on Stirling numbers [Tem93] (see (7)-(??) in Section??). Let us notice that the number of locall maximal solutions is alwas lower than or equal to the number of solutions. A consequence of the following result on χ F and Ψ F and our main result is that for an CSP, the exponential order of the expected number of locall maximal solutions is alwas strictl lower than the exponential order of the expected number of solutions. Proposition 4.3 For ever finite multi-set F of Boolean constraint functions of fixed arit k, and for an c > 0, χ F (γ 1 F (c)) > Ψ F(γ 1 F (c)). Proof: Since γ is decreasing, it suffices to prove that for ever in (1, 2], Ψ() < χ(). Observe that Ψ(2) = 0 and χ(2) = ln 2. Hence, after eas simplifications it comes down to prove that for ever (1, 2) [ S () 1 + S() One can use successivel that on (1, 2): ln ( 1) ln( 2( 1) ) ] > ln( S() S(1) ) ( 1). ( 1) ln( ) < ln 2 (2 ) and ln 2 (2 ) + ln ln 2 > 0. 2( 1) Hence, proving that ψ() < χ() comes down to show that

8 116 Nadia Creignou and Hervé Daudé and Olivier Dubois S () S() ( 1) + ( 2) ln(s() ) > 0. (6) S(1) As we have alread noted in the proof of Proposition 4.1, S ()S() + S()S () S 2 () > 0. Thus we obtain that the derivative of the left hand side of (6) is positive, this function being equal 0 when = 1 the proof of (6) follows. 4.2 Threshold upper bounds Let F be a finite multi-set of Boolean constraint functions of arit k, and S F be its sensitivit polnomial. Upper bounds for the satisfiabilit threshold can be obtained b appling the first moment method to the number of solutions or to the number of locall maximal solutions. Observe that χ F is strictl increasing from to ln 2. If # F denotes the unique solution of χ F() = 0 then from (2) or (5) the probabilit of satisfiabilit of a random F-formula with n variables and c n constraints tends to 0 as soon as c > c # F, where c # F = γ(# F ) = ln 2 ln( 2k F S F (1) ). In the same wa, since Ψ F is continuous, Ψ F (2) = 0 and lim Ψ F() =, let 1 + F = inf Ψ 1 F ({0}) be the least solution of Ψ F () = 0, then from (3) we get the following upper bound. Theorem 4.4 Let F be a finite multi-set of Boolean constraint functions of arit k, and S F be its sensitivit generating function. Let γ F and Ψ F be the functions defined on (1, 2] b ( ) γ F () = ln SF() 2( 1) S F () Ψ F () = ( 1) ln( 1) ln( ( ) 2 ) + γ SF () F() ln 2 k. F The probabilit of satisfiabilit of a random F-formula with n variables and c n constraints tends to 0 (Pr n,cn (SAT(F)) 0) for ever c > c F, where c F = γ F(F ) and F is the unique number such that for ever < F, Ψ F() < 0. Thus it follows from Proposition 4.3 that for ever smmetric Boolean CSP, the threshold upper bound for satisfiabilit given b the expected number of locall maximal solutions is alwas better than the one given b the expected number of solutions. The picture given in Figure 1 is indeed generic, for an F we have : c F < c # F. Let us now explain how we can appl our theorem to an specific smmetric CSP in order to get an upper bound for the corresponding threshold. Let us start with the function h(x,, z) such that h 1 (1) = {0, 1} 3 \ {000, 110}, i.e., h(x,, z) = (x z) ( x ȳ z). We stud the CSP generated b h, CSP({h}). According to Schaefer s theorem this problem is NP-complete, and according to Creignou

9 locall maximal solutions for random CSPs 117 Name f 1 (1) S f c # c 3-SAT {0, 1} 3 \ {000} NAE-3-SAT {0, 1} 3 \ {000, 111} XOR-SAT {000, 110, 011, 101} in-3-SAT {000, 110, 101} Tab. 1: Upper bounds for well-known problems and Daudé s result it has a sharp threshold [CD04b, Theorem 4.2]. The sensitivit generating function of h is S() = We have to solve the equation ( 1) ln( 1) ln( ( ) S() 2 ) + γ() ln 2 3 = 0 MAPLE provides the following numerical estimates: = , γ( ) = , and hence for ever c > we have Pr n,cn (SAT({h})) 0. In appling Theorem 4.2 to the well-known problems mentioned in Example 2.1, the upper bounds shown in Table 1 are obtained. Better upper bounds have alread been obtained for some of these problems with specific techniques. This is the case for 3-SAT, an upper bound has been computed at in [DBM00], or for 3-XOR-SAT, the exact threshold has even been computed in [DM02]. However the technique used for each of these problem has not been generalized to an class of problems. Our Theorem 4.2 is a general tool giving for an problem being a smmetric Boolean CSPs, an upper bound of the satisfiabilit threshold better than the one derived straightforwardl from the expected number of solutions. 5 Proof of Theorem 4.2 In this section we give more details on the estimate of the exponential number of locall maximal solutions. The first step is the following combinatorial result. k Proposition 5.1 Let F be a finite multi-set of Boolean constraint functions of arit k and let θ r (F) r be its sensitivit polnomial. The expected number of locall maximal solutions for a random F-formula with n variables and L constraints is ( n n ) i 2 i E n,l (MaxSol) = ( F 2 k (n) k ) L i=0 L 0 +L 1 + +L k =L L1+2L 2 + +kl k i ( L L 0... L k k ] Lr [θ r (F) (n r) (k r) r=0 ) T (L 1 + 2L kl k, i) where T (L 1 +2L 2 + +kl k, i) denotes the number of applications ϕ from {1,, L 1 +2L 2 + +kl k } onto {1,..., i}, with the additional requirement that for ever j {0,, k} and for each 0 i r=0

10 118 Nadia Creignou and Hervé Daudé and Olivier Dubois j(l j 1) with i 0(j), #{ϕ(l 1 + 2L (j 1)L j 1 + i + 1),..., ϕ(l 1 + 2L (j 1)L j 1 + i + j)} = j. Proof: E n,l (MaxSol) = (I,P) N (I, P) ( F 2 k (n) k ) L, where N (I, P) is the number of F-formula Φ with n variables and L constraints such that I Sol(Φ) and P s (I, Φ) = P. Our first analsis will be to estimate N (I, P) for fixed (I, P). Given a constraint C = f(x τ1 i 1,..., x τ k ik ) satisfied b I, we sa that the position j is a sensitive position in C with respect to I if f(i(x τ1 i 1 ),..., 1 I(x τj i j ),..., I(x τ k ik )) = 0. A constraint C is said to be l- sensitive with respect to I if P s (I, C) = l. Due to the smmetr of our model the number of such constraints is θ l (f) (n) k. In order to count the number of formulas Φ = (C 1,..., C L ) in CSP n,l (F) such that I Sol(Φ) and P s (I, Φ) = P, we partition the constraints of Φ in k + 1 classes, according to their degree of sensitivit. For j = 0,..., k, let L j be the number of j-sensitive constraints in Φ, L L k = L. The number of f-formulas of the form Φ = (C 0 1,..., C 0 L 0, C 1 L 1,..., C 1 L 1,..., C k 1,..., C k L k ), such that I Sol(Φ) and P s (I, Φ) = P, with P = i, is: T (L L 2 + k L k, i) k (θ r (f) (n r) (k r) ) Lr, where T (L 1 + 2L kl k, i) corresponds to the number of was of choosing the L L 2 + k L k variables from P in order to fill the sensitive positions so that all variables from P occur, with the additional requirement that variables occurring in the same constraint are pairwise distinct. This analsis can be extended to a finite set of constraint functions F in replacing the θ r (f) b θ r (F) = f F θ r(f). Thus we obtain: r=0 N (I, P) = L 0 +L 1 + +L k =L L1+2L 2 + +kl k i ( L L 0... L k k ] Lr [θ r (F) (n r) (k r) r=0 ) T (L 1 + 2L kl k, i) In taking into account the number of pairs (I, P) (there are ( n i) 2 i such pairs with P = i) and all the different possible distributions of the number of constraints according to their sensibilit, each requiring L 0 + L L k = L and L 1 + 2L kl k i, we get the expression of E n,l (MaxSol) as stated in the proposition.

11 locall maximal solutions for random CSPs 119 The asmptotical behavior of the number of applications from {1,, a} onto {1,, b} is wellstudied and based on Stirling numbers of the second kind. It follows from the above result that the exponential order of E n,l (MaxSol) is governed b exponential order of Stirling numbers of the second kind and of multinomial coefficients. Observe that in Proposition 5.1 onl indices r for which θ r 0 occur. Suppose that θ i0 0,..., θ is 0 and for all j / {i 0,..., i s }, θ j = 0. In setting L ij = β ij L and i = α n, in considering that β i0 = 1 β i1... β is and in using asmptotic estimates of Stirling numbers (see [Tem93]), we obtain that when L = cn and n tends to infinit: 1 n log E n,cn(maxsol) max D c Φ c, (7) where Φ c is a function of (s + 1) variables on a domain D c given b: with where B = Φ c (α, β i1,..., β is ) = α ln(e x0 1) + cb ln cb x 0 e s + c β ij ln θ i j (F) β ij j=0 α ln α (1 α) ln(1 α) + α ln 2 c ln(2 k F ), D c = (α, β ), 0 α 1; 0 β ij 1, s j=1 s i j β ij, and x 0 denotes the implicit solution of the equation j=0 β ij 1; c B α, 1 e x0 = α cb x 0. (8) It appears that the global maximum of Φ c on D c is located in the interior of D c, and thus is a point at which all the partial derivatives are 0. In using the change of variables introduced in Proposition 4.1 we show that there is a unique stationar point which is completel determined b c. Indeed, annihilating the partial derivates one gets and 2(1 α)(e x0 1) = α (9) (1 β 1 β k ) θ j (F) c j (β 1 + 2β kβ k ) j x j 0 θ 0(F) β j = 1 (10) Plugging these equations in (8) we get α and the β j s respectivel as a function of x 0 α = 2(ex0 1) 2e x0 1),

12 120 Nadia Creignou and Hervé Daudé and Olivier Dubois β j = θ j (F) (1 e x0 /2) j k j=0 θ for all j. j(f) (1 e x0 /2) j Therefore, in setting = 2ex 0 2e x 0 1, we get α and the β j s as a function of : α = 2, β j = θ j j k j=0 θ j j. Finall, from (8), c can also be expressed as a function of : ( ) c = ln S() 2( 1) S = γ(). (11) () In plugging in Φ c the above coordinates of the stationar point, elementar calculations reveal that the maximum value of Φ c is completel determined b c and can be expressed b the function Ψ occurring in our theorem: max Φ c = Φ c (2 γ 1 (c), θ i1 (F), θ i2 (F) γ 1 (c),..., θ is (F) (γ 1 (c)) s 1 ) = Ψ(γ 1 (c)). D c Thus, from (7) we get: References 1 n log E n,cn(maxsol) Ψ(γ 1 (c)) (12) [Ach00] D. Achlioptas. Setting 2 variables at a time ields a new lower bound for random 3-sat. In Proceedings of the 32nd ACM Smposium on Theor of Computing (STOC 2000), pages 28 37, [Ba05] [CD04a] [CD04b] [CF90] [CKS01] D. Bale. Phase transitions for generalized sat problems: upper bounds and experiments. Manuscript, N. Creignou and H. Daudé. Coarse and sharp transitions for random generalized satisfiabilit problems. In Proceedings of the third colloquium on mathematics and computer science, Vienna (Austria), pages Birkhäuser, N. Creignou and H. Daudé. Combinatorial sharpness criterion and phase transition classification for random csps. Information and Computation, 190(2): , M.T. Chao and J. Franco. Probabilistic analsis of a generalization of the unit-clause selection heuristics for the k-satisfiabilit problem. Information Science, 51(3): , N. Creignou, S. Khanna, and M. Sudan. Complexit classifications of Boolean constraint satisfaction problems, volume 7. SIAM Monographs on discrete mathematics and applications, 2001.

13 locall maximal solutions for random CSPs 121 [CR92] [DB97] [DBM00] [DM02] V. Chvátal and B. Reed. Mick gets some (the odds are on his side). In Proceedings of the 33rd Annual Smposium on Foundations of Computer Science (FOCS 92), pages , O. Dubois and Y. Boufkhad. A general upper bound for the satisfiabilit threshold of random r-sat formulae. Journal of Algorithms, 24(2): , O. Dubois, Y. Boufkhad, and J. Mandler. Tpical random 3-sat formulae and the satisfiabilit threshold. In Proceedings of the 11th ACM-SIAM Smposium on Discrete Algorithms (SODA 2000), pages , O. Dubois and J. Mandler. The 3-xor-sat threshold. In Proceedings 43rd Smposium on Foundations of Computer Science (FOCS 2002), Vancouver (British Columbia, Canada), pages , November [DMSZ01] O. Dubois, R. Monasson, B. Selman, and R. Zecchina, editors. Phase transitions for combinatorial problems, volume 265 of Theoretical Computer Science, [FP83] [FS96] [Goe96] [JSV00] J. Franco and M. Paull. Probabilistic analsis of the davis-putnam procedure for solving the satisfiabilit problem. Discrete Applied Mathematiques, 5:77 87, A.M. Frieze and S. Suen. Analsis of two simple heuristics on a random instance of the k-sat problem. Journal of Algorithms, 20(2): , A. Goerdt. A threshold for unsatisfiabilit. Journal of of Computer and Sstem Sciences, 53(3): , S. Janson, Y.C. Stamatiou, and M. Vamvakari. Bounding the unsatisfiabilit threshold of random 3-sat. Random Structures and Algorithms, 17(2):79 102, Erratum, Random Structures and Algorithms, 18(1):99 100, [KKKS98] L.M. Kirousis, E. Kranakis, D. Krizanc, and Y.C. Stamatiou. Approximating the unsatisfiabilit threshold of random formulas. Random Structures and Algorithms, 12(3): , [MdlV95] [Mol03] [MSL92] [Sch78] A. El Maftouhi and W. Fernandez de la Vega. On random 3-sat. Combinatorics, Probabilit and Computing, 4(3): , M. Mollo. Models for random constraint satisfaction problems. SIAM Journal on Computing, 32(4): , D. Mitchell, B. Selman, and H.J. Levesque. Hard and eas distributions of sat problems. In Proceedings 10th National Conference on Artificial Intelligence, San Jose (CA, USA), pages AAAI, T.J. Schaefer. The complexit of satisfiabilit problems. In Proceedings 10th STOC, San Diego (CA, USA), pages Association for Computing Machiner, [Tem93] N.M. Temme. Asmptotic estimates of stirling numbers. Stud. appl. Math., 89: , 1993.

14 122 Nadia Creignou and Hervé Daudé and Olivier Dubois