Second moment method for a family of boolean CSP

SECOND MOMENT METHOD F OR A F AMIL Y OF BOOLEAN CSP Y ACINE BOUFKHAD AND OLIVIER DUBOIS Abstract. The estimation of phase transitions in random bo olean Constraint Satisfaction Prob- lems (CSP) is based on tw o fundamen tal tools: the first and second momen t methods. While the first moment metho d on the num ber of solutions permits to compute upp er bounds on an y bo olean CSP , the second moment metho d used for computing lo w er bounds pro v es to be more tricky and in most cases giv es only the trivial low er b ound 0. In this paper, we define a subcl ass of bo olean CSP cov ering the monotone v ersions of many known NP-Complete bo olean CSPs. W e give a method for computing non trivial low er b ounds for any mem b er of this subclass. This is achiev ed thanks to an application of the second momen t method to some selected solutions called characteristic solutions that depend on the b oolean CSP considered. W e apply this metho d with a finer analysis to establish that the threshold r k (ratio : #constrains/#v ariables) of monotone 1-in-k-SA T is log k/k ≤ r k ≤ log 2 k/k . Introduction The empirical evidence has sho wn that random instances of b o olean Constraint Satisfaction Prob- lems C S P exhibit a phase transition i.e. a sudden c hange from SA T to UNSA T when the num b er of constrain ts increases: that is there exists a critical v alue r ∗ of the ratio r n umber of constrain ts to num b er of v ariables suc h that random instances are w.h.p. satisfiable if r < r ∗ and w.h.p. unsatisfiable if r > r ∗ . r ∗ is called the threshold v alue of the transition. The sharpness of the threshold has b een addressed in a series of works [12, 17, 5, 6]. Computing the threshold asso ciated to a C S P is at presen t out of reac h apart from some exceptions [3, 14, 7, 11] for polynomial sub classes. Since this cannot b e carried out, upper and lo wer b ounds of r ∗ are computed. These b ounds are almost all obtained by different applications of the probabilistic metho d to ols: the first and second moment methods and for some of them through anlaysis of algorithms[10, 9, 8, 16, 2] for 3-SA T. While the first moment metho d on the num b er of solutions p ermits to obtain an upp er b ound of the lo cation of the threshold for any bo olean C S P , the second momen t method on the num b er of solutions fails at an y ratio to estimate the probabilit y of satisfiability . In [1], an original metho d is presen ted to o vercome this problem in the case of k -SA T for which the direct calculus also fails. W e define a sub class of C S P and a metho d that allow to b ound the phase transition from b oth sides for this sub class. The latter is characterized b y constraints ha ving the prop ert y of b eing closed under permutations. It includes the monotone v ersions of many w ell known problems like : 1-in-kSA T, NAE-k-SA T... and then it includes man y NP-Complete b o olean C S P s. Roughly sp eaking, we show ho w the second method can b e made “to work” for every problem in this class. More precisely , we parameterize the v aluations b y their num b er of v ariables having the v alue 1 and we sho w that there exist precise v alues for this parameter dep ending on ev ery problem for whic h the second moment method gives a non trivial lo wer bound, the corresp onding solutions are called char acteristic solutions . W e prov e that the b ound given by this metho d is at least some w ell defined v alue for any problem in the sublclass. Ho wev er, the generality of this v alue is obtained at the cost of some w eakness. Better b ounds can be computed using the same scheme through a finer analysis on a case by case basis. T o illustrate this, w e do the full analysis for Key words and phrases. Constraint Satisfaction Problems, Phase transitions, Second momen t method. 1 2 Y. BOUFKHAD AND O. DUBOIS p ositiv e 1-in- k -SA T to deriv e the asymptotically optimal lo wer b ound with respect to our method that is log k/k . T o show that this lo wer bound is tight, we establish using the first moment method an upp er bound of log 2 k /k for k ≥ 7. 1. Basic definitions and main resul ts Giv en a set X of n b oolean v ariables, a v aluation σ is a mapping X → { 0 , 1 } that assigns to an y v ariable x ∈ X the v alue 0 or 1. k b eing an integer, a relation R of arit y k is a subset of { 0 , 1 } k . A relation is said to b e trivial if (0 , ..., 0) or (1 , ..., 1) is an element of R . W e consider throughout the paper only non trivial relations. A constrain t defined from a relation R of arit y k , is a tuple of k b oolean v ariables, denoted R ( x 1 , ..., x k ), which is said to b e satisfied under some v aluation σ iff ( σ ( x 1 ) , ..., σ ( x k )) ∈ R , otherwise it is said unsatisfied. Given a set of relations S , an instance of bo olean CSP with respect to S , denoted b y C S P ( S ), is a conjunction set of constrain ts R ( x 1 , ..., x k ) where R ∈ S . An instance C S P ( S ) is satisfied iff every constrain t is satisfied. In this pap er, we consider a sub class of C S P ( S )s defined as follo ws. A relation R is said to b e in v ariant b y permutation, iff any p erm utation of the coordinates of a tuple t ∈ R is also in R. Suc h a relation is denoted by R inv . The inv ariance prop ert y implies that for every tuple t ∈ R inv , all tuples having the same num b er of co ordinates equal to 1 or (0s) as t must b e also in R inv . This defines an equiv alence relation, t wo elements t and t 0 of R inv b elonging to the same equiv alence class iff they hav e the same num b er of co ordinates equal to 1s (or 0s). Thus the equiv alence classes partition R inv in to subsets eac h asso ciated to an integer i equal to the n umber of 1s of the elemen ts of the class. In this pap er, we will only consider b o olean C S P s with resp ect to a single non trivial inv ari- an t under permutation relation denoted C S P ( { R inv } ) . In order to designate more explicitly a C S P ( R inv ) w e will denote it in an equiv alent manner b y C S P ( I k ), where I k is the subset of in te- gers in { 1 , ..., k − 1 } asso ciated to all equiv alence classes. Th us an instance C S P ( I k ) is satisfiable with resp ect to I k iff there exits a v aluation σ suc h that the n um b er of 1s in every constraint of C S P ( I k ) 1 is an element of I k . Example 1. k = 4 and R = { 1000 , 0100 , 0010 , 0001 , 0111 , 1011 , 1101 , 1110 } . R is inv ariant b y p erm utation. The set of integers asso ciated to R is I 4 = { 1 , 3 } . A constraint ( x i 1 , x i 2 , x i 3 , x i 4 ) of an instance C S P ( R ) is satisfied iff exactly one or three of the four v ariables of the constraint has the v alue 1. The C S P s of the class defined abov e are NP-complete for an y relation of arit y greater or equal to 3 according to the Sc haefer classification. They include tw o w ell kno wn problems of this classification that are positive 1-in- k -SA T ( I k = { 1 } according to the abov e definition) and positive not-all-equal- k -SA T ( I k = { 1 , ..., k − 1 } ). The random v ersion of a C S P ( I k ) is as follows. Given a relation R inv , I k the set of integers asso ciated with R inv , a random C S P ( I k ) instance with m constrain ts ov er n b o olean v ariables is formed by drawing uniformly , indep enden tly and with replacemen t m tuples of k v ariables ov er the set of n v ariables. Suc h a random C S P ( I k ) instance is denoted b y I k ( m, n ) . This defines a probabilit y space denoted by Ω( I k , m, n ) in which instances I k ( m, n ) are equiprobable. Definition 1. A p -v aluation for some natural integer 0 ≤ p ≤ n is a v aluation suc h that |{ x i | σ ( x i ) = 1 }| = p . Let δ ∈ [0 , 1], for the sake of simplicity we will denote whenever it is non ambiguous a b δ n c -v aluation b y δ -v aluation. A δ -v aluation that is a solution of an instance I k ( m, n ) is said to b e a δ -solution. Let X δ b e the random v ariable asso ciating to eac h I k ( m, n ) the num b er of its δ -solutions. 1 Alternatively , this class can be seen as h ypergraph bi-coloring problem where the n umber of v ertices allow ed to hav e a certain color in some edge are taken only in I k . SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 3 Theorem 1. F or any C S P ( I k ) , ther e exist a r ∗ I k > 0 and 0 < δ < 1 such that for al l r < r ∗ I k , lim n →∞ E [ X δ ] 2 E [ X 2 δ ] > 0. Roughly sp eaking, the preceding Theorem states that for any problem in C S P ( I k ), there exists a δ that mak es the second moment to succeed in computing a low er b ound. Combined with the inequalit y of Cauch y-Sch wartz: P ( X δ > 0) ≥ E [ X δ ] 2 E [ X 2 δ ] and the sharpness of the threshold of the problems in this class [13] [4], we ha ve the follo wing consequence of Theorem 1. Corollary 1. F or any C S P ( I k ) , ther e exists a r e al r ∗ I k > 0 such that for r < r ∗ I k , lim n →∞ P r ( I k ( n, r n ) is satisfiable ) = 1 . The Theorem 1 states that the second moment succeeds for some v alues of δ using X δ as a random v ariable. How ever, the v alue of the b ound r ∗ I k men tioned in the theorem (and giv en in Section 2.2) is not the optimal bound that can b e deriv ed b y the method. This is the price of its generality . The full analysis establishing the optimal b ound can somehow b e done on a case by case basis. Note that the b est b ound that can b e obtained can not exceed the smallest ratio that mak e E [ X δ ] → 0 when n → ∞ . Indeed, b y Marko v inequality the probabilit y of X δ > 0 in that case is 0. The v alue of the b est b ound that can b e exp ected for p ositiv e 1-in-k-SA T is as ymptotically log k/k . This is precisely what we obtain through the full analysis of this particular problem. W e obtain : Theorem 2. L et 1 k = { 1 } . lim n →∞ Pr (1 k ( n, r n ) is satisfiable ) = 1 If r < log k /k and k ≥ 3 . Sp ecifically , for k = 3 a b etter low er b ound at 0.546 has b een computed in [15] analyzing the success to find out a solution with a sp ecific algorithm. How ever our aim is to provide a to ol yielding systematically a low er b ound for a large class of CSPs. W e give a rough upp er b ound at (log k ) 2 /k (v alid for k ≥ 7) whic h shows that our bounds are tight around the threshold. 2. Second moment and characteristic solutions W e first define the char acteristic solutions b efore w e giv e the second moment of their n umber. Definition 2. Char acteristic valuations for some C S P ( I k ) are δ -v aluations for which the proba- bilit y of satisfying a uniformly randomly drawn constraint is lo cally maximum with resp ect to δ . The solutions of an instance that are c haracteristic v aluations are said to b e char acteristic solutions for this instance. Giv en some δ -v aluation, the probability π i ( δ ) that a randomly selected k -tuple con tains i ones is π i ( δ ) =  k i  δ i (1 − δ ) k − i . So the probability that a δ -v aluation satisfies with resp ect to some set I k ⊂ { 1 , 2 , ..., k − 1 } a randomly selected k -tuple is obtained b y summing up, the m utually exclusiv e cases for different i ’s in I . This probability is g I k ( δ ) = P i ∈ I k π i ( δ ) = P i ∈ I k  k i  δ i (1 − δ ) k − i . Let ∆ I k b e the set of reals for which g I k ( δ ) is locally maxim um. Clearly , for any δ ∈ ∆ I k , δ -v aluations are by definition the c haracteristic v aluations of C S P ( I k ). Since I k ⊆ { 1 , ..., k − 1 } then g I k (0) = g I k (1) = 0. The function g I k ( δ ) b eing smo oth, strictly p ositiv e inside ]0 , 1[, it maximizes inside the interv al ]0 , 1[ at at least one stationary p oin t. Th us for an y I k ⊆ { 1 , ..., k − 1 } , ∆ I k 6 = ∅ . The fact that ev ery δ ∈ ∆ I k is a stationary p oin t for g I k ( δ ) will b e used later. 4 Y. BOUFKHAD AND O. DUBOIS 2.1. First and second momen t of num b er of characteristic solutions. The key idea used in the metho d presen ted in this pap er is instead of taking as a random v ariable the n umber of solutions, to consider as random v ariable the num b er of δ -solutions. The first moment of X δ is: E [ X δ ] =  n δ n  g I k ( δ ) rn ∼ 1 p 2 π δ (1 − δ ) n γ I k ,r ( δ ) n Where: γ I k ,r ( δ ) = g I k ( δ ) r δ δ (1 − δ ) 1 − δ Remark 1. It is easy to see that lim n →∞ γ I k ,r ( δ ) = 0 for any r > ˆ r I k ,δ = δ log δ +(1 − δ ) log(1 − δ ) log g I k ( δ ) . By Mark ov inequalit y , this means that the δ -solutions do es not exist for r > ˆ r I k ,δ and then the lo wer b ound that w e can get through δ -solutions is at most ˆ r I k ,δ . F or computing the second momen t, we consider tw o δ -v aluations σ 1 and σ 2 ha ving p v ariables assigned 1 in σ 1 and 0 in σ 2 . This defines all the other categories of v ariables. Indeed, the num b er of v ariables assigned 0 in σ 1 and 1 in σ 2 m ust be also p in order that σ 2 is a δ -v aluation. b δ n c − p is the num b er of v ariables assigned 1 in both solutions and n − b δ n c − p are assigned 0 in both solutions. First, we give the probability φ i,j,δ  p b δ n c  that a random k -tuple has i 1s under σ 1 and j 1s under σ 2 suc h that d v ariables of the k -tuple are equal to 1 in both assignments. d must range from d min = max (0 , i + j − k ) to d max = min ( i, j ). φ i,j,δ  p b δ n c  = d max X d min  k i  i d  k − i j − d   b δ n c − p n  d  p n  i + j − 2 d  n − b δ n c − p n  k − i − j + d (1) Summing up ov er couples ( i, j ), w e get G I k ,δ ( µ ), the probability that a couple of δ -v aluations ha ving µδ n v ariables taking a different v alue in σ 1 or σ 2 satisfies a random constraint: G I k ,δ  p b δ n c  = X i ∈ I k X j ∈ I k φ i,j,δ  p b δ n c  W e can now write the second moment b y summing up ov er all p ossible couples ( σ 1 , σ 2 ) : E [ X 2 δ ] = min ( b δ n c ,n −b δn c ) X p =0  n ( b δ n c − p ) p p ( n − b δ n c − p )  G I k ,δ  p b δ n c  rn W e now estimate E [ X 2 δ ] as a function of n , using a classical asymptotic estimate of the m ultinomial co efficien t. F or small multinomial num b ers the asymptotic estimate being also an upp er bound, it will b e sufficien t for the estimation we need. W e set : µ = p b δ n c .  n (1 − µ ) δ n µδ n µδ n (1 − δ − µδ ) n  ≤ (2 π n ) − 3 / 2 p 2 µδ (1 − µ ) δ (1 − δ − µδ ) ( t δ ( µ )) − n where : t δ ( µ ) = ((1 − µ ) δ ) (1 − µ ) δ ( µδ ) 2 µδ (1 − δ − µδ ) 1 − δ − µδ . W e ha v e : E [ X 2 δ ] ≤ X µ ∈{ 0 , 1 b δn c ,...,min ( 1 , n −b δn c b δn c ) } (2 π n ) − 3 / 2 p 2 µδ (1 − µ ) δ (1 − δ − µδ ) (Γ I k ,δ,r ( µ )) n (2) with: SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 5 (3) Γ I k ,δ,r ( µ ) = G I k ,δ ( µ ) r t δ ( µ ) Eac h term in (2) consisting of a p olynomial factor and an exp onen tial factor in n the sum can b e estimated with a discrete version of Laplace metho d. Thus : Lemma 1. if u and v ar e smo oth r e al-value d functions of one variable x , and if v has a single maximum on [ a, + ∞ [ , lo c ate d at ξ 0 with ξ > a and if further v 00 ( ξ 0 ) is not 0, then : 1 √ n ω n X i = aω n u ( i n ) exp ( nv ( i n )) ∼ g ( ξ 0 ) √ 2 π ω p | v 00 ( ξ 0 ) | exp ( nv ( ξ 0 )) W e will apply Lemma 1, setting v ( µ ) = log(Γ I k ,δ,r ( µ )) and u ( µ ) = (2 π ) − 3 2 √ 2 µδ 2 (1 − µ )(1 − δ − µδ ) . Then : (4) E [ X 2 δ ] ≤ n − 1 (2 π ) − 3 / 2 p 2(1 − δ ) 3 δ 3 × √ 2 π min ( δ, (1 − δ )) q | Γ 00 I k ,δ,r (1 − δ ) | max µ ∈ [0 ,min ( 1 , 1 − δ δ ) ] Γ I k ,δ,r ( µ ) ! n The success of the second moment method relies mainly on the b eha vior of Γ I k ,δ,r ( µ ) for µ ∈ [0 , min  1 , 1 − δ δ  ] The upp er b ound of the ratio E [ X δ ] 2 E [ X 2 δ ] will then dep end mainly on its exp onen tial part γ I k ,r ( δ ) 2 max µ ∈ [0 ,min ( 1 , 1 − δ δ ) ] Γ I k ,δ,r ( µ ) that must b e equal to 1 otherwise, all what we will get is the trivial relation E [ X δ ] 2 E [ X 2 δ ] ≥ 0. W e will see in the next Section that this achiev ed through characteristic solutions. F rom (1) we will write in the sequel of the pap er : φ i,j,δ ( µ ) = min ( i,j ) X d = max (0 ,i + j − k )  k i  i d  k − i j − d  κ i,j,d,δ ( µ ) (5) with : κ i,j,d,δ ( µ ) = ((1 − µ ) δ ) d ( µδ ) i + j − 2 d ((1 − δ − µδ )) k − i − j + d (6) and : G I k ,δ ( µ ) = X i ∈ I k X j ∈ I k φ i,j,δ ( µ ) (7) 2.2. Pro of of Theorem 1 and its Corollary. In the following, we sk etch first the pro of b y discussing its most imp ortant ingredients. A crucial p oin t for the success of the metho d is the p oin t where µ = 1 − δ or the independence point. T o understand this, consider t w o v aluations dra wn indep endently uniformly at random from the set of δ -v aluations. A v ariable is assigned 1 under one of the tw o δ -v aluations with probability δ and 0 with probability 1 − δ . Since the tw o v aluations are selected indep enden tly , the probability of b eing assigned 1 b y a δ -v aluation and 0 by the other is δ (1 − δ ). Th us, according to the notation of the Section 2.1, these pairs of δ -v aluations are c haracterized by Hamming distance 2 µ with µ = 1 − δ . These uncorrelated pairs of δ -v aluations pla y a central role in the success of the metho d. Indeed: 6 Y. BOUFKHAD AND O. DUBOIS G I k ,δ (1 − δ ) = X i ∈ I k X j ∈ I k φ i,j,δ (1 − δ ) = X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  X d  i d  k − i j − d  = X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  k j  = X i ∈ I k X j ∈ I k π i ( δ ) π j ( δ ) = g I ( δ ) 2 It is easy to see that t δ (1 − δ ) =  δ δ (1 − δ ) 1 − δ  2 , thus : (8) Γ I k ,δ,r (1 − δ ) = G I k ,δ (1 − δ ) r t δ (1 − δ ) = g I k ( δ ) 2 r  δ δ (1 − δ ) 1 − δ  2 = γ I k ,r ( δ ) 2 Since Γ I k ,δ,r (1 − δ ) /γ I k ,r ( δ ) 2 = 1, if µ = 1 − δ is not the global maxim um of Γ I k ,δ,r ( µ ), there exist some µ for which γ I k ( δ ) 2 / Γ I k ,δ,r ( µ ) < 1 making the metho d to fail. Consequen tly , a necessary condition for the success of the metho d is that µ = 1 − δ is a stationary p oin t. Lemma 2 states that this is the case only for the c haracteristic solutions. Let ρ = max µ ∈ [0 ,min (1 , 1 − δ δ )]  δ 1 − µ + 2 δ µ + δ 2 1 − δ − δµ  , ν = max µ ∈ [0 ,min (1 , 1 − δ δ )] (log ( G I k ,δ ( µ ))) 00 . Let: (9) r ∗ I k = ρ ν The Lemma 3 states that r ∗ I k is well defined and that it is strictly p ositiv e and for an y r < r ∗ I k , the second deriv ative of log (Γ I k ,δ,r ( µ )) is negativ e for any µ ∈ [0 , min (1 , (1 − δ ) /δ )]. This function is then concav e in the previous interv al. Combining the tw o lemmas, we can conclude that there is a range of ratios ]0 , r ∗ I k [ for which µ = 1 − δ is the global maxim um of Γ I k ,δ,r . Remark 2. In general, the p oint µ = 1 − δ con tinues to be the global maximum in a range b ey ond r ∗ I k after the function ceases to b e conca ve, allo wing through a more precise analysis to get b etter lo wer b ound than r ∗ I k . Ho wev er, a general bound b ey ond concavit y is hard to figure out for the class and w e do not need this fact for the pro of of Theorem 1 which aim is to give the conditions under which the second momen t succeeds regardless of the v alue of the b ound obtained. When one needs for a particular problem to compute the b est lo wer b ound with resp ect to δ -solutions, a finer analysis is required for this particular problem. This is what w e do to get the b est p ossible lo wer b ound with resp ect to δ -solutions for p ositiv e 1-in- k -SA T. Lemma 2. Γ 0 I k ,δ,r (1 − δ ) = 0 iff δ ∈ ∆ I k . Pr o of. Considering (3), it is easy to chec k that the deriv ative of t 0 δ ( µ ) (defined in (6)) is such that t 0 δ (1 − δ ) = 0. It is then necessary and sufficien t that G 0 I k ,δ (1 − δ ) = 0. It can b e sho wn (see App endix A), that: (10) G 0 I k ,δ (1 − δ ) = − δ  g I k ( δ ) 0  2 k (1 − δ ) 2 whic h is equal to 0 iff g I k ( δ ) 0 = 0 i.e. δ ∈ ∆ I k .  SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 7 Lemma 3. r ∗ I k (as define d in (9)) is strictly gr e ater than 0 and for every r < r ∗ I k , log (Γ I k ,δ,r ( µ )) 00 < 0 for µ ∈ [0 , min (1 , (1 − δ ) /δ )] . Pr o of. (log (Γ I k ,δ,r ( µ ))) 00 =  − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δ µ  + r (log ( G I k ,δ ( µ ))) 00 It can be sho wn (see Appendix B) that  − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δµ  is negative and b ounded from ab o ve b y − ρ and that (log ( G I k ,δ ( µ ))) 00 is p ositiv e and b ounded from ab ov e by ν , then (log (Γ I k ,δ,r ( µ ))) 00 ≤ − ρ + r ν < 0 if r < ρ/ν = r ∗ I k .  No w we are in position to give the pro of of Theorem 1. Pro of of Thorem 1 Thanks to Lemma 2, we kno w that µ = 1 − δ is a stationary p oin t for log(Γ I k ,δ,r ( µ )) and thanks to Lemma 3, we know that log (Γ I k ,δ,r ( µ )) is concav e for r < r ∗ I k . Combining these t w o facts, we deduce that µ = 1 − δ is a global maximum for log(Γ I k ,δ,r ( µ )). The inequality (4) b ecomes: E [ X 2 δ ] ≤ n − 1 (2 π ) − 3 / 2 p 2(1 − δ ) 3 δ 3 × √ 2 π min ( δ, (1 − δ )) q | Γ 00 I k ,δ,r (1 − δ ) | (Γ I k ,δ,r (1 − δ )) n On putting C 1 = (2 π ) − 3 / 2 √ 2(1 − δ ) 3 δ 3 × √ 2 π min ( δ , (1 − δ )) q | Γ 00 I k ,δ,r (1 − δ ) | and ha ving thanks to (8) Γ I k ,δ,r (1 − δ ) = γ I k ,r ( δ ) 2 , this yields : E [ X 2 δ ] ≤ n − 1 C 1 ( γ I k ,r ( δ ) 2 n ). Allo wing for the relation E [ X δ ] ≥ e − 1 / 6 √ 2 π δ (1 − δ ) n ( γ I k ,r ( δ )) n , w e deduce : E [ X δ ] 2 E [ X 2 δ ] ≥ C 2 C 2 b eing a positive constan t. Thus Theorem 1 is prov ed. F rom the Creignou and Daud´ e criterion [5, 6] for k ≥ 3 a C S P ( I k ) is neither dep ending on one comp onen t nor strongly dep ending on a 2XOR-relation, it can be stated according to the F riedgut’s theorem in [12] the following fact : F act 1. F or every k ≥ 3 and a r andom C S P ( I k ) , ther e exists a function λ k ( n ) such that for any  > 0 : lim n →∞ P r ( I k ( r n, n ) is satisf iabl e ) = ( 1 if r > λ k ( n )(1 −  ) 0 if r < λ k ( n )(1 −  ) It follows that for an y C S P ( I k ) , if r < r ∗ I k as defined (9), then : lim n →∞ Pr ( I k ( n, r n ) is satisfiable ) = 1. Thus Corollary 1 is pro v ed. 3. Positive 1 -in- k -SA T case: proof of Theorem 2 F or 1-in- k -SA T, w e denote the corresponding I k b y 1 k = { 1 } . The function g 1 k ( δ ) = kδ (1 − δ ) k − 1 . It is easy to c heck that ∆ 1 k = { 1 k } . W e note first that the b est low er bound that w e can hop e to get is ˆ r 1 k , 1 /k as defined in Remark 1. It is easy to c heck that lim k →∞ ˆ r 1 k , 1 /k log k/k = 1. Then asymptotically , the b est lo wer b ound that can b e obtained with respect to 1 /k -solutions is log k/k . F or the second momen t, as previously w e consider only δ -v aluations. Only the function G 1 k , 1 /k ( µ ) c hanges: G 1 k , 1 /k ( µ ) = k ( k − 1) κ 1 , 1 , 1 , 1 /k ( µ ) + k κ 1 , 1 , 0 , 1 /k ( µ ) = k 1 − k ( k − 1 − µ ) k − 2  k  1 − µ + µ 2  − 1  8 Y. BOUFKHAD AND O. DUBOIS Thanks to Lemma 2, w e know that µ = 1 − 1 /k is stationary p oin t of Γ 1 k , 1 k ,r ( µ ) and that Γ 1 k , 1 k ,r (1 − 1 /k ) = γ 1 k ,r (1 /k ) 2 . It remains to prov e that it is a global maxim um for r ≤ log k /k . This b ound go es in general b ey ond conca vity so we need a finer analysis. That is the purp ose of this Lemma. Lemma 1. F or any µ ∈ [0 , 1] , Γ 1 k , 1 k ,r ( µ ) ≤ γ 1 k ,r (1 /k ) 2 . Pr o of. W e giv e here just an outline of the pro of. A detailed one is giv en in the App endix C. The in terv al of µ is divided in to t wo parts [0 , 1 / 2] and [1 / 2 , 1] where the function Γ 1 k , 1 k ,r is bounded from ab o ve using t wo differen t techniques. First for µ ∈ [0 , 1 / 2]: In this interv al, we use mainly the fact that for some a ∈ [0 , 1 / 2], 1 − µ − µ 2 ≤ l a = 1 − a − a 2 for any µ ∈ [ a, 1 / 2]. Let τ a ( µ ) = log k log  k 1 − k ( k − 1 − µ ) k − 2 ( k l a − 1)  − k log t 1 /k ( µ ) τ a ( µ ) b ounds from ab o v e k log Γ 1 k , 1 k , log k k ( µ ) in the interv al [ a, 1 / 2]. It can b e shown that τ a ( µ ) is strictly increasing in the ab ov e interv al. Beginning with a 0 = 0, w e find a v alue a 1 suc h that τ a 0 ( a 1 ) < 2 k log γ 1 k ,r (1 /k ) proving the desired inequalit y for µ ∈ [ a 0 , a 1 ]. W e rep eat the same with τ a 1 and find a a 2 and so on... until an a i ≥ 1 / 2 whic h finishes this part of the pro of. In fact, only tw o steps are sufficient with a 1 = 0 . 15. Second for µ ∈ [1 / 2 , 1]: Recall that k log Γ 1 k , 1 k , log k k ( µ ) = log k log G 1 k , 1 k ( µ ) − k log t 1 /k ( µ ). W e pro ve first separately that the deriv atives of log G 1 k , 1 k ( µ ) and − log t 1 /k ( µ ) are conca ve in the whole considered interv al. Then we split the ab o ve interv al in tw o parts ]1 / 2 , 1 − 1 /k [ and ]1 − 1 /k , 1]. Considering their concavit y , b oth functions can b e b ound from b elo w in the first in terv al b y the linear functions represen ting the c hords joining the tw o p oints corresp onding to the tw o b ounds of the interv al. The sum of these tw o linear functions b eing p ositiv e, this prov es that the deriv ative of k log Γ 1 k , 1 k , log k k ( µ ) is p ositiv e in the first interv al and then that the v alue of the function at µ = 1 − 1 /k is maximum. F or the second interv al, the functions are b ounded from ab o v e by the linear functions represen ting the tangen t lines at µ = 1 − 1 /k . The sum of these t wo linear functions b eing negativ e, the deriv ativ e of k log Γ 1 k , 1 k , log k k ( µ ) is negative in the second interv al and then µ = 1 − 1 /k is also the maxim um in the second interv al. Summing up, Γ 1 k , 1 k , log k k (1 − 1 /k ) = γ 1 k ,r (1 /k ) 2 is the maximum of Γ 1 k , 1 k , log k k ( µ ) within [1 / 2 , 1].  3.1. A general upp er b ound for p ositive 1 -in- k -SA T. X is the random v ariable asso ciating to each 1 k ( m, n ) the num b er of its solutions.. W e hav e: E X = n X p =0  n p   k p n  1 − p n  k − 1  rn ∼  max δ ∈ [0 , 1] ( γ 1 k ,r ( δ ))  n poly ( n ) F or the upp er b ound, we prov e that for k ≥ 7 and r = log 2 k /k , max δ ∈ [0 , 1] ( γ 1 k ( δ )) < 1. This is the purp ose of the follo wing F act. F act 2. for k ≥ 7 , max δ ∈ [0 , 1]  γ 1 k , log 2 k/k ( δ )  < 1 . Pr o of. W e prov e it first in the in terv al δ ∈ [1 / 2 , 1]. Both g 1 k ( δ ) r and 1 δ δ (1 − δ ) 1 − δ are decreasing in δ in this interv al. γ 1 k ,r (1 / 2) = 2 g 1 k (1 / 2) r = 2  k 2 k  log 2 k/k < 1 then γ 1 k , log 2 k/k ( δ ) < 1 in the in terv al δ ∈ [1 / 2 , 1] . g 1 k ( δ ) = k δ (1 − δ ) k − 1 increases from 0 until δ = 1 /k . In the same in terv al δ δ (1 − δ ) 1 − δ is decreasing. Then γ 1 k , log 2 k/k ( δ ) = g 1 k ( δ ) log 2 k/k δ δ (1 − δ ) 1 − δ ≤ g 1 k (1 /k ) log 2 k/k (1 − 1 /k ) 1 − 1 /k (1 /k ) 1 /k ≤ k e (-1+k) (-1+log k) (1+logk) k < 1 for k ≥ 7. SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 9 It remains to handle the function within the in terv al [1 /k, 1 / 2]. Since log 1 δ δ (1 − δ ) 1 − δ is concav e it can b e b ound from ab o ve b y the line of slop e its deriv ative 1 δ δ (1 − δ ) 1 − δ ≤ ( − 1 + k ) − 1+ δ k . g 1 k ( δ ) log 2 k/k δ δ (1 − δ ) 1 − δ ≤ ( − 1 + k ) − 1+ δ k g 1 k ( δ ) log 2 k/k . ( − 1 + k ) − 1+ δ k g 1 k (1 /k ) log 2 k/k is less than 1 within [1 /k , s ] where s = − log (1 − 1 /k )  ( k − 1) log ( k ) 2 − k  / ( k log ( k − 1)). Finally , we b ound from abov e log g 1 k ( δ ) b y log g 0 1 k ( s ) ( δ − s ) + log g 1 k ( s ). The upp er b ound is less than 1 for k > 7 in [ s, 1 / 2].  The application of Marko v inequality finishes the pro of of the upp er bound. 4. Discussion An y elemen t of ∆ I k is necessary and sufficien t to make the second momen t metho d to b e successful as stated by theorem 1. An interesting question that is raised b y the fact that ∆ I k ma y hav e many v alues is : what v alue gives the b etter lo w er b ound? Precisely , is there a simple criterion that p ermits to select the δ ∈ ∆ I k that gives the best low er b ound? In the example of Figure 1, the function g I 13 is represented for I 13 = { 1 , 8 , 12 } . It has three lo cal maxima and so ∆ I 13 = { δ 1 , δ 2 , δ 3 } with g I 13 ( δ 2 ) < g I 13 ( δ 1 ) < g I 13 ( δ 3 ). As said b efore, the second momen t method succeeds only for those three v alues of δ . An immediate candidate for this choice of the b est v alue could b e δ 3 since it is the one for which the probabilit y of satisfying a randomly selected constraint is maximum. In fact, the b est lo wer b ound is obtained using δ 2 . The latter is the one that maximizes the first moment of X δ i.e. that corresp onds to max δ ∈ ∆ I k ( γ I k ( δ )). Since γ I k ( δ ) = δ − δ (1 − δ ) δ − 1 g I k ( δ ), the entrop y term δ − δ (1 − δ ) δ − 1 cen tered on 1 / 2 tends to fav or v alues of δ near 1 / 2. W e hav e v erified this fact for many problems. W e conjecture that for any problem defined b y the set I k , the b est v alue of δ for the second moment method is the δ ∗ ∈ ∆ I k that maximizes γ I k ( δ ). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 ! ! 1 ! 2 ! 3 g I 13 " I 13 Figure 1. An example of the functions g I k and γ I k for I 13 = { 1 , 8 , 12 } for r = 0 . 64. References [1] D Achlioptas and Y P eres. The Threshold for Random k- { SA T } is 2ˆ {\ mbox { k }} ln2 - O(k). J. A mer. Math. So c. , 17(4):947–973, 2004. [2] Y acine Boufkhad and Thomas Hugel. Non Uniform Selection of Solutions for Upp er Bounding the 3-SA T Threshold. In SA T , pages 99–112, 2010. [3] V Ch v´ atal and B Reed. Mick Gets Some (the Odds Are on His Side). In FOCS , pages 620–627, 1992. 10 Y. BOUFKHAD AND O. DUBOIS [4] N Creignou and H Daud´ e. Generalized satisfiability problems: minimal elemen ts and phase transitions. The or. Comput. Sci. , 1-3(302):417–430, 2003. [5] N Creignou and H Daud´ e. Combinatorial sharpness criterion and phase transition classification for random CSPs. Inf. Comput. , 190(2):220–238, 2004. [6] Nadia Creignou and Herv´ e Daud´ e. The SA T-UNSA T transition for random constraint satisfaction problems. Discr ete Mathematics , 309(8):2085–2099, 2009. [7] W F ernandez de la V ega. On random 2-SA T 1992. unpublished man uscript. [8] Josep D ´ ıaz, Lefteris M Kirousis, Dieter Mitsc he, and Xavier P´ erez-Gim´ enez. On the satisfiabilit y threshold of formulas with three literals p er clause. The or. Comput. Sci. , 410(30-32):2920–2934, 2009. [9] O Dubois, Y Boufkhad, and J Mandler. Typical random 3- { SA T } formulae and the satisfiability threshold. In Symp osium on Discr ete A lgorithms , pages 126–127, 2000. [10] O Dubois, Y Boufkhad, and J Mandler. Typical random 3- { SA T } formulae and the satisfiability threshold. T echnical Report TR03-007, ECCC, 2003. [11] O Dub ois and J Mandler. The 3- { X ORSA T } threshold. In Pr o c e e dings of the 43rd Annual IEEE Symp osium on F oundations of Computer Scienc e, FOCS’2002 (V anc ouver, BC, Canada, Novemb er 16-19, 2002) , pages 769–778, Los Alamitos-W ashington-Brussels-T okyo, 2002. IEEE Computer So ciet y , IEEE Computer Society Press. [12] E F riedgut. Sharp thresholds for graph properties and the k- { SA T } problem. J. Amer. Math. So c. , 12:1017– 1054, 1999. [13] Ehud F riedgut and Gil Kalai. Every Monotone Graph Prop erty has a Sharp Threshold. Pro c e e dings of the Americ an Mathematic al So ciety , 124(10):2993–3002, 1996. [14] A Goerdt. A Threshold for Unsatisfiability . J. Comput. Syst. Sci. , 53(3):469–486, 1996. [15] V amsi Kalapala and Cris Mo ore. The Phase T ransition in Exact Cov er. CoRR , abs/cs/050, 2005. [16] A C Kaporis, L M Kirousis, and E G Lalas. The probabilistic analysis of a greedy satisfiability algorithm. R andom Struct. Algorithms , 28(4):444–480, 2006. [17] Michael Molloy . Mo dels and thresholds for random constraint satisfaction problems. In STOC , pages 209–217, 2002. SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 11 Appendix A. Proof of the equality (10) Pr o of. Considering (7) : G 0 I k ,δ (1 − δ ) = X i ∈ I k X j ∈ I k φ 0 i,j,δ (1 − δ ) = X i ∈ I k X j ∈ I k X d  i d  k − i j − d  κ 0 i,j,d,δ (1 − δ ) Recall that κ i,j,d,δ ( µ ) = ((1 − µ ) δ ) d ( µδ ) i + j − 2 d ((1 − δ − µδ )) k − i − j + d and then κ 0 i,j,d,δ ( µ ) = κ i,j,d,δ ( µ )  − d 1 − µ + i + j − 2 d µ − δ ( k − i − j + d ) (1 − δ − δ µ )  Noting that κ i,j,d,δ (1 − δ ) = δ i + j (1 − δ ) 2 k − i − j w e get: φ 0 i,j,δ (1 − δ ) = δ i + j (1 − δ ) 2 k − i − j  k i  P d  i d  k − i j − d   − d δ + i + j − 2 d 1 − δ − δ ( k − i − j + d ) (1 − δ ) 2  . Using the mean of the h yp ergeometric distribution of parameters k , i and j ( P j d =0 d  i d  k − i j − d  /  k j  = ij k ) and V andermonde iden tit y , we get: φ 0 i,j,δ (1 − δ ) = δ i + j (1 − δ ) 2 k − i − j  k i  k j   − ij kδ + i + j − 2 ij /k 1 − δ − δ ( k − i − j + ij /k ) (1 − δ ) 2  . Denoting the quantit y h I k ( δ ) = P i ∈ I i  k i  δ i (1 − δ ) k − i X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  X d  i d  k − i j − d  ij = h I k ( δ ) 2 X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  X d  i d  k − i j − d  ij = h I k ( δ ) g I k ( δ ) X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  X d  i d  k − i j − d  = g I k ( δ ) 2 G 0 I k ,δ (1 − δ ) = X i ∈ I k X j ∈ I k δ i + j (1 − δ ) 2 k − i − j  k i  k j  − ij k δ + i + j − 2 ij /k 1 − δ − δ ( k − i − j + ij /k ) (1 − δ ) 2 ! = − h I k ( δ ) 2 k δ + 2 h I k ( δ ) g I k ( δ ) − 2 h I k ( δ ) 2 /k 1 − δ − δ  k g 2 I k ( δ ) − 2 h I k ( δ ) g I k ( δ ) + h I k ( δ ) 2 /k  (1 − δ ) 2 = − ( h I k ( δ ) − k δ g I k ( δ )) 2 δ (1 − δ ) 2 Noting that b ecause of: g 0 I k ( δ ) = X i ∈ I  k i  δ i (1 − δ ) k − i  i δ − k − i 1 − δ  = 1 δ h I k ( δ ) − k g I k ( δ ) h I k ( δ ) − k δ g I k ( δ ) = δ g 0 I k ( δ ) allo wing for the desired relation.  12 Y. BOUFKHAD AND O. DUBOIS Appendix B. Proof of Lemma 3 Pr o of. The second deriv ative of log (Γ I k ,δ,r ( µ )) is: (log (Γ I k ,δ,r ( µ ))) 00 =  − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δ µ  + r (log ( G I k ,δ ( µ ))) 00 =  − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δ µ  + r G I k ,δ ( µ ) G 00 I k ,δ ( µ ) − G 0 I k ,δ ( µ ) 2 G I k ,δ ( µ ) 2 It is easy to c hec k that the second deriv ative of − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δµ is negative and that its deriv ative tends to ∞ when µ tends to 0 and to −∞ on the other side then − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δµ increases from −∞ attains a maximum at a negativ e v alue then decreases to −∞ . Let − ρ ( ρ > 0) b e its maxim um v alue. In the second part G I k ,δ ( µ ) 2 is b ounded and strictly p ositiv e. Indeed it is formed b y a sum of p ositiv e terms some of which are strictly p ositiv e. Indeed all κ i,j,d,δ ( µ ) > 0 for ev ery µ ∈ ]0 , min  1 , 1 − δ δ  [. Moreov er, κ i,i,i,δ (0) > 0 and if δ ≤ 1 / 2 κ i,i, 0 ,δ (1) > 0 otherwise κ i,i, 2 i − k,δ ((1 − δ ) /δ ) > 0. G I k ,δ ( µ ) G 00 I k ,δ ( µ ) − G 0 I k ,δ ( µ ) 2 is a p olynomial in µ . It is also b ounded for µ ∈ [0 , min  1 , 1 − δ δ  ]. The second part hav e no singular p oin t and it it b ounded. Le ν b e its maxim um v alue. W e prov e that ν > 0. W e know thanks to Lemma 2 that G 0 I k ,δ (1 − δ ) = 0. Moreov er the second deriv ative G 00 I k ,δ (1 − δ ) = δ 2 g 00 I k ( δ ) 2 k ( k − 1) and as seen before G I k ,δ (1 − δ ) = g I k ( δ ) 2 . W e deduce that ν > 0. Indeed, ν ≥ G I k ,δ (1 − δ ) G 00 I k ,δ (1 − δ ) − G 0 I k ,δ (1 − δ ) 2 G I k ,δ (1 − δ ) 2 = δ 2 g 00 I k ( δ ) 2 k ( k − 1) g I k ( δ ) 2 > 0 . Finally :  − δ 1 − µ − 2 δ µ − δ 2 1 − δ − δ µ  + r G I k ,δ ( µ ) G 00 I k ,δ ( µ ) − G 0 I k ,δ ( µ ) 2 G I k ,δ ( µ ) 2 ≤ − ρ + r.ν The second deriv ative if then negativ e ov er [0 , min  1 , 1 − δ δ  ] for every r < r ∗ I k = ρ/ν .  Appendix C. Det ailed proof of Lemma 1 Pr o of. µ ∈ [ 0 , 1 / 2 ]: F or µ ∈ [0 , 1]he second deriv ative of − log t 1 /k ( µ ) is negativ e and so is the second deriv ative of log ( k − 1 − µ ). This permits to conclude that τ 0 a ( µ ) is decreasing. τ 0 a (1 / 2) = log(2 k − 3) − (2( k − 2) log ( k )) / (2 k − 3) > 0 for every k > 3 . So τ 0 a ( µ ) > 0 for µ ∈ [0 , 1 / 2] and then τ a ( µ ) is strictly increasing in the same interv al. It is easy to chec k that τ 0 (0 . 15) − 2 k log  γ 1 k , log k k  1 k   < 0 for ev ery k > 3. Consequently log Γ 1 k , 1 k , log k k ( µ ) < 2 log  γ 1 k , log k k  1 k   for every µ ∈ [0 , 0 . 15]. Similarly τ 0 . 15 (0 . 5) − 2 k log  γ 1 k , log k k (1 /k )  < 0 for an y k > 3. Concluding that log Γ 1 k , 1 k , log k k ( µ ) < 2 log  γ 1 k , log k k  1 k  in the interv al [0 , 1 2 ]. µ ∈ [ 1 / 2 , 1 [: The second deriv ative of − k log t 1 /k ( µ ) is − 1 1 − µ − 1 k − 1 − µ − 2 µ . It can b e c hec ked easily that its third deriv ative is negative in [1 / 2 , 1]. Then the first deriv ativ e of k log t 1 /k ( µ ) is conca ve. log G 1 k , 1 k ( µ ) hav e also the same prop erties. SECOND MOMENT METHOD FOR A F AMIL Y OF BOOLEAN CSP 13 The v alue of  − k log t 1 /k ( µ )  0 at the p oin t µ = 1 / 2 is log (2 k − 3). The line joining the points (1 − 1 /k , 0) to (1 / 2 , log (2 k − 3)) b ounds from below this first deriv ative. So  k log t 1 /k ( µ )  0 ≥ log(2 k − 3) − 1 2 + 1 k ( µ − 1 + 1 /k ). Similarly  log k log G 1 k , 1 k ( µ )  0 ≥ (2 − k ) log k ( k − 3 2 )( − 1 2 + 1 k ) ( µ − 1 + 1 /k ). Summing up  k log Γ 1 k , 1 k , log k k ( µ )  0 ≥  (2 − k ) log k ( k − 3 2 )( − 1 2 + 1 k ) + log(2 k − 3) − 1 2 + 1 k  ( µ − 1 + 1 /k ) ≥ 0 for µ ∈ [1 / 2 , 1 − 1 /k [. As a consequence: log Γ 1 k , 1 k , log k k ( µ ) is increasing in this in terv al and then log Γ 1 k , 1 k , log k k ( µ ) < Γ 1 k , 1 k , log k k (1 − 1 /k ) = 2 log γ 1 k , log k k (1 /k ) for µ ∈ [1 / 2 , 1 − 1 /k [. W e prov e in the following that in this interv al, k log Γ 1 k , 1 k , log k k ( µ ) is strictly decreasing. As already seen the first deriv ative of − k log t 1 /k ( µ ) is concav e and can b e b ounded from ab ov e by its tangent in 1 − 1 /k . Then  k log t 1 /k ( µ )  0 ≤ − k 3 ( k − 1) 2 ( µ − 1 + 1 /k ). log k log G 1 k , 1 k ( µ ) hav e also the same prop erties  log k log G 1 k , 1 k ( µ )  0 ≤ k 3 log k ( k − 1) 3 ( µ − 1 + 1 /k ). Summing up  k log Γ 1 k , 1 k , log k k ( µ )  0 ≤ k 3 ( k − 1) 2  log k k − 1 − 1  ( µ − 1 + 1 /k ) < 0 for k ≥ 3. As a conse- quence: log Γ 1 k , 1 k , log k k ( µ ) < Γ 1 k , 1 k , log k k (1 − 1 /k ) = 2 log  γ 1 k , log k k (1 /k )  for µ ∈ ]1 − 1 /k , 1].  (Y. Boufkhad) LIAF A, Universit ´ e P aris Diderot, CNRS UMR 7089 P aris, France E-mail address : Yacine.Boufkhad@liafa.jussieu.fr (O. Dubois) LIP6, CNRS UMR7606, Universit ´ e Pierre et Marie Curie P aris, France E-mail address : Olivier.Dubois@lip6.fr