Bounds on Thresholds Related to Maximum Satisfiability of Regular Random Formulas

Bounds on Thresholds Relate d to Maximum Satisfiability of Re gular Rando m F ormulas V ishwambhar Rathi ∗ † , Erik Aurell ∗ ‡ , La rs Rasmus sen ∗ † , Mikae l Skoglund ∗ † { vish@, eaurell@, lars.rasmusse n@ee, skoglund@ee } .kth.se ∗ School of Electrical Eng ineering † KTH Linna eus Centre ACCESS KTH-Royal Institute o f T echno logy , Stockholm, Sweden ‡ Dept. Informatio n and Comp uter Science, TKK-Helsinki University of T echnology , Espoo, Finland Abstract —W e consider the regular balanced model of f ormula generation in conjunctiv e normal fo rm (CNF) introduced by Boufkhad, Dubois, Interian, and Selman. W e say that a fo rmula is p -satisfying if ther e is a truth assignment sa tisfying 1 − 2 − k + p 2 − k fraction of clauses. Using the first moment method we determine upper bound on the threshold clause density such that there are no p -satisfying assignments with high p robability abov e th is upper boun d. There ar e two aspects in deriving the lower bound using the second moment method. The first aspect is, given any p ∈ ( 0 , 1 ) and k , evaluate the lower bound on the threshold. This ev aluation is numerical in nature. The second aspect i s to derive the lower bound as a function of p for large enough k . W e address the first aspect an d evaluate the lower bound on the p -satisfying threshold using the second moment method. W e ob serv e that as k increases the lower bound seems to con ver ge to the asymptotically derive d lower b ound fo r uniform model of for mula generation by Achlioptas, Naor , and Peres. I . R E G U L A R F O R M U L A S A N D M O T I V AT I O N A literal of a bo olean variable is the variable itself o r its ne gation. A clause is a disjunction (OR) of k literals. A formu la is a co njunctio n ( AND) of a finite set of clauses. A k -SA T f ormula is a f ormula whe re each clause is a disjunction of k literals. A legal clause is one in wh ich the re are no repeated or complem entary literals. Using the terminolo gy of [5], we say that a for mula is simple if it consists of only legal clauses. A co nfigu ration formula is n ot necessarily legal. A satisfying (SA T) assignment of a formu la is a truth assign ment of v ariables for wh ich the formula evaluates to true i.e. a ll the clauses ev aluates to true. W e d enote the numb er of variables by n , the nu mber of clauses by m , and the clause density i.e. the ratio o f clauses to variables by α = m n . W e de note the binary entr opy fun ction by h ( x ) , − x ln ( x ) − ( 1 − x ) ln ( 1 − x ) , where the logarithm is th e n atural logarith m. The popu lar , un iform k-SA T model g enerates a f ormula by selecting unifor mly and inde penden tly m - clauses f rom th e set of all 2 k  n k  k -clau ses. In th is model, the literal degree can vary . W e are interested in the mod el whe re the literal degree is con stant, which was intro duced in [5]. Su ppose each literal has degree r . Then 2 nr = km , w hich giv es α = 2 r / k . Hence α can on ly take values fr om a discrete set o f p ossible values. This pro blem can b e circum vented by allowing each literal to take two possible v alues for a d egree. Howe ver, in this paper we consider th e case where all th e literals ha ve d egree r due to space restriction. A formu la is represented by a bip artite graph. Th e lef t vertices r epresent th e litera ls and right vertices represent the clau ses. A literal is co nnected to a clau se if it appears in the clau se. There are k α n edges coming out fr om all the literals and k α n ed ges co ming ou t fr om the clauses. W e assign the labels fr om the set E = { 1 , . . . , k α n } to edges o n both sides of the bipartite gr aph. In order to generate a formula, we gener ate a rando m p ermutatio n Π o n E . Now we co nnect an edge i on th e literal node side to an ed ge Π ( i ) on the clause node side. This gives rise to a regular ran dom k -SA T for mula. Note that no t all th e formulas g enerated by this pro cedure are simple. Ho we ver , it was shown in [5] that the threshold is same fo r th is collection o f formu las and th e collection of simple formu las. Thus, we can work with the collection of configur ation formulas gene rated by this pr ocedur e. No te that this pr ocedure is similar to the pr ocedur e of genera ting regular LDPC cod es [11]. The regular r andom k -SA T formulas are of interest because such instanc es are co mputation ally harder than the un iform k -SA T instances. This was experimenta lly observed in [5], where the au thors also d eriv ed upper and lower bound s on the satisfiability th reshold for regular rand om 3-SA T . In [9], upper bound on the satisfiability threshold for any k ≥ 3 was derived usin g the first moment method. It was shown that as k increases, the upper bou nd c onv erge to the correspo nding bound s on the thresho ld of the unifor m model [1], [3]. For unifor m mo del, in a series of bre akthrou gh papers, Achlioptas and Peres in [3] and Achliopta s and Mo ore in [1], derived almost matching lower bounds with the upp er bound by carefully applying the second moment method to balanced satisfying assignmen ts. In [1], based on their belief that the simple ap plication of second mom ent method should work for symmetric p roblem s, Achliop tas and Moore po sed the qu estion of its success f or regular ran dom k -SA T . In an attempt to answer this question, the lower bound using the second mom ent m ethod was ev a luated in [9]. As the ev alu ation of the lower boun d w as num erical in natur e (though exact), it was observed that th e lower bou nd also conver ges to th e co rrespon ding lower bound fo r th e u niform m odel as k increases. In this work we are interested in the maximu m satisfiability problem over regu lar rand om form ulas. W e say that a f ormula is p -satisfying if ther e is a truth assignment satisfy ing c ( p ) , 1 − 2 − k + p 2 − k fraction of its clauses, p ∈ ( 0 , 1 ) . Th e num ber of p -satisfying truth assignments is den oted b y N ( n , α, p ) . W e define the following qua ntities related to p -satisfiability: α ( p ) , sup { α : A r egular rando m form ula , is p - satisfiable w .h. p. } , (1) α ∗ ( p ) , inf { α : A regula r r andom formu la , is not p -satisfiable w .h.p . } . (2) Note that α ( p ) ≤ α ∗ ( p ) . In [2], for the uniform mod el Ac hliop- tas, Naor, an d Peres derived lower b ound on α ( p ) which almost matches with the u pper bound on α ∗ ( p ) derived via the first mo ment method. The lower boun d was ob tained by a careful ap plication of the second moment method . W e derive up per bo und o n α ∗ ( p ) by applying the first moment method to N ( n , α, p ) . The obtained upper bou nd matches with the correspond ing boun d f or the unifo rm model. W e e valuate a lower bo und on α ( p ) by applyin g the second moment method to N ( n , α, p ) . W e observe that f or increasing k the lower bou nd seems to con verge to the corresp onding bound for the unifo rm model. In the next section, we obtain upper bo und on α ∗ ( p ) by the first mo ment method . Due to sp ace limitatio ns, some of the argum ents are acc om- panied b y short explan ations. Further details c an be f ound in the f orthcom ing journ al sub mission [1 0]. I I . U P P E R B O U N D O N T H R E S H O L D V I A F I R S T M O M E N T Let X be a non -negative integer-v a lued random variable and E ( X ) be its expectation. Then th e first mo ment metho d giv es: P ( X > 0 ) ≤ E ( X ) . No te that b y ch oosing X to be th e number o f solution s of a ran dom formula, we can obtain an upper boun d on the thresho ld α ∗ ( p ) beyond which no p -satisfying solution exists with proba bility one. This upper bound correspon ds to th e largest value of α at which the av erage numb er of p -satisfying solution s goes to z ero as n tends to infinity . In the fo llowing lemma, we der iv e the first moment of N ( n , α, p ) for r egular rando m k -SA T f or k ≥ 2. Lemma 2.1 : Let N ( n , α, p ) be the number of p -satisfying assignments fo r a random regular k-SA T formula . Th en 1 , E ( N ( n , α, p )) = 2 n  α n c ( p ) α n   k α n 2  !  2 ( k α n ) ! × coef  s ( x ) x  c ( p ) α n , x ( k 2 − c ( p ) ) α n ! , (3) where s ( x ) = ( 1 + x ) k − 1 and coef   s ( x ) x  c ( p ) α n , x ( k 2 − c ( p ) ) α n  denotes th e coefficient of x ( k 2 − c ( p ) ) α n in th e expansion of  s ( x ) x  c ( p ) α n . Pr oof: Due to symmetry of the form ula generation, any assignment of variables has the same pro bability o f being p - satisfying. Th is implies E ( N ( n , α, p )) = 2 n P ( X = { 0 , . . . , 0 } is p -satisfying. ) . 1 W e assume that k α n is an e ven integ er . The probab ility o f the all-zero vector being p -satisfying is giv en by P ( X = { 0 , . . . , 0 } is p -satisfy ing ) = Number of formulas for which X = { 0 , . . . , 0 } is p -satisfying T otal number o f formulas . The total numb er of f ormulas is given by ( k α n ) !. The total number o f for mulas fo r which the all-zer o assignmen t is a p -satisfying is given by  α n c ( p ) α n    k α n 2  !  2 coef  s ( x ) c ( p ) α n , x k α n 2  . The binomial term correspo nds to ch oosing c ( p ) fr action of clauses bein g satisfied by the all-zero assignment. Th e factorial term s co rrespon d to permutin g the edges among true an d false li terals. Note that there are equal numbe rs of true and false literals. The gen erating fu nction s ( x ) corr e- sponds to placing at least one positive literal in a clause. W ith these results a nd ob serving that coef  s ( x ) c ( p ) α n , x k α n 2  = coef  ( s ( x ) / x ) c ( p ) α n , x ( k 2 − c ( p ) ) α n  , we obtain ( 3). Remark: The compu tation of the first m oment of N ( n , α, p ) is similar to the com putation of the first mom ent for weight distribution of regular LDPC ensembles. There is large body of work dealing with first m oment of weigh t d istribution. So, we re fer to [ 11] and the ref erences there in . W e now state the Haym an metho d to appro ximate the coef- term which is asympto tically correct [ 11]. Lemma 2.2 (Hayman Method ): Let q ( y ) = ∑ i q i y i be a polyno mial with no n-negative coef ficients such that q 0 6 = 0 and q 1 6 = 0. Define a q ( y ) = y d q ( y ) d y 1 y , b q ( y ) = y d a q ( y ) d y . (4) Then, coef ( q ( y ) n , y ω n ) = q ( y ω ) n ( y ω ) ω n p 2 π nb q ( y ω ) ( 1 + o ( 1 )) , (5) where y ω is the un ique positive solution of the saddle p oint equation a q ( y ) = ω . The solution y ω also satisfies y ω = inf y > 0 q ( y ) n y ω n . (6) W e now use Lem ma 2.2 to compute the expe ctation of the total num ber of p -satisfying assignments. Lemma 2.3 : Let N ( n , α, p ) denote the total number of p - satisfying assignmen ts of a regular random k-SA T formula. Let t ( x ) = s ( x ) x , wh ere s ( x ) is defined in Lemma 2.1. Then, E ( N ( n , α, p )) = s k 8 π c ( p ) 2 ( 1 − c ( p )) b t ( x k ) α n e n (( 1 − k α ) ln ( 2 )) e n ( α h ( c ( p ))+ c ( p ) α ln ( t ( x k )) − ( k α 2 − α c ( p ) ) ln ( x k ) ) ( 1 + o ( 1 )) , (7) where x k is the solu tion of a t ( x ) = k 2 c ( p ) − 1. The qu antity a t ( x ) and b t ( x ) are defined according to ( 4). Pr oof: Using Stirling’ s ap proxim ation for the b inomial terms (see [11, p. 513]) an d Haym an appro ximation for the coef term from Le mma 5 gi ves the desired result. In the following lemma we derive e xplicit upp er boun ds on the clause den sity f or the existence of p - satisfying assign - ments. Lemma 2.4 (Upper bo und): Let α ∗ ( p ) be as defined in (2). Define α ∗ u ( p ) to be the u pper boun d o n α ∗ ( p ) o btained by the first mo ment method . T hen, α ∗ ( p ) ≤ α ∗ u ( p ) , 2 k ln ( 2 ) p + ( 1 − p ) ln ( 1 − p ) . (8) Pr oof: Using (6), we ob tain the following u pper bo und on the exponent of E ( N ( n , α, p ))) f or any x > 0 , lim n → ∞ ln ( E ( N ( n , α ))) n ≤ ( 1 − k α ) ln ( 2 ) + α h ( c ( p ))+ c ( p ) α ln ( t ( x )) − α  k 2 − c ( p )  ln ( x ) . (9) W e substitute x = 1 in (9) and obtain the fo llowing upp er bound o n the p -satisfiability threshold , α ∗ ( p ) ≤ ln ( 2 ) k ln ( 2 ) − h ( c ( p )) − c ( p ) ln ( 2 k − 1 ) . (10) W e co mplete th e proof by showing that the denominator in (10) is lower boun ded by 2 − k ( p + ( 1 − p ) ln ( 1 − p )) . This can be easily shown by consid ering th eir d ifference an d th en showing its positivity . Note th at th e up per b ound coin cides with the u pper b ound derived for unifo rm formu las in [ 2]. In th e next section we use the s econd moment m ethod to ob tain lower b ound on α ( p ) defined in (1). I I I . S E C O N D M O M E N T In [2], almost match ing lower bounds on the p -satisfiability threshold of uniform fo rmulas were d erived using the second moment method . The second m oment method is governed by the f ollowing equation P ( X > 0 ) ≥ E ( X ) 2 E ( X 2 ) , (11) where X is a no n-negative rand om v ariable. Before we use the second m oment m ethod, we present the following theorem from [6] an d its corollary g iv en in [ 2]. Pro of of both theorem and the corollary are identical for the unifor m mod el and the regular model. Theor em 3.1 ( [6]): Let U k ( n , α ) b e th e maximu m nu mber of clauses satisfied (over all the tru th assign ments) of a r egular random f ormula with n variables and m clauses. Th en P ( | U k ( n , α ) − E ( U k ( n , α )) | > t ) < 2 exp  − 2 t 2 α n  . Cor ollary 3.1 ( [2]): Assume th at ther e exists c = c ( k , p , r ) such that for n large enough, a r egular ran dom f ormula is p -satisfiable with pro bability grea ter than n − c . Th en a regular random formula is p ′ -satisfiable with h igh probability f or e very constant p ′ < p . W e now apply the second moment method to N ( n , α, p ) . The calculation for p = 1 i.e. for n umber o f satisfy ing assignm ents was done in [9]. Our co mputation of the second m oment is inspired b y the computatio n of the second mom ent for the weight and stoppin g set distributions of regular L DPC cod es in [7], [8] (see also [4]). W e compu te the second mo ment in the next lemm a. Lemma 3.1 : Consider regular rand om satisfiability fo rmu- las with literal degree r . Then the second momen t of N ( n , α, p ) is given by E  N ( n , α, p ) 2  = n ∑ i = 0 c ( p ) α n ∑ j = α n ( c ( p ) − ( 1 − p ) 2 − k ) 2 n  n i  α n c ( p ) α n   c ( p ) α n j  α n ( 1 − c ( p )) c ( p ) α n − j  (( r ( n − i )) ! ) 2 (( ri ) ! ) 2 ( k α n ) ! coef  f ( x 1 , x 2 , x 3 ) j ( s ( x 1 ) s ( x 3 )) c ( p ) α n − j , ( x 1 x 3 ) r ( n − i ) x ri 2  , (12) where the generating fu nction f ( x 1 , x 2 , x 3 ) is given by f ( x 1 , x 2 , x 3 ) = ( 1 + x 1 + x 2 + x 3 ) k − ( 1 + x 1 ) k − ( 1 + x 3 ) k + 1 . (13) The gen erating fu nction s ( x ) is d efined as s ( x ) , ( 1 + x ) k − 1 , which is same as defined in Lemma 2.1. Pr oof: For truth assignments X an d Y , define the indicator variable 1 1 X Y which e valuates to 1 if the tru th assign ments X and Y are p -satisfying. Then , E ( N ( n , α, p ) 2 ) = ∑ X , Y ∈{ 0 , 1 } n E ( 1 1 XY ) , = 2 n ∑ Y ∈{ 0 , 1 } n P ( 0 and Y are p -satisfyin g ) . Due to the symm etry in r egular form ula generatio n, the number o f f ormulas f or which both X and Y are p -satisfy ing depend s only on the number of variables on which X and Y agree. This explain s the last simplificatio n where we fix X to be the all-zero vector . W e want to e valuate the pro bability o f the e vent that the truth assignm ents 0 an d Y p -satisfy a randomly cho sen regular formu la. This probability depend s only on the overlap , i.e., the number of variables wher e the two truth assignmen ts agree. Thus for a given overlap i , we can fix Y to b e eq ual to zero in the first i variables and equal to 1 in th e remaining v ariables i.e. Y = { 0 , · · · , 0 | {z } i times , 1 , · · · , 1 | {z } n − i times } . This giv es, E ( N ( n , α, p ) 2 ) = n ∑ i = 0 2 n  n i  P ( 0 and Y are p -satisfying ) . (14) In ord er to ev aluate the p robab ility tha t both 0 and Y are p - satisfying, define C = { 1 , · · · , α n } to be the set of clauses and C 0 and C Y to b e th e set of clauses satisfied by 0 and Y resp ectiv ely . Clearly , | C 0 | = | C Y | = c ( p ) α n . Then, P ( 0 and Y are p -satisfyin g ) = ∑ C 0 , C Y ⊂ C P ( 0 only satisfies C 0 and Y only satisfies C Y ) . (15) Again fr om the symm etry of the regular for mula gener ation, we fix C 0 = { 1 , · · · , c ( p ) α n } . For | C 0 ∩ C Y | = j , we fix C Y = { 1 , · · · , j , c ( p ) α n + 1 , . . . , 2 c ( p ) α n − j } . T his gives, P ( 0 and Y are p -satisfying ) =  α n c ( p ) α n  c ( p ) α n ∑ j = α n ( c ( p ) − ( 1 − p ) 2 − k )  c ( p ) α n j  α n ( 1 − c ( p )) c ( p ) α n − j  × P ( 0 o nly satisfies C 0 and Y o nly satisfies C Y ) . (16) Note that j ≥ α n ( c ( p ) − ( 1 − p ) 2 − k ) as | C 0 | + | C Y | − | C 0 ∩ C Y | ≤ α n . For a g iv en overlap i between 0 an d Y , we observe that ther e are fou r different ty pes of edges conn ecting the literals and the clauses. There are r ( n − i ) type 1 edges which are connected to true literals w .r . t. the 0 tru th a ssignment an d false w .r .t. to the Y tru th assignme nt. The ri type 2 edges are con nected to true literals w .r .t. both the tr uth assignm ents. There are r ( n − i ) ty pe 3 edges which are con nected to false literals w .r .t. the 0 truth assignm ent a nd true literals w .r .t. to the Y truth assignment. Th e r i type 4 ed ges are co nnected to false literals w .r .t. both the truth assignme nts. Let f ( x 1 , x 2 , x 3 ) be the generating function counting the number of possible edge conn ections to a clau se such tha t the clause is satisfied by both 0 an d Y . In f ( x 1 , x 2 , x 3 ) , the p ower of x i giv es the number of e dges of type i , i ∈ { 1 , 2 , 3 } . A clau se is satisfied by bo th 0 and Y if it is connec ted to at least one type 2 edge, else it is conn ected to at least one ty pe 1 and at least one type 3 edge. Then the gener ating fu nction f ( x 1 , x 2 , x 3 ) is given as in ( 13). Using th is, we obtain P ( 0 only satisfies C 0 and Y o nly satisfies C Y ) = (( r ( n − i )) ! ) 2 (( ri ) ! ) 2 ( k α n ) ! × coef  f ( x 1 , x 2 , x 3 ) j ( s ( x 1 ) s ( x 3 )) c ( p ) α n − j , ( x 1 x 3 ) r ( n − i ) x ri 2  , (17) where s ( x 1 ) is the generating function for clauses satisfied by 0 and not satisfied by Y ( similarly we define s ( x 3 ) ). The term ( k α n ) ! is the total nu mber of fo rmulas. Consider a gi ven formu la which is satisfied by bo th truth assignmen ts 0 and Y . If we per mute the positions of type 1 edges on the clause side, we obtain another form ula having 0 and Y as solutions. The argum ent holds true fo r the type i ed ges, i ∈ { 2 , 3 , 4 } . This explains the term ( r ( n − i )) ! in (1 7) which co rrespon ds to permutin g the type 1 edges (it is squ ared because of the same contribution from type 3 edge s). Similar ly , ( r i ) ! 2 correspo nds to p ermuting type 2 an d type 4 edges. As | C 0 ∩ C Y | = j , there are j clauses which satisfied by bo th 0 and Y . This explains the factor f ( x 1 , x 2 , x 3 ) j in the coef term. Th ere are | C 0 \ ( C 0 ∪ C Y ) | ) = | C Y \ ( C 0 ∪ C Y ) | = c ( p ) α n − j clau ses which are satisfied by 0 (r esp. Y ) and not satisfied b y Y (r esp. 0 ). This explains the factor ( s ( x 1 ) s ( x 3 )) c ( p ) α n − j in the co ef term. W e complete the proof b y substituting (17) in ( 16), then (1 6) in ( 14). In order to ev alua te th e second mom ent, we now presen t the m ultidimensio nal saddle point metho d in the n ext lemma. A detailed techn ical exposition o f th e multidim ensional saddle point m ethod can b e fou nd in Appendix D o f [11]. Theor em 3.2: Let i : = ( r ( n − i ) , r i , r ( n − i )) , x = ( x 1 , x 2 , x 3 ) , and 0 < lim n → ∞ i / n < 1 . W e define g n , j ( x ) = f ( x ) j ( s ( x 1 ) s ( x 3 )) c ( p ) α n − j , where f ( x ) and s ( x ) ar e defined in Lemma 3 .1. W e define the normalizatio ns η , i / n and γ , j / ( α n ) . Let t = ( t 1 , t 2 , t 3 ) be a po siti ve solution of the saddle po int equa tions a g ( x ) ,  x i n ∂ ln ( g n , j ( x )) ∂ x i  3 i = 1 = { r ( 1 − η ) , r η , r ( 1 − η ) } . (1 8) Then coef  g n , j ( x ) , x i  can be appro ximated as , coef  g n , j ( x ) , x i  = g n , j ( t ) ( t ) i p ( 2 π n ) 3 | B g ( t ) | ( 1 + o ( 1 )) , using the sad dle point meth od fo r multi variate polynomials, where B g ( x ) is a 3 × 3 matr ix who se elements are gi ven by B i , j = x j ∂ a gi ( x ) ∂ x j = B j , i and a gi ( x ) is the i th coordin ate of a g ( x ) . In the following theorem we deriv e lower bound on the p -satisfiability thre shold by evaluating the second moment of N ( n , α, p ) with the help o f Theo rem 3.2. Theor em 3.3: Consider regular ran dom k -SA T formu las with litera l degree r . Let S ( i , j ) denote the ( i , j ) th summation term in (12). Define the norm alization η = i / n and γ = j / ( α n ) . If S ( n / 2 , n α c ( p ) 2 ) is the do minant ter m i. e. f or η ∈ [ 0 , 1 ] , γ ∈ [( c ( p ) − 2 − k ( 1 − p )) , c ( p )] , η 6 = 1 2 , an d γ 6 = c ( p ) 2 lim n → ∞ ln  S  n 2 , n α c ( p ) 2  n > lim n → ∞ ln ( S ( η n , γ α n )) n , (19) then fo r some p ositiv e constants c , c ′ P ( N ( n , α, p ) > 0 ) ≥ c ′ n − c . (20) Let r ∗ ( p ) b e the largest literal degree for wh ich S ( n / 2 , n α c ( p ) 2 ) is the dominan t term, i.e. (1 9) ho lds. Then th e threshold α ( p ) defined in ( 1) is lower b ound ed by α ( p ) ≥ α ∗ l ( p ) , 2 r ∗ ( p ) k . Pr oof: Assuming (19) holds, then for n large enough E  N ( n , α, p ) 2  ≤ ( n + 1 )( α ( 1 − p ) 2 − k n + 1 ) S  n 2 , n α c ( p ) 2  . (21) From (12) and Theo rem 3.2, the gr owth rate of S ( η n , γ α n ) is giv en by , s ( η , γ ) , lim n → ∞ ln ( S ( η n , γ α n )) n = ( 1 − k α )( ln ( 2 ) + h ( η )) + α h ( c ( p )) + α c ( p ) h  γ c ( p )  + α ( 1 − c ( p )) h  c ( p ) − γ ( 1 − c ( p ))  + γ α ln ( f ( t 1 , t 2 , t 3 )) + α ( c ( p ) − γ ) ( ln ( s ( t 1 )) + ln ( s ( t 3 ))) − r ( 1 − η )( ln ( t 1 ) + ln ( t 3 )) − r η ln ( t 2 ) , (2 2) where t 1 , t 2 , t 3 is a po siti ve solution of th e saddle p oint equa- tions as defined in Theorem 3.2, a g ( t ) = { r ( 1 − η ) , r η , r ( 1 − η ) } . (23) In or der to compute the maximum exponen t of the summatio n terms, we compute its p artial deri vati ves with respe ct to η and γ and eq uate them to zero, which result in the fo llowing respective equations. ( 1 − k α ) ln 1 − η η + r ln  t 1 t 3 t 2  = 0 , (2 4) ln  ( c ( p ) − γ ) 2 γ ( 1 − 2 c ( p ) + γ )  + ln  f ( t 1 , t 2 , t 3 ) s ( t 1 ) s ( t 3 )  = 0 . (2 5) Note that the partial d eriv ati ves of t 1 , t 2 and t 3 w .r .t. η an d γ vanish b ecause of the saddle point equation s gi ven in (23). Every positive so lution ( t 1 , t 2 , t 3 ) of ( 23) satisfies t 1 = t 3 as (23) and f ( t 1 , t 2 , t 3 ) are symmetric in t 1 and t 3 . If η = 1 / 2 , γ = c ( p ) 2 is a m aximum, the n th e vanishing der iv ative in (24) an d equality of t 1 and t 3 imply t 2 = t 2 1 . W e substitute η = 1 / 2, t 1 = t 3 , and t 2 = t 2 1 in (23). Th is reduces (2 3) to the saddle point e quation corre sponding to the polynom ial t ( x ) d efined in Lemma 2. 3 whose solution is den ote by x k . By observing f ( x k , x 2 k , x k ) = s ( x k ) 2 , we have S  n 2 , α c ( p ) 2 n  = k 3 / 2 32 π 2 c ( p ) 2 ( 1 − c ( p )) 2 q | B g ( x k , x 2 k , x k ) | n 2 × e 2 n ( ( 1 − k α ) ln ( 2 )+ α h ( c ( p ) )+ α c ( p ) ln ( s ( x k )) − k α 2 ln ( x k ) ) ( 1 + o ( 1 )) . (26) Using the relation that t ( x ) = s ( x ) x , we note that the exponent of S ( n / 2 , α c ( p ) 2 n ) is twice the exponent of the first mo ment of the total nu mber of solutio ns as given in (7). W e substitute (26) in to (21). Th en we use Lemm a 2.3 and (21) in the secon d moment meth od: P ( N ( n , α, p ) > 0 ) ≥ E ( N ( n , α, p ) 2 ) E ( N ( n , α, p ) 2 ) , ≥ 4 π q | B g ( x k , x 2 k , x k ) | b t ( x k ) α 2 √ kn ( 1 + o ( 1 )) . Clearly , if the maximum of th e growth rate of S ( η n , γ α n ) is not achieved at η = 1 / 2 and γ = c ( p ) 2 , then the lo wer b ound gi ven by the second mome nt method con verges to zero exponentially fast. By using Coro llary 3.1, we obtain the desired lower bo und on α ( p ) . T his pr oves the theorem. In the next section we discuss the ob tained lower and u pper bound s on the p -satisfiability threshold . I V . B O U N D S O N T H R E S H O L D A N D D I S C U S S I O N In T ab le I , we compute the ratio of α ⋆ l ( p ) and α ⋆ u ( p ) for p = 0 . 1 , · · · , 0 . 9 and k = 3 , 6 , 12, where α ⋆ l ( p ) is th e lower bound on α ( p ) obtained from Theorem 3.3 and α ∗ u ( p ) is th e upper bou nd on α ∗ ( p ) d efined in Lemm a 2 .4. Note that the case p = 1 w as alrea dy solved in [9]. I n ord er to apply the second mome nt method, we have to verify that s ( η , γ ) , defined in ( 22), attains its maximu m at η = 1 2 , γ = c ( p ) 2 over [ 0 , 1 ] × [ c ( p ) − ( 1 − p ) 2 − k , c ( p )] . Th is requ ires that η = 1 2 , γ = c ( p ) 2 is the only positive solution of the system of equatio ns consisting of (23), (2 4) an d (25) which co rrespon ds to a max imum. The system of eq uations ( 23), (24), and (25) is equ i valent to a p k = 3 k = 6 k = 12 0 . 1 0 . 252 0 . 717 0 . 977 0 . 2 0 . 258 0 . 720 0 . 979 0 . 3 0 . 272 0 . 738 0 . 980 0 . 4 0 . 281 0 . 755 0 . 981 0 . 5 0 . 295 0 . 765 0 . 983 0 . 6 0 . 308 0 . 782 0 . 986 0 . 7 0 . 325 0 . 801 0 . 988 0 . 8 0 . 344 0 . 822 0 . 990 0 . 9 0 . 402 0 . 855 0 . 993 T ABLE I: V alue of the ratio α ∗ l ( p ) / α ∗ u ( p ) . system o f p olynom ial eq uations. For small values of k and p close to on e, we can solve this system o f polyno mial equations and verify the desired co nditions. For larger v alues of k , the degree of mono mials in (24) gr ows expo nentially in k . Thus, solving ( 23), ( 24), an d (25) becomes comp utationally difficult. Howe ver, s ( η , γ ) can be easily computed as its comp utation requires solving o nly (23), w here the maximum mon omial degree is linear in k . Thu s, the de sired co ndition for m aximum of s ( η, γ ) at η = 1 2 , γ = c ( p ) 2 can be verified nu merically in an efficient manner . From the T able I, we see th at as k b ecomes larger the ratio α ∗ l ( p ) /α ∗ u ( p ) gets closer to one. Ou r b elief is that indeed as k becom es larger and larger, the ratio α ∗ l ( p ) /α ∗ u ( p ) conver ges to o ne. Our main future g oal is to deriv e explicit expression for α ∗ l ( p ) as k bec omes larger . A C K N O W L E D G E M E N T This work has b een suppor ted by Swedish r esearch coun cil (VR) throu gh KTH Lin naeus cen ter A CCESS. R E F E R E N C E S [1] D . A C H L I O P TA S A N D C . M O O R E , Random k-SAT: T wo moments suffice to cr oss a sharp thre shold , SIAM J. COMPUT ., 36 (2006), pp. 740–762. [2] D . A C H L I O P TA S , A . N AO R , A N D Y . P E R E S , On the maximum satisfi- abilit y of random formula s , Journal of the Associati on of Computing Machina ry (J A CM), 54 (2007). [3] D . A C H L I O P TA S A N D Y . P E R E S , The thre shold for random k-SAT is 2 k ln ( 2 ) − O ( k ) , Journal of the American Mathema tical Society , 17 (2004), pp. 947–973. [4] O . B A R A K A N D D . B U R S H T E I N , Lower bounds on the spectrum and err or rate of LDPC code ensembles , in Internat ional Symposium on Information Theory , Adela ide, Australia, 2005. [5] Y . B O U F K H A D , O . D U B O I S , Y . I N T E R I A N , A N D B . S E L M A N , Re gular random k-SA T: P r operti es of balanced formulas , Journal of Automated Reasonin g, (2005). [6] A . Z . B R O D E R , A . M . F R I E Z E , A N D E . U P FA L , On the satisfiabili ty and maximum satisfiabil ity of random 3-CNF formulas , Proc. 4th Annual Symposium on Discrete Algorit hms, (1993), pp. 322–330. [7] V . R A T H I , On the asymptotic wei ght and stopping set d istrib utions of re gular LDPC ensembles , IEEE Trans. Inform. Theory , 52 (2006), pp. 4212–4218. [8] , Non-binary LDPC codes and EXIT lik e functions , PhD thesis, Swiss Federal Institute of T echnolo gy (EPFL), Lausanne, 2008. [9] V . R A T H I , E . A U R E L L , L . R A S M U S S E N , A N D M . S KO G L U N D , Bounds on thr esholds of re gular random k-SA T . Accepted to Interna tiona l Conferen ce on Theory and Applications of S atisfiabi lity T esting (SA T ) 2010. [10] , Satisfiabili ty and maximum satisfiability of re gular random k- sat: B ounds on threshol ds . in prepa ratio n for submission to IEEE Tra nsactio ns on Information Theory . [11] T. R I C H A R D S O N A N D R . U R B A N K E , Modern Coding Theory , Cam- bridge Univ ersity Press, 2008.