A Note on Random k-sat for Moderately Growing k

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1 A Noe o Radom k-sat for Moderaely Growig k Ju Liu LMIB ad School of Mahemaics ad Sysems Sciece, Beihag Uiversiy, Beijig, , P.R. Chia juliu@smss.buaa.edu.c Zogsheg Gao LMIB ad School of Mahemaics ad Sysems Sciece, Beihag Uiversiy, Beijig, , P.R. Chia zshgao@buaa.edu.c Ke Xu Sae Key Laboraory of Sofware Developme Evirome, Deparme of Compuer Sciece, Beihag Uiversiy, Beijig, , P.R. Chia kexu@lsde.buaa.edu.c Submied: Aug 1, 011; Acceped: Ja 15, 01; Published: Ja 1, 01 Mahemaics Subjec Classificaios: 05D40, 68Q5 Absrac Cosider a radom isace I of k-sat wih variables ad m clauses. Suppose ha θ, c > 0 are ay fixed real umbers. Le k = k 1 + θ log. We prove ha 1 m 1 c lim P ri is saifiable = k l 0 m 1 + c k l. Keywords: k-sat, phase rasiio, he secod mome mehod. 1 Iroducio Le C k V be he se of all possible k k k-clauses o V, where a k-clause is a disjucio of k boolea variables or heir egaios ad V is a se of boolea variables. A radom isace I of k-sat is formed by selecig uiformly, idepedely ad wih replaceme m clauses from C k V ad akig heir cojucio [1, 3]. A. Frieze ad N.C. Wormald [3] proved he followig resul. he elecroic joural of combiaorics 19 01, #P4 1

2 Theorem A. Suppose k log. Le m 0 = l ad le ε l1 k > 0 be such ha ε. T he 1 m 1 ε m lim P ri is saisfiable = 0 0 m 1 + ε m 0. No log aferwards, A. Coja-Oghla ad A. Frieze [] proved he followig resul. Theorem B. Suppose k log bu k log = ol. Le m = k l + c for a absolue cosa c. The lim P ri is saisfiable = 1 e e c. For a lo of radom Cosrai Saisfacio Problem CSP for shor models, he secod mome mehod is haressed o esimae he desired lower bouds o he saisfiabiliy hreshold. Ulimaely, we ofe eed o boud sums which have commo srucure of =0 p 1 p Z, m. Take he radom CSP model proposed i [3] for example, specifically, he sum is 4. Le Gτ = gτ m, 1 ad he global maximum is G max = Gτ max. I [3], Frieze ad Wormald esimaed 4 by locaig he global maximum G max = Gτ max, ad he esimaig he coribuio of he erms close o τ max by G max. I his paper, by usig he properies of he Gamma Fucio Γ ad he iequaliy 1 [4] < 1 +1 i=1 γ l < 1 where γ is Euler-Mascheroi Cosa, we ca i aalyze he moooiciy of G very close o τ max. Thus, we ca divide he ifiiely small eighbourhood of τ max io several smaller iervals, he esimae he coribuio of each ierval, respecively, by usig he moooiciy of G. Theorem 1. Suppose ha θ, c > 0 are ay fixed real umbers. Le k 1 + θ log ad le m 0 = l l1 k. T he lim P ri is saisfiable = 1 m 1 c m0 0 m 1 + c m0. I his oe log x meas log x, ad l x meas he aural logarihm. Proof of Theorem 1 Le X = XI be he umber of saisfyig assigmes for I ad le τ =. The [3] E[X] = 1 k m, 1 E[X ] = 1 k + k τ k m. 3 =0 he elecroic joural of combiaorics 19 01, #P4

3 Simple calculaio yields E[X ] E[X] = =0 where gτ = τ k 1 k 1 1 k The upper boud: By simple calculaio k. E[X] = 1 k m 1 k 1+ c gτ m, 4 l l1 k = c, by he Markov Iequaliy P ri is saisfiable EX, lim P ri is saisfiable = 0 whe m 1 + c m 0. The lower boud: Sice θ > 0 is a arbirarily small cosa, we require ha θ < 1 4 i he followig of his paper. Le = 1 + θ log, which is he smalles clause legh permied. Le m 1 = 1 c l. Defie a pariio of he ierval [0, 1]: τ 1 = /, where ζ = 1 θ ; ζ τ = 1 l ; τ 3 = 1 α, where α 0, l1+θ is a cosa; τ 4 = 1 1 ; τ 5 = 1 1 1/+θ ad τ 6 = 1 1. We require ha m 1 specified. E[X ] E[X] c m0 i he followig of his paper uless oherwise.1. A rough esimae. Firs we will give a rough upper boud for he sum i 4, which is easier o aalyse. Lemma 1. Le Φ = [ ] fτ m 1, where fτ = τ 1. T he o1 =/ Φ. To prove Lemma 1, firs we will give he followig wo claims. Claim 1 is used o prove Claim, ad Claim is used o prove Lemma 1. Claim 1. F or ay posiive real umber x, 1 + x l1 + x < x + x /. Claim. Le ϕ r x = x 1 l1 + x 1+r + x +r x. T he here exiss a cosa ε i.e., idepede of r, such ha for ay r [0, 1], ϕ rx > 0, x 0, ε. Proof. For ay r [0, 1], defie u r ad v r o 0, + as u r x = x r 1 + x r 1, Wih Claim 1 i mid, ad oe ha u r > 0, he v r x = rx r rx r x u r l1 + x u r < x u r + x 4 u r/. 6 he elecroic joural of combiaorics 19 01, #P4 3

4 Simple calculaio yields ϕ r = u r + v r 1 + x u r l1 + x u r x. 7 By 6 ad 7, ϕ r > v r x u r/ 1 + x u r. 8 For ay x > 0, defie u x r = u r x ad v x r = v r x o [0, 1]. The u x = 1 + xx r 1 l x, v x = [ 1 + x1 + l x + r1 + x l x ] x r 1. 9 Noe ha lim x 0 + x1 + l x = 0, here exiss a cosa ε 0, 4 1 i.e., idepede of r such ha v x 0 > 0, x 0, ε. u x < 0, x 0, ε. The for ay x 0, ε i For ay r [ 0, ] 1, u r x = u x r u x 0 = x, 1 } v r x = v x r mi v x 0, v x = Noe ha x < ε < 1, by 8 ad 10, 4 ϕ r > 1 x / x = x / > / 4 > x u r 1 + x u r 1 + x u r ii Keep x < 1 4 i mid, he for ay r 1, 1], 1 u r x = u x r u x = 1 + x 1 < 1, x x 1 } v r x = v x r mi v x, v x 1 = 4x. 1 By 8 ad 1, ϕ r > 4x x 1 x / 1 + x u r > Proof of Lemma 1. For ay /, gτ 1. By 4, E[X ] E[X] 1 =/ gτ m. 14 he elecroic joural of combiaorics 19 01, #P4 4

5 Keep Claim i mid, ad oe ha m 1 c m0 < 1 c k l, he E[X ] E[X] 1 =/ 1 gτ c k l, where 1 gτ c k l = [1 + 1 τ k ] k k + O 1 c k l k k = [1 + τ k 1 ] 1 } 1 c k l + o k k k k = 1 + o [ 1 + τ k 1 ]} 1 c k l k k k = 1 + o1 exp 1 c [ 1 } ϕ log τ ] l k 1 + o1 exp 1 c [ 1 } ϕ log τ ] l = 1 + o1 fτ m 1, where o1 is idepede of τ i.e., idepede of. The E[X ] E[X] o1 =/ Φ..3. The moooiciy of Φ. Geerally, he geeral erm of he sum i 4, Gτ, as defied i 1, has wo local maxima, oe approaches 1, ad he oher approaches 1 see [3]. We ca regard 1 ad 1 as sigulariies of G, sice he proporio of each erm ad moooiciy of erms close o he wo pois chage suddely, also he sum i 4 is mosly coribued by o erms very close o he wo pois. I his secio, by sudyig he moooiciy of G, we show some asympoic srucure of he fucio close o is sigulariies, ad hus yields Theorem 1. Lemma. Defie Φ c o [0, ] as Φ c x = Γ + 1 x m1, Γx + 1Γ x + 1 f 15 where Γ is Gamma Fucio. The Φ c = Φ, = 1,,..., ad Φ c < 0, x τ 1, τ ; Φ c > 0, x τ 3, τ 4 ; Φ c < 0, x τ 5, τ he elecroic joural of combiaorics 19 01, #P4 5

6 Proof. Takig he logarihm of boh sides of 15, ad differeiaig, [ x [l Φ c ] = [l Γx + 1] [l Γ x + 1] + m 1 l f. 17 ] We ca rewrie 17 as where Φ c Φ c = Ax + Bx, 18 Ax = Γ x + 1 Γx Γ x + 1 Γ x + 1, Bx = m 1 [ l f For ay real posiive umber x, where γ is Euler-Mascheroi Cosa. If x is a Ieger = 0, 1,,..., he where [4] By 0 ad 1, x ]. x Γ Γx = 1 x + γ + 1 i + x 1, 19 i i=1 Γ + 1 Γ + 1 = γ l 1 < γ i=1 i=1 1 i, 0 1 i < l l τ 1 + R < A < l τ 1 + R + 1, where R = , = 1,,,. Simple calculaio yields Bx = l + o1 x 1. 3 τ Choose a cosa ξ such ha ζ < 1 + θ log ξ < 1, he ξ 1, 1. Defie = 1 l. he elecroic joural of combiaorics 19 01, #P4 6

7 By 19, Ax is decreasig i 0,, hece we ca hadle τ 1, ξ, τ, τ, τ 3, τ 4, τ 5, τ 6 as iegers i he followig. Noe ha R < R + 1, = 1,,...,. i If x τ 1, τ, he a If x τ 1, ξ, by 1 1 Ax Aτ 1 l 1 + Rτ l 1 + Rξ + 1 τ 1 τ 1 1 ζ 1 = l + O ζ + O = 1 ζ + o 1 ζ, ζ Bx l + o1 ξ = l + o θ log ξ = o. 4 ζ The Φ c < 0 by 18 ad 4. b If x [ξ, τ ], by 1 1 Ax Aξ l ξ 1 + Rξ + 1 = l ξ 1 + o1, Bx l + o1 τ = l + o1 1 l = l + o1 exp = l + o1 1 l 1 l } l + o1 exp l } = o1. 5 Noe ha ξ 1, 1, he l 1 ξ 1 < 0. The Φ c < 0 by 18 ad 5. c If x τ, τ, by 1 Ax Aτ l τ 1 + Rτ + 1 l l + R = 1 + o1 l, 1 l Bx l + o1 τ = l + o1 1 l = l + o1 exp l 1 l } l + o1 exp l } = l + o1. 6 The Φ c < 0 by 18 ad 6. Φ c < 0, x τ 1, τ follows from a, b ad c. ii If x τ 3, τ 4, by 1 Ax Aτ 4 l 1 l τ 4, Bx l + o1 [ 1 ] τ3 = + θ e α + o1 l. 7 he elecroic joural of combiaorics 19 01, #P4 7

8 Noe ha α < l1 + θ, hece 1 + θe α 1 > 0, he Φ c > 0 follows from 18 ad 7. iii If x τ 5, τ 6, by 1 Ax Aτ 5 l 1 τ 5 Bx l + o1 = 1 + R = + θ + o1 l, 1 + θ + o1 l. 8 Hece Φ c < 0 by 18 ad 8. I order o simplify he proof of he followig several Lemmas, we iroduce he followig hree claims. Claim 3. = oψτ provided ha τ1 τ as, where Ψτ = 1/τ τ 1 τ 1 τ. Proof. By usig Sirlig s formula, ad oe ha τ1 τ is equivale o τ ad 1 τ, he = 1 + o1 1 τ πτ1 τ τ τ τ = τ oψτ. 9 Claim 4. fτ 1 m 1 = 1 + o1. Proof. Noe ha 1 3 = o 1 m 1 ad l1 + x < x, x 0, +, he fτ 1 = [ 1 + τ 1 1 ] = [ [ exp l } ζ } ] exp 1 ζ ] ζ + o 1 m 1 1 = 1 + O + o ζ m 1 1 = 1 + O + o 1+θ/ m 1 1 = 1 + o. 30 m 1 Noe ha fτ 1 > 1, by 30, fτ 1 m 1 = 1 + o1. Claim 4 solves he crucial puzzle of esimaig he sum i 4 close o 1 successfully. As he claim shows, τ 1 is a urig poi, by i, we divide he eighborhood of 1 io wo pars, ad he esimae he wo pars separaely see Claim 5, Lemmas 3 ad 6. If k 1 log k as, he he sum i 4 diverges ad he secod mome mehod failed o obai orivial resul. The reasos are as follows: he elecroic joural of combiaorics 19 01, #P4 8

9 Arbirarily fix wo posiive umbers h > h 1. Le τ = 1 + h, h h 1, h. The gτ = k τ k 1 k 1 1 k k 1 + h k 1 = k exp k l hk exp 4 k 1 + hk } h } 1 4 k 1 + h 1k 1 k. 31 Hece gτ m diverges i he ierval 1 + h 1, 1 + h uiformly as. O he oher had, by he de Moivre-Laplace heorem lim Hece he sum i 4 diverges. Claim 5. Φτ 1 = o h = 1 + h 1 Proof. Noe ha 3ζ > 1 follows from θ < 1 4, he l [ Ψτ 1 ] = = [1 + 1 ζ = 1 h e x dx. 3 π h 1 l ζ 1 ζ 1 ζ [ ζ 1 ζ 1 ζ + l 1 1 ] ζ 1 ζ 1 ζ ] + o1 = θ + o1. By Claims 3 ad 4, Φτ 1 = o Ψτ 1 = o exp } θ = o 1. Lemma 3. τ =τ 1 Φ = o1. Proof. By Lemma ad Claim 5, τ =τ 1 Φ Lemma 4. τ 4 =τ 3 Φ = o1. τ =τ 1 Φτ 1 Φτ 1 = o1. he elecroic joural of combiaorics 19 01, #P4 9

10 Proof. Noe ha l Ψτ 4 = 1 + o1 l, he Ψτ 4 = 1 +o1 log. 33 τ 4 = exp l 1 1 } exp } = 1 + o 1, he fτ [ o 1 ] = o. 34 The fτ 4 m 1 [ ] m1 + o = exp m 1 l [ ]} + o m1 exp = exp 1 1 } + o 1 c 1 } + o l exp 1 } l =. 35 By Lemma ad Claim 3, 33 ad 35, τ 4 Φ Φτ 4 Ψτ 4 fτ 4 m1 θ+o1 log = o1. =τ 3 Lemma 5. 1 =τ 5 Φ = o1. Proof. Noe ha Φτ 5 = o 1 see 47, by Lemma 1 =τ 5 Φ 1 =τ 5 Φτ 5 Φτ 5 = o Bouds of he sum i 4 i oher iervals. To boud he sum i 4, excep for he wo ifiiely small eighbourhoods of 1 ad 1, radiioal mehods, such as Sirlig s formula, he moooiciy of, ec., are eough o deal wih i. Lemma 6. τ 1 =/ Φ 1 + o1. he elecroic joural of combiaorics 19 01, #P4 10

11 Proof. Keep Claim 4 i mid, he τ 1 =/ τ 1 Φ = Lemma 7. τ 3 =τ Φ = o1. =/ = 1 + o1 τ 1 τ 1 fτ m 1 =/ =/ 1 + o1. fτ 1 m 1 Proof. lim τ 1 Ψτ = 1, hece Ψτ = o1. By Claim 3, Ψτ = o1. 37 τ Choose a cosa ε > 0 such ha e α < 1 ε, he The fτ 3 = [ 1 + τ 3 1 ] = 1 + e α + o ε. 38 By 37 ad 39, ε fτ m 1 fτ 3 m 1 } = exp l l exp 1 ε l } = 1 ε. 39 τ 3 Lemma 8. τ 5 =τ 4 Φ = o1. } l ε =τ Φ o1 1 ε = ε+o1 = o1. 40 Proof. For ay τ [τ 4, τ 5 ], here exiss a uique β [0, θ] such ha τ = 1 1. The 1/+β l Ψτ = 1 + β + o1 l, hece 1/+β Ψτ = 1 log +β+o1 β. 41 τ = 1 1 = exp 1/+β exp l 1 1 } 1/+β } = 1 1 1/+β + o. 4 1/+β he elecroic joural of combiaorics 19 01, #P4 11

12 The The fτ [ /+β + o 1 1 ] = [ 1 1/+β + o 1 ]. 43 fτ m 1 [ = exp m 1 l 1 ] + o m1 1/+β [ ]} + o 1/+β m1 exp 1 1 } + o 1/+β = c+ β +o1. 44 Defie Λ o [0, θ] as Λβ = log[ Ψτ fτ m 1 ]. 45 By 41 ad 44, Λβ c + β θ + o1 log β + o1 c + o1. 46 By 9, = oψτ is uiformly for all τ 4 τ 5. By 45 ad 46, Φ c+o1. 47 Hece τ 5 Φ c+o1 = o1. 48 =τ 4 Lemma 9. Φ = o1. Proof. Φ = f1 m 1 = = exp = exp m } exp m } m 1 l 1 c } l = c+o1 = o1. The proof of he lower boud ow follows from lemmas 1, 3, 4, 5, 6, 7, 8 ad 9. he elecroic joural of combiaorics 19 01, #P4 1

13 Ackowledgmes This research was suppored by Naioal Naural Sciece Fud of Chia Gra No , Refereces [1] D. Achliopas ad Y. Peres, The hreshold for radom k-sat is k log Ok, J Amer Mah Soc , [] A. Coja-Oghla ad Ala Frieze, Radom k-sat: he limiig probabiliy for saisfiabiliy for moderaely growig k, E-JC , #N. [3] A. Frieze ad N.C. Wormald, Radom k-sat: A igh hreshold for moderaely growig k, Combiaorica 5 005, [4] R.M. Youg, Euler s Cosa, Mah Gaz , he elecroic joural of combiaorics 19 01, #P4 13