On the maximum satisfiability of random formulas

Maximum satisfiability is a canonical NP-complete problem that appears empirically hard for random instances. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random k-CNF formula. In this paper we give asymptotically tight estimates for this quantity. Let us say that a k-CNF formula is p-satisfiable if there exists a truth assignment satisfying 1-2^-k +p2^-k fraction of all clauses (every k-CNF is 0-satisfiable). Let F_k(n, m) denote a random k-CNF formula on n variables formed by selecting uniformly, independently and with replacement m out of all (2n)^k possible k-clauses. Finally, let τ(p) = 2^kln 2/(p + (1 - p) ln(1 - p)). It is easy to prove that for every k ≥ 2 and p ∈ (0, 1), if r ≥ τ (p) then the probability that F_k(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δ_k → 0 such that if r ≤ (1 - δ_k)τ(p) then the probability that F_k(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δ_k tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the fraction of satisfiable clauses in a random k-CNF. In particular, for k > 2 it improves upon all previously known such bound.