Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT
Theoretical Computer Science
Strong Refutation Heuristics for Random k-SAT
Combinatorics, Probability and Computing
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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 paperwe 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). LetFk (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 t(p) = 2^k In 2/(p + (1 - p) In (1 - p)).It is easy to prove that for every k 驴 2 and p \in (0,1), if r 驴 t(p) then the probability that Fk(n,m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence \delta _k\to 0 such that if r \leqslant (1 - \delta _k )t(p) then the probability that Fk(n,m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence \delta _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.