On the Maximum Satisfiability of Random Formulas

Maximum satisfiability is a canonical NP-hard optimization problem that appears empirically hard for random instances. Let us say that a Conjunctive normal form (CNF) formula consisting of \(k\)-clauses is \(p\)-satisfiable if there exists a truth assignment satisfying \(1-2_{-k}+p2_{-k}\) of all clauses (observe that every \(k\)-CNF is 0-satisfiable). Also, let \(F_k(n,m)\) denote a random \(k\)-CNF on \(n\)variables formed by selecting uniformly and independently \(m\) out of all possible \(k\)-clauses. It is easy to prove that for every \(k>1\) and every \(p\) in \((0,1]\), there is \(R_k(p)\) such that if \(r>R_k(p)\), then the probability that \(F_k(n,rn)\) 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 \(r<(1-{\delta }_k)R_k(p)\) then the probability that \(F_k(n,rn)\)is \(p\)-satisfiable tends to 1 as \(n\) tends to infinity. The sequence \({\delta }_k\) tends to 0 exponentially fast in \(k\).