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\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
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 -clauses is -satisfiable if there exists a truth assignment satisfying of all clauses (observe that every -CNF is 0-satisfiable). Also, let denote a random -CNF on variables formed […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Dimitris Achlioptas\t","user_id":0},{"type":"text","value":"Assaf Naor","user_id":0},{"type":"user_nicename","value":"peres","user_id":"33234"}],"msr_publishername":"ACM","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Journal of the 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