{"id":358760,"date":"2017-01-27T10:45:33","date_gmt":"2017-01-27T18:45:33","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=358760"},"modified":"2018-10-16T20:03:07","modified_gmt":"2018-10-17T03:03:07","slug":"maximum-satisfiability-random-formulas","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/maximum-satisfiability-random-formulas\/","title":{"rendered":"On the Maximum Satisfiability of Random Formulas"},"content":{"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 k<\/span><\/span><\/span><\/span>-clauses is p<\/span><\/span><\/span><\/span>-satisfiable if there exists a truth assignment satisfying 1<\/span>\u2212<\/span>2<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span>+<\/span>p<\/span>2<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span> of all clauses (observe that every k<\/span><\/span><\/span><\/span>-CNF is 0-satisfiable). Also, let F<\/span>k<\/span><\/span>(<\/span>n<\/span>,<\/span>m<\/span>)<\/span><\/span><\/span><\/span> denote a random k<\/span><\/span><\/span><\/span>-CNF on n<\/span><\/span><\/span><\/span>variables formed by selecting uniformly and independently m<\/span><\/span><\/span><\/span> out of all possible k<\/span><\/span><\/span><\/span>-clauses. It is easy to prove that for every k<\/span>><\/span>1<\/span><\/span><\/span><\/span> and every p<\/span><\/span><\/span><\/span> in (<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span>, there is R<\/span>k<\/span><\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> such that if r<\/span>><\/span>R<\/span>k<\/span><\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span>, then the probability that F<\/span>k<\/span><\/span>(<\/span>n<\/span>,<\/span>r<\/span>n<\/span>)<\/span><\/span><\/span><\/span> is p<\/span><\/span><\/span><\/span>-satisfiable tends to 0 as n<\/span><\/span><\/span><\/span> tends to infinity. We prove that there exists a sequence \u03b4<\/span>k<\/span><\/span>\u2192<\/span>0<\/span><\/span><\/span><\/span> such that if r<\/span><<\/span>(<\/span>1<\/span>\u2212<\/span>\u03b4<\/span>k<\/span><\/span>)<\/span>R<\/span>k<\/span><\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> then the probability that F<\/span>k<\/span><\/span>(<\/span>n<\/span>,<\/span>r<\/span>n<\/span>)<\/span><\/span><\/span><\/span>is p<\/span><\/span><\/span><\/span>-satisfiable tends to 1 as n<\/span><\/span><\/span><\/span> tends to infinity. The sequence \u03b4<\/span>k<\/span><\/span><\/span><\/span><\/span> tends to 0 exponentially fast in k<\/span><\/span><\/span><\/span>.<\/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 k-clauses is p-satisfiable if there exists a truth assignment satisfying 1\u22122\u2212k+p2\u2212k of all clauses (observe that every k-CNF is 0-satisfiable). Also, let Fk(n,m) denote a random k-CNF on […]<\/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":null,"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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