Upper Bounds for MaxSat: Further Improved

Abstract

Given a boolean CNF formula F of length |F| (sum of the number of variables in each clause) with m clauses on n variables, we prove the following results.

• The MAXSAT problem, which asks for an assignment satisfying the maximum number of clauses of F, can be solved in O(1:341294^m|F|) time.

• The parameterized version of the problem, that is determining whether there exists an assignment satisfying at least k clauses of the formula (for some integer k), can be solved in O(k ²1:380278^k + |F|) time.

• MAXSAT can be solved in O(1:105729^|F||F|) time.

These bounds improve the recent bounds of respectively O(1:3972^m|F|), O(k ²1:3995^k + |F|) and O(1:1279^|F||F|) due to Niedermeier and Rossmanith [11] for these problems. Our last bound comes quite close to the O(1:07578^|F||F|) bound of Hirsch[6] for the Satisfiability problem (not MAXSAT).

The work was done while the first author was at IIT Mumbai and visited IMSc Chennai as a summer student.