A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Application

Abstract

For a formula F in conjunctive normal form (CNF), let sat(F) be the maximum number of clauses of F that can be satisfied by a truth assignment, and let m be the number of clauses in F. It is well-known that for every CNF formula F, sat(F) ≥ m/2 and the bound is tight when F consists of conflicting unit clauses (x) and $(\bar{x})$. Since each truth assignment satisfies exactly one clause in each pair of conflicting unit clauses, it is natural to reduce F to the unit-conflict free (UCF) form. If F is UCF, then Lieberherr and Specker (J. ACM 28(2):411-421, 1981) proved that ${\rm sat}(F)\ge {\hat{\phi}} m$, where ${\hat{\phi}} =(\sqrt{5}-1)/2$.

We introduce another reduction that transforms a UCF CNF formula F into a UCF CNF formula F′, which has a complete matching, i.e., there is an injective map from the variables to the clauses, such that each variable maps to a clause containing that variable or its negation. The formula F′ is obtained from F by deleting some clauses and the variables contained only in the deleted clauses. We prove that ${\rm sat}(F) \ge {\hat{\phi}} m + (1-{\hat{\phi}})(m-m') + n'(2-3{\hat{\phi}})/2$, where n′ and m′ are the number of variables and clauses in F′, respectively. This improves the Lieberherr-Specker lower bound on sat(F).

We show that our new bound has an algorithmic application by considering the following parameterized problem: given a UCF CNF formula F decide whether sat(F) $\ge {\hat{\phi}} m + k,$ where k is the parameter. This problem was introduced by Mahajan and Raman (J. Algorithms 31(2):335–354, 1999) who conjectured that the problem is fixed-parameter tractable, i.e., it can be solved in time f(k)(nm)^O(1) for some computable function f of k only. We use the new bound to show that the problem is indeed fixed-parameter tractable by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $\lfloor (7+3\sqrt{5})k\rfloor$ variables.

Research of Gutin, Jones and Yeo was supported in part by an EPSRC grant. Research of Gutin was also supported in part by the IST Programme of the European Community, under the PASCAL 2 Network of Excellence.