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

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Anders Yeo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6478)

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Anders Yeo
    • 1
  1. 1.Royal HollowayUniversity of LondonUnited Kingdom

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