A new bound for 3-satisfiable MaxSat and its algorithmic application

  • Authors:

  • Gregory Gutin;Mark Jones;Dominik Scheder;Anders Yeo

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK;Department of Computer Science, Aarhus University, DK-8000 Aarhus C, Denmark;Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

  • Venue:
  • Information and Computation
  • Year:
  • 2013

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Abstract

Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 23 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment @t will satisfy at least 23m+13m"U+@rn^' clauses, where m is the number of clauses of F, m"U is the number of clauses of F containing a variable from U, n^' is the total number of variables in clauses not containing a variable in U, and @r is a positive absolute constant. Both U and @t can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MaxSat-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 23m+k clauses, where k is the parameter.