Adding cardinality constraints to integer programs with applications to maximum satisfiability

  • Authors:

  • Markus Bläser;Thomas Heynen;Bodo Manthey

  • Affiliations:

  • Universität des Saarlandes, FR Informatik, Postfach 151150, 66041 Saarbrücken, Germany;ETH Zürich, Departement Informatik, ETH Zentrum, 8092 Zürich, Switzerland;Universität des Saarlandes, FR Informatik, Postfach 151150, 66041 Saarbrücken, Germany

  • Venue:
  • Information Processing Letters
  • Year:
  • 2008

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Abstract

Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most @? literals, we obtain the problem Max-@?SAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398-405] designed a (1-e^-^1)-approximation algorithm for Max-SAT-CC. This result is tight unless P=NP [U. Feige, J. ACM 45 (4) (1998) 634-652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-@?SAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1-(1-1@?)^@?-@e. To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space.