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Approximation Algorithms for MAX 4-SAT and Rounding Procedures for Semidefinite Programs

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Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

Abstract

Karloff and Zwick obtained recently an optimal 7/8-approximation algorithm for MAX 3-SAT. In an attempt to see whether similar methods can be used to obtain a 7/8-approximation algorithm for MAX SAT, we consider the most natural generalization of MAX 3-SAT, namely MAX 4-SAT. We present a semidefinite programming relaxation of MAX 4-SAT and a new family of rounding procedures that try to cope well with clauses of various sizes. We study the potential, and the limitations, of the relaxation and of the proposed family of rounding procedures using a combination of theoretical and experimental means. We select two rounding procedures from the proposed family of rounding procedures. Using the first rounding procedure we seem to obtain an almost optimal 0.8721-approximation algorithm for MAX 4-SAT. Using the second rounding procedure we seem to obtain an optimal 7/8-approximation algorithm for satisfiable instances of MAX 4-SAT. On the other hand, we show that no rounding procedure from the family considered can yield an approximation algorithm for MAX 4-SAT whose performance guarantee on all instances of the problem is greater than 0.8724.

Although most of this paper deals specifically with the MAX 4-SAT problem, we believe that the new family of rounding procedures introduced, and the methodology used in the design and in the analysis of the various rounding procedures considered would have a much wider range of applicability.

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Authors and Affiliations

  1. Department of Computer Science, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Eran Halperin & Uri Zwick

Authors
  1. Eran Halperin
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  2. Uri Zwick
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Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Halperin, E., Zwick, U. (1999). Approximation Algorithms for MAX 4-SAT and Rounding Procedures for Semidefinite Programs. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_16

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  • DOI: https://doi.org/10.1007/3-540-48777-8_16

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  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

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