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A Refined Branching Algorithm for the Maximum Satisfiability Problem

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Abstract

The Maximum satisfiability problem (MaxSAT) is a fundamental NP-hard problem which has significant applications in many areas. Based on refined observations, we derive a branching algorithm of running time \(O^{*}(1.2989^m)\) for the MaxSAT problem, where m denotes the number of clauses in the given CNF formula. Our algorithm considerably improves the previous best result \(O^*(1.3248^m)\) published in 2004. For our purpose, we derive improved branching strategies for variables of degrees 3, 4, and 5. The worst case of our branching algorithm is at certain degree-4 variables. To serve the branching rules, we also propose a variety of reduction rules which can be exhaustively applied in polynomial time.

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Notes

  1. As the MaxSAT problem is self-reducible [24, Chapter 9.1], its decision version is polynomial-time equivalent to its optimization version where the task is to find an assignment satisfying the maximum number of clauses.

  2. A kernelization algorithm for a problem with respect to a parameter is a polynomial-time algorithm which transforms every instance of the problem into an equivalent instance of the same problem such that the size of the new instance is bounded by a function of the parameter. See [5] for a comprehensive introduction.

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Authors and Affiliations

  1. Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, Changsha, China

    Wenjun Li & Chao Xu

  2. Chair of Economic Theory, Faculty of Human and Business Sciences, Saarland University, Saarbrücken, Germany

    Yongjie Yang

  3. Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA

    Jianer Chen

  4. School of Computer Science and Engineering, Central South University, Changsha, China

    Jianxin Wang

Authors
  1. Wenjun Li
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  2. Chao Xu
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  3. Yongjie Yang
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  4. Jianer Chen
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  5. Jianxin Wang
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Correspondence to Jianxin Wang.

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This work was supported by the National Natural Science Foundation of China under Grants 61872048, 61672536, and 61702557, and the Research Foundation of Education Bureau of Hunan Province, China under Grant 21B0305. This work was done when the second author was affiliated with School of Computer Science and Engineering, Central South University, Changsha, China.

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Li, W., Xu, C., Yang, Y. et al. A Refined Branching Algorithm for the Maximum Satisfiability Problem. Algorithmica 84, 982–1006 (2022). https://doi.org/10.1007/s00453-022-00938-8

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