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In this paper, we investigate the approximation and parameterized complexities of MAXNAESAT variants. We begin by presenting a simple yet rigorous proof establishing the APX-completeness of the MAXNAE2SAT problem. Notably, APX-completeness holds even when the repetition factor of each variable is bounded by 3, i.e., each variable appears in at most three clauses in the MAXNAE2SAT instance. Our APX-completeness proof is a strict reduction that directly establishes a new inapproximability bound for the MAXNAE2SAT problem. The decision version of MAXNAE2SAT remains NP-complete when the repetition factor of each variable is bounded by 3, mirroring the NP-completeness of MAXCUT in cubic graphs. We further establish a tight computational dichotomy by proving that the MAXNAE2SAT problem is solvable in linear time when the repetition factor of each variable is bounded by 2. Finally, we present a fixed-parameter tractable algorithm for MAXNAE2SAT instances where the repetition factor of each variable is bounded by 3.

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From MAXCUT to MAXNAESAT: Elegant Proofs and Algorithmic Advances

  • 会议论文

  • 发表日期：

    2025年

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  • Sangram K. Jena,
  • K. Subramani

摘要

In this paper, we investigate the approximation and parameterized complexities of MAXNAESAT variants. We begin by presenting a simple yet rigorous proof establishing the APX-completeness of the MAXNAE2SAT problem. Notably, APX-completeness holds even when the repetition factor of each variable is bounded by 3, i.e., each variable appears in at most three clauses in the MAXNAE2SAT instance. Our APX-completeness proof is a strict reduction that directly establishes a new inapproximability bound for the MAXNAE2SAT problem. The decision version of MAXNAE2SAT remains NP-complete when the repetition factor of each variable is bounded by 3, mirroring the NP-completeness of MAXCUT in cubic graphs. We further establish a tight computational dichotomy by proving that the MAXNAE2SAT problem is solvable in linear time when the repetition factor of each variable is bounded by 2. Finally, we present a fixed-parameter tractable algorithm for MAXNAE2SAT instances where the repetition factor of each variable is bounded by 3.