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Maximum And- vs. Even-SAT

Authors Tamio-Vesa Nakajima , Stanislav Živný

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • approximation
  • promise constraint satisfaction
  • max and
  • max even
  • max cut
  • max dicut
  • max acyclic

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Abstract

A multiset of literals, called a clause, is strongly satisfied by an assignment if no literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies ρ of the clauses, an assignment in which at least ρ of the clauses are weakly satisfied, in the sense that an even number of literals evaluate to false.
In particular, this implies an efficient algorithm for finding an undirected cut of value ρ in a graph G given that a directed cut of value ρ in G is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of G with ρ edges under the same promise.

Cite As Get BibTex

Tamio-Vesa Nakajima and Stanislav Živný. Maximum And- vs. Even-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 3:1-3:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.3

Author Details

Tamio-Vesa Nakajima
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Nakajima, Tamio-Vesa: UKRI EP/X024431/1, Clarendon Fund Scholarship
  • Živný, Stanislav: UKRI EP/X024431/1

Acknowledgements

We thank anonymous reviewers of an earlier version of this paper for useful comments and a simplification of our algorithm. We also thank the anonymous reviewers of APPROX 2025 for feedback and suggestions for changes.

References

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