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Optimal Inapproximability of Promise Equations over Finite Groups

Authors Silvia Butti , Alberto Larrauri , Stanislav Živný

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • promise constraint satisfaction
  • approximation
  • linear equations

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Abstract

A celebrated result of Håstad established that, for any constant ε > 0, it is NP-hard to find an assignment satisfying a (1/|G|+ε)-fraction of the constraints of a given 3-LIN instance over an Abelian group G even if one is promised that an assignment satisfying a (1-ε)-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

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Silvia Butti, Alberto Larrauri, and Stanislav Živný. Optimal Inapproximability of Promise Equations over Finite Groups. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.38

Author Details

Silvia Butti
  • Department of Computer Science, University of Oxford, UK
Alberto Larrauri
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Butti, Silvia: UKRI EP/X024431/1.
  • Larrauri, Alberto: UKRI EP/X024431/1.
  • Živný, Stanislav: UKRI EP/X024431/1.

Acknowledgements

We thank the anonymous reviewers for their feedback.

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