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Undefinability of Approximation of 2-To-2 Games

Authors Anuj Dawar , Bálint Molnár

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Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Hardness of Approximation
  • Unique Games
  • Descriptive Complexity
  • Fixed-Point Logic with Counting

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Abstract

Recent work by Atserias and Dawar [Albert Atserias and Anuj Dawar, 2019] and Tucker-Foltz [Jamie Tucker-Foltz, 2024] has established undefinability results in fixed-point logic with counting (FPC) corresponding to many classical complexity results from the hardness of approximation. In this line of work, NP-hardness results are turned into unconditional FPC undefinability results. We extend this work by showing the FPC undefinability of any constant factor approximation of weighted 2-to-2 games, based on the NP-hardness results of Khot, Minzer and Safra. Our result shows that the completely satisfiable 2-to-2 games are not FPC-separable from those that are not ε-satisfiable, for arbitrarily small ε. The perfect completeness of our inseparability is an improvement on the complexity result, as the NP-hardness of such a separation is still only conjectured. This perfect completeness enables us to show the FPC undefinability of other problems whose NP-hardness is conjectured. In particular, we are able to show that no FPC formula can separate the 3-colourable graphs from those that are not t-colourable, for any constant t.

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Anuj Dawar and Bálint Molnár. Undefinability of Approximation of 2-To-2 Games. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.16

Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Bálint Molnár
  • Department of Computer Science and Technology, University of Cambridge, UK

Funding

  • Dawar, Anuj: Funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee: grant number EP/X028259/1.

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