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Undefinability of Approximation of 2-to-2 Games

Anuj Dawar, Bálint Molnár

TL;DR

This work establishes unconditional undefinability in fixed-point logic with counting (FPC) for approximating the value of weighted 2-to-2 games, showing that no FO formula can separate completely satisfiable instances from those that are δ-satisfiable for any fixed δ > 0. The authors construct an FO-definable reduction chain from regular 3XOR to transitive 2-to-2 games and then to a final weighted 2-to-2 game, preserving perfect completeness and enabling FO definability of the entire process. As a corollary, FO undefinability results extend to problems such as 3-colorability (no FO separation from any fixed t-colorability) and strengthen known undefinability results for Unique Games and Vertex Cover. The results bridge hardness-of-approximation and logical definability, suggesting further exploration into PCSPs and the FO-definability of reductions to simpler, unweighted instances.

Abstract

Recent work by Atserias and Dawar (J. Log. Comp 2019) and Tucker-Foltz (LMCS 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 epsilon-satisfiable, for arbitrarily small epsilon. 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.

Undefinability of Approximation of 2-to-2 Games

TL;DR

This work establishes unconditional undefinability in fixed-point logic with counting (FPC) for approximating the value of weighted 2-to-2 games, showing that no FO formula can separate completely satisfiable instances from those that are δ-satisfiable for any fixed δ > 0. The authors construct an FO-definable reduction chain from regular 3XOR to transitive 2-to-2 games and then to a final weighted 2-to-2 game, preserving perfect completeness and enabling FO definability of the entire process. As a corollary, FO undefinability results extend to problems such as 3-colorability (no FO separation from any fixed t-colorability) and strengthen known undefinability results for Unique Games and Vertex Cover. The results bridge hardness-of-approximation and logical definability, suggesting further exploration into PCSPs and the FO-definability of reductions to simpler, unweighted instances.

Abstract

Recent work by Atserias and Dawar (J. Log. Comp 2019) and Tucker-Foltz (LMCS 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 epsilon-satisfiable, for arbitrarily small epsilon. 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.

Paper Structure

This paper contains 25 sections, 14 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardness of Approximation in Optimization
  4. Label Cover Games
  5. Undefinability of Approximation
  6. The Reduction
  7. Regular 3XOR
  8. Reducing to Transitive Games
  9. The final (weighted) 2-to-2 game
  10. Irregular soundness case
  11. $2 \leftrightarrow 2$ games
  12. Definability
  13. Perfect completeness
  14. Vocabularies
  15. Undefinability of Regular 3XOR
  16. ...and 10 more sections

Key Result

Theorem 2.3

For any $\delta,\epsilon$ with $0 < \delta < 1-\epsilon < 1$, there exists a positive integer $q$ so that $\mathrm{Gap2to2}_q(1-\epsilon,\delta)$ is NP-Hard.

Theorems & Definitions (24)

  • Definition 2.1: $d$-to-$d$ games
  • Conjecture 2.2: Unique Games Conjecture (UGC) KHOT2002
  • Theorem 2.3: Khot-Minzer-Safra
  • Definition 2.4: Weighted $d$-to-$d$ games
  • Theorem 2.5: Atserias-Dawar atserias2019definable
  • Theorem 3.1
  • Definition 3.2: Transitive 2-to-2 games
  • Definition 3.3
  • Definition 3.4: Irregular Values
  • Theorem 3.5: Definable 2-to-2 Games Theorem with irregular soundness
  • ...and 14 more