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Statistical Physics Approaches to Unique Games

Authors Matthew Coulson, Ewan Davies , Alexandra Kolla, Viresh Patel, Guus Regts

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Author Details

Matthew Coulson
  • Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Ewan Davies
  • Department of Computer Science, University of Colorado, Boulder, CO, USA
Alexandra Kolla
  • Department of Computer Science, University of Colorado, Boulder, CO, USA
Viresh Patel
  • Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Guus Regts
  • Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands


We would like to thank Tyler Helmuth for insightful discussions, and the anonymous referees for comments.

Cite AsGet BibTex

Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. Statistical Physics Approaches to Unique Games. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 13:1-13:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Algorithm design techniques
  • Unique Games Conjecture
  • approximation algorithm
  • Potts model
  • cluster expansion
  • polynomial interpolation


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