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DOI: 10.4230/LIPIcs.CCC.2020.13

Statistical Physics Approaches to Unique Games

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

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

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.

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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)

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@InProceedings{coulson_et_al:LIPIcs.CCC.2020.13,
  author =	{Coulson, Matthew and Davies, Ewan and Kolla, Alexandra and Patel, Viresh and Regts, Guus},
  title =	{{Statistical Physics Approaches to Unique Games}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{13:1--13:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.13},
  URN =		{urn:nbn:de:0030-drops-125650},
  doi =		{10.4230/LIPIcs.CCC.2020.13},
  annote =	{Keywords: Unique Games Conjecture, approximation algorithm, Potts model, cluster expansion, polynomial interpolation}
}

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Statistical Physics Approaches to Unique Games

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