Playing unique games on certified small-set expanders

Mitali Bafna, Boaz Barak, Pravesh K. Kothari, Tselil Schramm, David Steurer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

9 Scopus citations

Abstract

We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive inequality. Our algorithm is in fact more versatile, and succeeds even when the constraint graph is not a small-set expander as long as the structure of non-expanding small sets is (informally speaking) "characterized"by a low-degree sum-of-squares proof. Our results are obtained by rounding low-entropy solutions - measured via a new global potential function - to sum-of-squares (SoS) semidefinite programs. This technique adds to the (currently short) list of general tools for analyzing SoS relaxations for worst-case optimization problems. As corollaries, we obtain the first polynomial-time algorithms for solving any UG instance where the constraint graph is either the noisy hypercube, the short code or the Johnson graph. The prior best algorithm for such instances was the eigenvalue enumeration algorithm of Arora, Barak, and Steurer (2010) which requires quasi-polynomial time for the noisy hypercube and nearly-exponential time for the short code and Johnson graphs. All of our results achieve an approximation of 1-? vs ?for UG instances, where ?>0 and ?> 0 depend on the expansion parameters of the graph but are independent of the alphabet size.

Original languageEnglish (US)
Title of host publicationSTOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
EditorsSamir Khuller, Virginia Vassilevska Williams
PublisherAssociation for Computing Machinery
Pages1629-1642
Number of pages14
ISBN (Electronic)9781450380539
DOIs
StatePublished - Jun 15 2021
Externally publishedYes
Event53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy
Duration: Jun 21 2021Jun 25 2021

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/TerritoryItaly
CityVirtual, Online
Period6/21/216/25/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • approximation algorithms
  • Computational complexity theory
  • Sum of Squares algorithms
  • Unique games conjecture

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