Approximate Lasserre integrality gap for unique games

Subhash Khot, Preyas Popat, Rishi Saket

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

Abstract

In this paper, we investigate whether a constant round Lasserre Semi-definite Programming (SDP) relaxation might give a good approximation to the Unique Games problem. We show that the answer is negative if the relaxation is insensitive to a sufficiently small perturbation of the constraints. Specifically, we construct an instance of Unique Games with k labels along with an approximate vector solution to t rounds of the Lasserre SDP relaxation. The SDP objective is at least 1-ε whereas the integral optimum is at most γ, and all SDP constraints are satisfied up to an accuracy of δ>0. Here ε, γ>0 and t ∈ℤ+ are arbitrary constants and k=k(ε, γ) ∈ℤ+. The accuracy parameter δ can be made sufficiently small independent of parameters ε, γ, t, k (but the size of the instance grows as δ gets smaller).

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings
Pages298-311
Number of pages14
DOIs
StatePublished - Nov 15 2010
Event13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010 - Barcelona, Spain
Duration: Sep 1 2010Sep 3 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6302 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010
CountrySpain
CityBarcelona
Period9/1/109/3/10

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Khot, S., Popat, P., & Saket, R. (2010). Approximate Lasserre integrality gap for unique games. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings (pp. 298-311). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6302 LNCS). https://doi.org/10.1007/978-3-642-15369-3_23