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: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings
Pages298-311
Number of pages14
Volume6302 LNCS
DOIs
StatePublished - 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)03029743
ISSN (Electronic)16113349

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

Fingerprint

Semidefinite Programming Relaxation
Integrality
Semidefinite Programming
Labels
Game
Small Perturbations
Arbitrary
Approximation

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

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 (Vol. 6302 LNCS, 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

Approximate Lasserre integrality gap for unique games. / Khot, Subhash; Popat, Preyas; Saket, Rishi.

Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. Vol. 6302 LNCS 2010. p. 298-311 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6302 LNCS).

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

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. vol. 6302 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6302 LNCS, pp. 298-311, 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010, Barcelona, Spain, 9/1/10. https://doi.org/10.1007/978-3-642-15369-3_23
Khot S, Popat P, Saket R. 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. Vol. 6302 LNCS. 2010. p. 298-311. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-15369-3_23
Khot, Subhash ; Popat, Preyas ; Saket, Rishi. / Approximate Lasserre integrality gap for unique games. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. Vol. 6302 LNCS 2010. pp. 298-311 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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