Rounding parallel repetitions of unique games

Boaz Barak, Moritz Hardt, Oded Regev, Ishay Haviv, Anup Rao, David Steurer

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

Abstract

We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically, denoting by val(G) the value of a two-prover unique game G, and by sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every ℓ ∈ N, if sdpval(G) ≥ 1 - δ, then Val(G) ≥ 1 - √sℓδ. Here, G denotes the ℓ-fold parallel repetition of G, and s = O(log(k/δ)), where k denotes the alphabet size of the game. For the special case where G is an XOR game (i.e., k = 2), we obtain the same bound but with s as an absolute constant. Our bounds on s are optimal up to a factor of O(log1/δ)). For games with a significant gap between the quantities val(G) and sdpval(G), our result implies that val(G) may be much larger than val(G), giving a counterexample to the strong parallel repetition conjecture. In a recent breakthrough, Raz (FOCS '08) has shown such an example using the max-cut game on odd cycles. Our results are based on a generalization of his techniques.

Original languageEnglish (US)
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages374-383
Number of pages10
DOIs
StatePublished - 2008
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: Oct 25 2008Oct 28 2008

Other

Other49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
CountryUnited States
CityPhiladelphia, PA
Period10/25/0810/28/08

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Barak, B., Hardt, M., Regev, O., Haviv, I., Rao, A., & Steurer, D. (2008). Rounding parallel repetitions of unique games. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 (pp. 374-383). [4690971] https://doi.org/10.1109/FOCS.2008.55

Rounding parallel repetitions of unique games. / Barak, Boaz; Hardt, Moritz; Regev, Oded; Haviv, Ishay; Rao, Anup; Steurer, David.

Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. p. 374-383 4690971.

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

Barak, B, Hardt, M, Regev, O, Haviv, I, Rao, A & Steurer, D 2008, Rounding parallel repetitions of unique games. in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008., 4690971, pp. 374-383, 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, United States, 10/25/08. https://doi.org/10.1109/FOCS.2008.55
Barak B, Hardt M, Regev O, Haviv I, Rao A, Steurer D. Rounding parallel repetitions of unique games. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. p. 374-383. 4690971 https://doi.org/10.1109/FOCS.2008.55
Barak, Boaz ; Hardt, Moritz ; Regev, Oded ; Haviv, Ishay ; Rao, Anup ; Steurer, David. / Rounding parallel repetitions of unique games. Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. pp. 374-383
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