A counterexample to strong parallel repetition

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

28 Scopus citations

Abstract

The parallel repetition theorem states that for any two-prover game, with value 1 - ε (for, say, ε ≤ 1/2), the value of the game repeated in parallel n times is at most (1 - εc)Ω(n/s), where s is the answers' length (of the original game) and c is a universal constant [13]. Several researchers asked wether this bound could be improved to (1 - ε)Ω(n/s); this question is usually referred to as the strong parallel repetition problem. We show that the answer for this question is negative. More precisely, we consider the odd cycle game of size m; a two-prover game with value 1 - 1/2m. We show that the value of the odd cycle game repeated in parallel n times is at least 1 - (1/m) · O(√n). This implies that for large enough n (say, n ≥ Ω(m2)), the value of the odd cycle game repeated in parallel n times is at least (1 - 1/4m2)O(n). Thus: 1. For parallel repetition of general games: the bounds of (1 - εc)Ω(n/s) given in [13, 7] are of the right form, up to determining the exact value of the constant c ≥ 2. 2. For parallel repetition of XOR games, unique games and projection games: the bounds of (1 - ε2)Ω(n) given in [5] (for XOR games) and in [12] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game: the bound of 1- (1/m) · Ω̃(√n) given in [5] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least 1 - ε2 satisfiable from instances that are at most 1 - (2/π) · ε satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.

Original languageEnglish (US)
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages369-373
Number of pages5
DOIs
StatePublished - Dec 31 2008
Externally publishedYes
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: Oct 25 2008Oct 28 2008

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

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

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

Fingerprint Dive into the research topics of 'A counterexample to strong parallel repetition'. Together they form a unique fingerprint.

  • Cite this

    Raz, R. (2008). A counterexample to strong parallel repetition. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 (pp. 369-373). [4690970] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2008.49