### 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 - ε^{c})ω^{(n/s)}, where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803]. Several researchers asked whether 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 to 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 (1-1/4m^{2})O(n). Thus the following hold. 1. For parallel repetition of general games, the bounds of (1 - ε ^{c})ω^{(n/s)}, given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767-775] 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 - ε ^{2})Ω ^{(n)} given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179-192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1-10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of 1 - (1/m) Ω(√n) given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179-192] 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 - ε ^{2} 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 language | English (US) |
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Pages (from-to) | 771-777 |

Number of pages | 7 |

Journal | SIAM Journal on Computing |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - Jul 20 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Keywords

- Odd cycle game
- Parallel repetition theorem
- Two-prover games

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## Cite this

*SIAM Journal on Computing*,

*40*(3), 771-777. https://doi.org/10.1137/090747270