SIAM Journal on Computing
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the Resemblance and Containment of Documents
SEQUENCES '97 Proceedings of the Compression and Complexity of Sequences 1997
Error Reduction by Parallel Repetition—A Negative Result
Combinatorica
Consequences and Limits of Nonlocal Strategies
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Parallel repetition: simplifications and the no-signaling case
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Understanding Parallel Repetition Requires Understanding Foams
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Parallel repetition in projection games and a concentration bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Rounding Parallel Repetitions of Unique Games
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Parallel repetition of the odd cycle game
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
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The parallel repetition theorem states that, for any two-prover game with value $1-\epsilon$ (for, say, $\epsilon\leq1/2$), the value of the game repeated in parallel $n$ times is at most $(1-\epsilon^c)^{\Omega(n/s)}$, 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 $(1-\epsilon)^{\Omega(n/s)}$; 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 $1-(1/m)\cdot O(\sqrt{n})$. This implies that, for large enough $n$ (say, $n\geq\Omega(m^2)$), 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 $(1-\epsilon^c)^{\Omega(n/s)}$ 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 $c\geq2$. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of $(1-\epsilon^2)^{\Omega(n)}$ 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 $1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})$ 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 $1-\epsilon^2$ satisfiable from instances that are at most $1-(2/\pi)\cdot\epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.