Strong Parallel Repetition Theorem for Free Projection Games
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Approximations for the isoperimetric and spectral profile of graphs and related parameters
Proceedings of the forty-second ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Improved rounding for parallel repeated unique games
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Unique Games with Entangled Provers Are Easy
SIAM Journal on Computing
Parallel repetition of entangled games
Proceedings of the forty-third annual ACM symposium on Theory of computing
A Counterexample to Strong Parallel Repetition
SIAM Journal on Computing
Derandomized Parallel Repetition Theorems for Free Games
Computational Complexity
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We show a connection between the semidefinite relaxation of uniquegames and their behavior under parallel repetition. Specifically,denoting by $val(G)$ the value of a two-prover unique game $G$, andby $sdpval(G)$ the value of a natural semidefinite program toapproximate val(G), we prove that for every $\ell\in\mathbb N$, if$sdpval(G) \geq 1-\delta$, then $val(G^{\ell}) \geq 1 -\sqrt{s\ell\delta\,}.$ Here, $G^{\ell}$ denotes the $\ell$-foldparallel repetition of $G$, and $s=O(\log(k/\delta))$, 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 $s$ asan absolute constant. Our bounds on $s$ are optimal up to a factor of$O(\log(1/\delta))$.For games with a significant gap between the quantities $val(G)$ and$sdpval(G)$, our result implies that $val(G^{\ell})$ may be muchlarger than $val(G)^{\ell}$, giving a counterexample to the strongparallel repetition conjecture. In a recent breakthrough, Raz (FOCS'08) has shown such an example using the max-cut game on oddcycles. Our results are based on a generalization of his techniques.