Rounding Parallel Repetitions of Unique Games

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, andby sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every l epsi N, if sdpval(G) ges 1-delta, then val(G^l) ges 1-radicsldelta. Here, G^l denotes the l-fold parallel 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 as an 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^l) may be much larger than val(G)^l, 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 oddcycles. Our results are based on a generalization of his techniques.