Parallel Repetition in Projection Games and a Concentration Bound

• Authors: Anup Rao
• Venue: SIAM Journal on Computing
• Year: 2011

A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples $n$ independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than $(1-\epsilon)$, what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141-172], who showed that the players cannot win all copies in the repeated game with probability better than $(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}$ (here $c$ is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is $(1-\epsilon/2)^{\Omega(\epsilon n)}$ as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant $0Proceedings of the $49$th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369-373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every $k$, one can convert any $q$ query PCP with answer length $c$, size $sc$, and soundness $(1-\epsilon)$ into a two-query PCP with answer length $ck$, size $O(ck (2s)^k)$, and soundness $(1-\epsilon/2q)^{\Omega(\epsilon k/q)}$. Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the $34$th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that for every $\epsilon 0$, there exists an alphabet size $M(\epsilon)$ for which it is NP-hard to distinguish a unique game with alphabet size $M$ in which a $(1-\epsilon^2)$ fraction of the constraints can be satisfied from one in which a $(1-\epsilon f(1/\epsilon))$ fraction of the constraints can be satisfied.