A Counterexample to Strong Parallel Repetition

• Authors: Ran Raz
• Venue: FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2008

The parallel repetition theorem states thatfor any {\em two-prover game}, with value $1- \epsilon$(for, say, $\epsilon \leq 1/2$), the value of the game repeatedin parallel$n$ times is at most$(1- \epsilon^c)^{\Omega(n/s)}$, where $s$ is the answers' length(of the original game) and $c$ is a universalconstant~\cite{R}.Several researchers asked wether this bound could be improvedto $(1- \epsilon)^{\Omega(n/s)}$; this question is usually referred toas the {\em strong parallel repetition problem}.We show that the answer for this question is negative.More precisely, we consider the{\em odd cycle game} ofsize $m$; a two-prover game with value $1-1/2m$. We show that thevalue of the odd cycle game repeated in parallel $n$ times is at least$1- (1/m) \cdot O(\sqrt{n})$. This implies thatfor large enough $n$ (say, $n \geq \Omega(m^2)$), thevalue of the odd cycle game repeated in parallel $n$ times is at least$(1- 1/4m^2)^{O(n)}$.Thus:\begin{enumerate}\item For parallel repetition of general games:the bounds of $(1- \epsilon^c)^{\Omega(n/s)}$ given in~\cite{R,Hol} areof the right form, up to determining the exact value of the constant$c \geq 2$.\item For parallel repetition of XOR games, unique gamesand projection games:the bounds of $(1- \epsilon^2)^{\Omega(n)}$ givenin~\cite{FKO} (for XOR games) andin~\cite{Rao} (for unique and projection games) are tight.\item For parallel repetition of the odd cycle game:the bound of $1- (1/m) \cdot \tilde{\Omega}(\sqrt{n})$ givenin~\cite{FKO} is almost tight.\end{enumerate}A major motivationfor the recent interest in the strong parallel repetitionproblem is that a strong parallel repetition theoremwould have implied that the{\em unique game conjecture} is equivalentto the NP hardness of distinguishing between instances of Max-Cutthat are at least $1 - \epsilon^2$ satisfiable from instancesthat are at most $1 - (2/\pi) \cdot \epsilon$ satisfiable.Our results suggest that this cannot be proved just by improvingthe known bounds on parallel repetition.