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### Paper:

TR08-018 | 28th February 2008 00:00

#### A Counterexample to Strong Parallel Repetition TR08-018
Authors: Ran Raz
Publication: 1st March 2008 22:36
Keywords:

Abstract:

The parallel repetition theorem states that for any two-prover game,
with value $1- \epsilon$ (for, say, $\epsilon \leq 1/2$), the value of
the game repeated in 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 universal constant.
Several researchers asked wether 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 for 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:

1) For parallel repetition of general games:
the bounds of $(1- \epsilon^c)^{\Omega(n/s)}$ given in~\cite{R,Hol} are
of the right form, up to determining the exact value of the constant
$c \geq 2$.

2) For parallel repetition of XOR games, unique games and projection games: the bounds of $(1- \epsilon^2)^{\Omega(n)}$ given in~\cite{FKO}
(for XOR games) and in~\cite{Rao} (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~\cite{FKO} 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.

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