A Counterexample to Strong Parallel Repetition

The parallel repetition theorem states that for any two-prover game, with value 1 − (for, say, ≤ 1/2), the value of the game repeated in parallel n times is at most (1 − c)Ω(n/s), where s is the answers’ length (of the original game) and c is a universal constant [R95]. Several researchers asked wether this bound could be improved to (1 − )Ω(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) · O( √ n). This implies that for large enough n (say, n ≥ Ω(m2)), the value of the odd cycle game repeated in parallel n times is at least (1 − 1/4m2)O(n). Thus:

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Ran Raz
• Keywords: FOCS 2008, Odd Cycle Game, Parallel Repetition, Parallel Repetition Theorem, Theoretical Computer Science