 ## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, October 29, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Felix Weissenberger

## Two-Prover Games for Parallel Repetition

In a two-prover game, a verifier chooses questions (x,y) according to a publicly known distribution. He sends x to Alice and y to Bob. They answer with a(x) and b(y). The game is won if a publicly known predicate V(x,y,a(x),b(y)) is satisfied. The value of the game is the winning probability if Alice and Bob play optimally. In the parallel repetition of a two-prover game, independent copies of the game are played in parallel, and it is won if and only if all copies are won.

Two-prover games and their parallel repetition are motivated by the error reduction of two prover one round proof systems, which are related to the PCP Theorem and inapproximability.

The best parallel repetition theorem states that for a game with value 1- ε, for ε > 0, and answer alphabet size 2s, the value of its k-fold parallel repetition is at most e-f(ε,s) · k, with f(ε,s)∈ Ω (ε3/s). Moreover, there are two independent examples of games which show that f(ε,s) ∈ O(ε2) and f(ε,s) ∈ O(log(s)/s).

We present a parallel repetition theorem for cycle games where f(ε,s) is independent of s. In cycle games the questions are drawn from the uniform distribution and the question graph is a cycle.

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