Hardness Amplification in Two Prover Game and Communication Complexity

Abstract

Whether repeating a task n-times in parallel amplifies the complexity measure in question

by n-times is a key technical challenge in complexity theory.

As a prelude to alternatives to parallel repetition, we investigate set-based repetition

techniques. First we explore the applications of Birthday repetition, first introduced in [2]

for quasi-polynomial time hardness results. These include hardness of computing best

approximate Nash Equilibria in strategic bimatrix game [53], hardness of finding Densest

k-subgraph [50] and hardness of obtaining best signaling scheme in zero-sum bayesian

game [68].

From computational perspective, we first investigate the power and limits of parallel

repetition of two-prover game [49]. We characterize the amortized behavior of the game

via how much information one needs to win the game with good probability providing

an intuitive explanation on why odd-cycle game introduced by [156] serves as a counterexample

to strong parallel repetition.

Then as an alternative to parallel repetition, we introduce symmetric parallel repetition,

that is parallel repetition without coordinates. We show that symmetric parallel

repetition indeed beats parallel repetition in some regimes and odd-cycle game is not

a counterexample for symmetric parallel repetition. This sheds some light on possible

attack route for infamous Unique Games Conjecture.

From communication complexity perspective, we first tackle round tradeoff for Disjointness

with Quantum communication. Then we introduce semi-direct sum as a mean

to amplify hardness for asymmetric communication. As an application, we reprove and

improve the lower bound for \ell_∞ approximate nearest neighbor search.