Strong Parallel Repetition for Unique Games on Small Set Expanders

TL;DR

This paper demonstrates how enlarging the alphabet and fortifying unique games on small set expanders enables strong parallel repetition, advancing understanding of the Unique Games Conjecture.

Contribution

It introduces a method to bypass previous impossibility results by enlarging the alphabet and fortifying unique games on small set expanders.

Findings

Strong parallel repetition holds for fortified unique games on small set expanders.

Enlarging the alphabet before repetition helps overcome previous limitations.

Fortification ensures bounded value in large induced sub-games.

Abstract

Strong Parallel Repetition for Unique Games on Small Set Expanders The strong parallel repetition problem for unique games is to efficiently reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1 is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of unique games over large alphabet. Due to its importance to the Unique Games Conjecture, this problem garnered a great deal of interest from the research community. There are positive results for certain easy unique games (e.g., unique games on expanders), and an impossibility result for hard unique games. In this paper we show how to bypass the impossibility result by enlarging the alphabet sufficiently before repetition. We consider the case of unique games on small set expanders for two setups: (i) Strong small set expanders that yield easy unique games. (ii) Weaker small set expanders underlying possibly hard unique games as long as the game is mildly fortified. We show how to fortify unique games in both cases, i.e., how to transform the game so sufficiently large induced sub-games have bounded value. We then prove strong parallel repetition for the fortified games. Prior to this work fortification was known for projection games but seemed hopeless for unique games.