Reduction From Non-Unique Games To Boolean Unique Games

TL;DR

This paper presents an efficient reduction from the Boolean Unique Games Conjecture to a non-unique game PCP theorem, introducing new concentration results in Gaussian space to establish soundness.

Contribution

It provides the first efficient reduction with a proof of soundness for the Boolean Unique Games Conjecture, advancing the understanding of hardness of approximation.

Findings

Established an efficient reduction with soundness proof

Developed a new concentration theorem in Gaussian space

Bounded Euclidean distances between function restrictions

Abstract

We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-delta vs. 1-C*delta, for any C> 1, and sufficiently small delta>0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known. Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.