Statistical physics approaches to Unique Games

TL;DR

This paper introduces statistical physics techniques to solve a variant of the Unique Games problem, called Count Unique Games, which could have implications for the longstanding Unique Games Conjecture.

Contribution

It adapts statistical physics methods to develop efficient algorithms for Count Unique Games, a variant with stronger guarantees than standard Unique Games.

Findings

Efficient algorithms for Count Unique Games based on partition function approximation

Potential to refute the Unique Games Conjecture with modest parameter improvements

Application of zero-free regions and cluster expansion techniques

Abstract

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would refute the Unique Games Conjecture.