How to Play Unique Games on Expanders

TL;DR

This paper presents an improved algorithm for solving the Unique Games problem on expander graphs, achieving better approximation guarantees under certain spectral and expansion conditions.

Contribution

It refines previous results by providing a more effective algorithm with improved approximation bounds for Unique Games on expanders.

Findings

Algorithm finds assignments satisfying a large fraction of constraints.

Performance depends on graph expansion and spectral gap.

Results improve upon recent bounds for Unique Games on expanders.

Abstract

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a $(1-\varepsilon)$-satisfiable instance of Unique Games with the constraint graph $G$, our algorithm finds an assignment satisfying at least a $1- C \varepsilon/h_G$ fraction of all constraints if $\varepsilon < c \lambda_G$ where $h_G$ is the edge expansion of $G$, $\lambda_G$ is the second smallest eigenvalue of the Laplacian of $G$, and $C$ and $c$ are some absolute constants.