Spectral Algorithms for Unique Games

TL;DR

This paper introduces a spectral algorithm for Unique Games that outperforms SDP-based methods by leveraging the full spectrum of the associated graph, providing better guarantees and efficiency on certain instances.

Contribution

The paper presents a novel spectral algorithm for Unique Games that uses the entire spectrum of the graph, improving upon previous SDP-based approaches and handling specific hard instances efficiently.

Findings

Algorithm recovers good assignments for highly satisfiable instances.

Runs in quasi-polynomial time on Khot-Vishnoi instances and detects unsatisfiability.

Succeeds where SDP relaxations fail, especially on expander graphs.

Abstract

We give a new algorithm for Unique Games which is based on purely {\em spectral} techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the {\em Label-Extended} graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasi-polynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games {\em fails}. As a special case, we also re-derive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.