Spectral Algorithms for Unique Games

Abstract

We present a new algorithm for Unique Games which is based on purely 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 Label-Extended graph associated with the instance of Unique Games. In particular, we show how our techniques imply a quasi-polynomial time algorithm that decides satisfiability of a game on the Khot-Vishnoi [14] integrality gap instance. Notably, when run on that instance, the standard SDP relaxation of Unique Games fails. As a special case, we also show how to re-derive a polynomial time algorithm for Unique Games on expander constraint graphs (similar to [2]) and a sub-exponential time algorithm for Unique Games on the Hypercube.