Playing Unique Games on Certified Small-Set Expanders

TL;DR

This paper introduces a new algorithm for solving unique games on certain graphs by leveraging low-degree sum-of-squares proofs and hypercontractive inequalities, achieving polynomial-time solutions for specific graph classes.

Contribution

It develops a versatile rounding technique for low-entropy solutions in sum-of-squares relaxations, enabling polynomial-time algorithms for unique games on graphs like the noisy hypercube, short code, and Johnson graph.

Findings

First polynomial-time algorithms for specific UG instances.

Achieves approximation guarantees independent of alphabet size.

Extends the applicability of sum-of-squares methods to worst-case problems.

Abstract

We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive inequality. Our algorithm is in fact more versatile, and succeeds even when the constraint graph is not a small-set expander as long as the structure of non-expanding small sets is (informally speaking) "characterized" by a low-degree sum-of-squares proof. Our results are obtained by rounding \emph{low-entropy} solutions -- measured via a new global potential function -- to sum-of-squares (SoS) semidefinite programs. This technique adds to the (currently short) list of general tools for analyzing SoS relaxations for \emph{worst-case} optimization problems. As corollaries, we obtain the first polynomial-time algorithms for solving any UG instance where the constraint graph is either the \emph{noisy hypercube}, the \emph{short code} or the \emph{Johnson} graph. The prior best algorithm for such instances was the eigenvalue enumeration algorithm of Arora, Barak, and Steurer (2010) which requires quasi-polynomial time for the noisy hypercube and nearly-exponential time for the short code and Johnson graphs. All of our results achieve an approximation of $1-\epsilon$ vs $\delta$ for UG instances, where $\epsilon>0$ and $\delta > 0$ depend on the expansion parameters of the graph but are independent of the alphabet size.