Playing Unique Games on Certified Small-Set Expanders

with Mitali Bafna, Boaz Barak, Pravesh Kothari, Tselil Schramm. STOC 2021.

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 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 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 noisy hypercube, the short code or the 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.

keywords

Unique Games Conjecture
small-set expansion
sum-of-squares