TY - GEN

T1 - Playing unique games on certified small-set expanders

AU - Bafna, Mitali

AU - Barak, Boaz

AU - Kothari, Pravesh K.

AU - Schramm, Tselil

AU - Steurer, David

N1 - Publisher Copyright:
© 2021 ACM.

PY - 2021/6/15

Y1 - 2021/6/15

N2 - 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-? vs ?for UG instances, where ?>0 and ?> 0 depend on the expansion parameters of the graph but are independent of the alphabet size.

AB - 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-? vs ?for UG instances, where ?>0 and ?> 0 depend on the expansion parameters of the graph but are independent of the alphabet size.

KW - approximation algorithms

KW - Computational complexity theory

KW - Sum of Squares algorithms

KW - Unique games conjecture

UR - http://www.scopus.com/inward/record.url?scp=85108144836&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85108144836&partnerID=8YFLogxK

U2 - 10.1145/3406325.3451099

DO - 10.1145/3406325.3451099

M3 - Conference contribution

AN - SCOPUS:85108144836

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1629

EP - 1642

BT - STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Khuller, Samir

A2 - Williams, Virginia Vassilevska

PB - Association for Computing Machinery

T2 - 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021

Y2 - 21 June 2021 through 25 June 2021

ER -