A note on Unique Games

Abstract

We give a tighter analysis of the algorithm of Khot [S. Khot, On the power of unique 2-prover 1-round games, in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montreal, Quebec, Canada, 2002, pp. 767–775] which shows that given a unique 2-prover-1-round game with value 1 − ε , one can find in polynomial time an assignment to the game with an expected weight of 1 − O ( k 6 / 5 ε 1 / 5 ( log 1 ε k ) 2 / 5 ) , where k is the size of the answer domain. This shows that if the Unique Games Conjecture is true then the domain size k, must be at least Ω ( ( ε 1 / 6 log 1 / 3 ( 1 / ε ) ) −1 ) , which is an improvement over the previous Ω ( ( ε 1 / 10 log 1 / 4 ( 1 / ε ) ) −1 ) bound.