We give an algorithm for solving unique games (UG) instances whose constraints correspond to edges of graphs with a sum-of-squares (SoS) small-set-expansion certificate. As corollaries, we obtain the first polynomial-time algorithms for solving UG on the noisy hypercube and the short code graphs. 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 graph. All of our results achieve an approximation of $1-\epsilon$ vs $\delta$ for UG instances, where $\delta > 0$ depends on the expansion parameters of the graph but is independent of the alphabet size. Specifically, say that a regular graph $G=(V,E)$ is a $(\mu,\eta)$ small-set expander (SSE) if for every subset $S \subseteq V$ with $|S| \leq \mu |V|$, the edge-expansion of $S$ is at least $\eta$. We say that $G$ is a $d$-certified $(\mu,\eta)$-SSE if there is a degree-d SoS certificate for this fact (based on 2 to 4 hypercontractivity). We prove that there is a $|V|^{f(d,\mu,\eta)}$ time algorithm $A$ (based on the SoS hierarchy) such that for every $\eta>0$ and $d$-certified $(\mu, \eta)$-SSE $G$, if $I$ is a $1-\eta^2/100$ satisfiable affine UG instance over $G$ then $A(I)$ is an assignment satisfying at least some positive fraction $\delta = \delta(\mu,\eta)$ of $I$'s constraints. As a corollary, we get a polynomial-time algorithm $A$ such that if $I$ is a $1-\epsilon$ satisfiable instance over the $\alpha$-noisy hypercube or short code graph, then $A(I)$ outputs an assignment satisfying an $\exp(-O(\sqrt{\epsilon}/\alpha))$ fraction of the constraints. Our techniques can be extended even to graphs that are not SSE, and in particular we obtain a new efficient algorithm for solving UG instances over the Johnson graph.
翻译:我们用一个算法来解决独特的游戏( UG) 。 其限制与图表的边缘值( SoS) 平价( square) 平价( some- square) 平价( suder- sqeet- sqeet- exacation) 的值( UG) 。 作为调序,我们获得了第一个在噪音超立方体和短代码图上解决 UG的多元时间算法( UG) 。 之前最好的算法是 Arora、 Barak 和 Steurer (2010) 平价( muto- motemoomy) 的算法( roup mo- morecudeal $( suder- squetoal) $( $( $) 美元) 的值( 美元 美元) 。 我们所有结果都近似地( 美元) 平价( 美元) 的比值( 美元) 美元比值( 美元) 。