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.

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在经过认证的小型扩展器上玩独特的游戏

我们给出了一种用于解决唯一游戏 (UG) 实例的算法，其约束对应于具有平方和 (SoS) 小集扩展证书的图的边。作为推论，我们获得了第一个多项式时间算法，用于在有噪声的超立方体和短代码图上解决 UG。此类实例的先前最佳算法是 Arora、Barak 和 Steurer (2010) 的特征值枚举算法，该算法要求噪声超立方体的准多项式时间和短代码图的近指数时间。对于 UG 实例，我们的所有结果都实现了 $1-\epsilon$ 与 $\delta$ 的近似值，其中 $\delta > 0$ 取决于图的扩展参数，但与字母表大小无关。具体来说，假设正则图 $G=(V,E)$ 是 $(\mu, \eta)$ 小集扩展器 (SSE) 如果对于每个子集 $S \subseteq V$ 和 $|S| \leq \mu |V|$，$S$ 的边展开至少为 $\eta$。我们说 $G$ 是 $d$-certified $(\mu,\eta)$-SSE，如果有这个事实的 degree-d SoS 证书（基于 2 到 4 的超收缩性）。我们证明存在 $|V|^{f(d,\mu,\eta)}$ 时间算法 $A$（基于 SoS 层次结构）使得对于每个 $\eta>0$ 和 $d$ -certified $(\mu, \eta)$-SSE $G$，如果 $I$ 是 $1-\eta^2/100$ 在 $G$ 上的可满足仿射 UG 实例，则 $A(I)$ 是一个赋值至少满足 $I$ 约束的一些正分数 $\delta = \delta(\mu,\eta)$。作为推论，我们得到多项式时间算法 $A$，如果 $I$ 是 $\alpha$-noisy 超立方体或短代码图上的 $1-\epsilon$ 可满足实例，然后 $A(I)$ 输出满足 $\exp(-O(\sqrt{\epsilon}/\alpha))$ 部分约束的赋值。我们的技术甚至可以扩展到非 SSE 的图，特别是我们获得了一种新的有效算法来解决 Johnson 图上的 UG 实例。