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 \emph{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 \emph{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 \emph{noisy hypercube}, the \emph{short code} or the \emph{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.
翻译:当低度总和(UG) 校验证明通过超分性不平等对下限限制图形进行小设置扩展时,我们提供一种解决独特游戏(UG) 的算法。我们的算法实际上更具有多功能性,即使约束图不是小设置扩展器,只要非扩展小组的结构是(非正式地说)“通过低度总和证明“字符化 ” 。我们的结果是通过四舍五入 0-entropy} 解决方案 — 通过新的全球潜在功能测量 — — 将底限图的缩放缩成 缩放(S) 半确定性程序。这种技术增加了(目前很短的)用于分析 SoS 松动小组结构的缩放量,只要非扩展小组的结构是(非正式地说) “ 非正式地说 ” 由低度总和平方平面总和时间算来解决任何 UG 的缩略图的缩略图。 硬度图的缩略图要么是 emph{noíal_ 超值 平面的缩数, 和前数的算法则需要前数。