带有硬约束的SOS下界：全局思考，局部行动
SOS lower bounds with hard constraints: think global, act local
摘要：
许多以前的平方和（SOS）的CSP下界有两个缺陷，涉及到全球的限制。首先，他们不能支持“基数约束”，比如说，在最小二分法问题。其次，虽然目标函数的伪期望被证明具有某个值$β$，但它不一定实际上“满足”约束“objective = $β$“。在本文中，我们将展示如何弥补这两个缺陷的情况下，随机CSP，通过将\emph{global}的约束转换为\emph{local}的约束。使用这些想法，我们还表明，度$Ω(\sqrt{n})$ SOS不提供一个$(\frac{4}{3} - ε)$ -近似的最小二分法，度$Ω(n)$ SOS不提供一个$(\frac{11}{12} + ε)$ -近似的最大二分法或$(\frac{5}{4} - ε)$ -近似的最小二分法。这些问题的SOS下限是未知的。
英文摘要：
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value $β$, it did not necessarily actually "satisfy" the constraint "objective = $β$". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating \emph{global} constraints into \emph{local} constraints. Using these ideas, we also show that degree-$Ω(\sqrt{n})$ SOS does not provide a $(\frac{4}{3} - ε)$-approximation for Min-Bisection, and degree-$Ω(n)$ SOS does not provide a $(\frac{11}{12} + ε)$-approximation for Max-Bisection or a $(\frac{5}{4} - ε)$-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.
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