SOS lower bounds with hard constraints: think global, act local

Abstract

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 ${\hat{I}}^2$, it did not necessarily actually "satisfy" the constraint "objective = ${\hat{I}}^2$". 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-$\hat{I}\text{c◯}(\sqrt{n})$ SOS does not provide a $(\frac{4}{3}-\hat{I}\mu )$-approximation for Min-Bisection, and degree-$\hat{I}\text{c◯}(n)$ SOS does not provide a $(\frac{11}{12}+\hat{I}\mu )$-approximation for Max-Bisection or a $(\frac{5}{4}-\hat{I}\mu )$-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.