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

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.