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 global constraints into local constraints. Using these ideas, we also show that degree-Ω(n) SOS does not provide a ( 3 − ε)-approximation for Min-Bisection, and degree-Ω(n) SOS does not provide a ( 11 12 + ε)-approximation for Max-Bisection or a ( 4 − ε)-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.