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

Understand

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 $\beta$, it did not necessarily actually "satisfy" the constraint "objective = $\beta$".
• 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.