Approximating CSPs with Global Cardinality Constraints Using SDP Hierarchies

This work reviews an approach for rounding constraint satisfaction problem (CSP) instances under cardinality constraints which is developed by Raghavendra and Tan [1]. In the CSP problems, we are given a set of variables over a fixed finite discrete domain and the goal is to find an assignment for these variables to satisfy a set of given local constraints. A constraint is local in the sense that its satisfaction only depends on a constant number of variables1. The goal is to satisfy the most possible number of constraints. It turns out that adding a single non-local (global) cardinality constraint on the variables makes the problem much harder. In this case, the goal is still the same as before, but we want to maintain the added global constraint at the same time. The idea of this paper is to find a set of “uncorrelated” vectors using higher order sum of squares (SOS) hierarchy. While all the presented results in this note can be applied to general CSPs, we only focus on the particular MAX BISECTION instance.