Approximating CSPs with Global Cardinality Constraints Using SDP Hierarchies

This work is concerned with approximating constraint satisfaction problems (CSPs) with an additional global cardinality constraints. For example, \maxcut is a boolean CSP where the input is a graph $G = (V,E)$ and the goal is to find a cut $S \cup \bar S = V$ that maximizes the numberof crossing edges, $|E(S,\bar S)|$. The \maxbisection problem is a variant of \maxcut with an additional global constraint that each side of the cut has exactly half the vertices, i.e., $|S| = |V|/2$. Several other natural optimization problems like \minbisection and approximating Graph Expansion can be formulated as CSPs with global constraints. In this work, we formulate a general approach towards approximating CSPs with global constraints using SDP hierarchies. To demonstrate the approach we present the following results: Using the Lasserre hierarchy, we present an algorithm that runs in time $O(n^{{poly(1/\epsilon)})$} that given an instance of \maxbisection with value $1-\epsilon$, finds a bisection with value $1-O(\sqrt{\epsilon})$. This approximation is near-optimal (up to constant factors in $O()$) under the Unique Games Conjecture. By a computer-assisted proof, we show that the same algorithm also achieves a 0.85-approximation for \maxbisection, improving on the previous bound of 0.70 (note that it is \uniquegames hard to approximate better than a 0.878 factor). The same algorithm also yields a 0.92-approximation for \maxtwosat with cardinality constraints. For every CSP with a global cardinality constraints, we present a generic conversion from integrality gap instances for the Lasserre hierarchy to a {\it dictatorship test} whose soundness is at most integrality gap. Dictatorship testing gadgets are central to hardness results for CSPs, and a generic conversion of the above nature lies at the core of the tight Unique Games based hardness result for CSPs. \cite{Raghavendra08}