Rounding Semidefinite Programming Hierarchies via Global Correlation

We show a new way to round vector solutions of semi definite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every $2$-CSP instance $\Ins$, a rounding algorithm for $r$ rounds of the Lasserre SDP hierarchy for $\Ins$ that obtains an integral solution which is at most $\e$ worse than the relaxation's value (normalized to lie in $[0,1]$), as long as\[ r >, k\cdot\rank_{\geq \theta}(\Ins)/\poly(\e) \;,\]where $k$ is the alphabet size of $\Ins$, $\theta=\poly(\e/k)$, and $\rank_{\geq \theta}(\Ins)$ denotes the number of eigen values larger than $\theta$ in the normalized adjacency matrix of the constraint graph of $\Ins$. In the case that $\Ins$ is a \unique games instance, the threshold $\theta$ is only a polynomial in $\e$, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for \emph{every} instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent sub exponential algorithm of Aurora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in some cases $r$ rounds of our program can be evaluated in time$2^{O(r)}\poly(n)$.