  We show a new way to round vector solutions of semidefinite 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 I a rounding algorithm for
r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution
that is at most \eps 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 I, $\theta=\poly(\e/k)$, and $\rank_{\geq \theta}(\Ins)$
denotes the number of eigenvalues larger than $\theta$ in the normalized
adjacency matrix of the constraint graph of $\Ins$.
  In the case that $\Ins$ is a \uniquegames 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 subexponential algorithm
of Arora, 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 $2^{O(r)}\poly(n)$.
