Convex relaxations based on different hierarchies of
linear/semi-definite programs have been used recently to devise
approximation algorithms for various optimization problems. The
approximation guarantee of these algorithms improves with the number
of {\em rounds} $r$ in the hierarchy, though the complexity of solving
(or even writing down the solution for) the $r$'th level program grows
as $n^{\Omega(r)}$ where $n$ is the input size.
In this work, we observe that many of these algorithms are based on
{\em local} rounding procedures that only use a small part of the SDP
solution (of size $n^{O(1)} 2^{O(r)}$ instead of $n^{\Omega(r)}$). We
give an algorithm to find the requisite portion in time polynomial in
its size. The challenge in achieving this is that the required portion
of the solution is not fixed a priori but depends on other parts of
the solution, sometimes in a complicated iterative manner.
Our solver leads to $n^{O(1)} 2^{O(r)}$ time algorithms to obtain the
same guarantees in many cases as the earlier $n^{O(r)}$ time
algorithms based on $r$ rounds of the Lasserre hierarchy. In
particular, guarantees based on $O(\log n)$ rounds can be realized in
polynomial time. For instance, one can (i) get $O(1/\lambda_r)$
approximations for graph partitioning problems such as minimum
bisection and small set expansion in $n^{O(1)}
2^{O(r)}$ time, where $\lambda_r$ is the $r$'th smallest eigenvalue of
the graph's normalized Laplacian; (ii) a similar guarantee in
$n^{O(1)} k^{O(r)}$ for Unique Games where $k$ is the number of labels (the polynomial dependence on $k$ is new); and (iii) find an independent set of size $\Omega(n)$ in $3$-colorable graphs in $(n 2^r)^{O(1)}$ time provided $\lambda_{n-r} \le 1.11$.
We develop and describe our algorithm in a fairly general abstract
framework. The main technical tool in our work, which might be of
independent interest in convex optimization, is an efficient ellipsoid
algorithm based separation oracle for convex programs that can output
a {\em certificate of infeasibility with restricted support}. This is
used in a recursive manner to find a sequence of consistent points in
nested convex bodies that ``fools'' local rounding algorithms.
