Faster SDP Hierarchy Solvers for Local Rounding Algorithms

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 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^Ω(r) where n is the input size. In this work, we observe that many of these algorithms are based on local rounding procedures that only use a small part of the SDP solution (of size n^O(1)2^O(r) instead of n^Ω(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/λ_r) approximations for graph partitioning problems such as minimum bisection and small set expansion in n^O(1)2^O(r) time, where λ_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 Ω(n) in 3-colorable graphs in (n2^r)^O(1) time provided λ_n-r <; 17/16. 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 op- imization, is an efficient ellipsoid algorithm based separation oracle for convex programs that can output a 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.