A Fast Approximation Algorithm for the Minimum Balanced Vertex Separator in a Graph

We present a family of fast pseudo-approximation algorithms for the minimum balanced vertex separator problem in a graph. Given a graph G=(V,E) with n vertices and m edges, and a (constant) balance parameter c∈(0,1/2), where G has some (unknown) c-balanced vertex separator of size OPT_c, we give a (Monte-Carlo randomized) algorithm running in O(n^O(ε)m^1+o(1)) time that produces a Θ(1)-balanced vertex separator of size O( OPT_c·√((log n)/ε)) for any value ε∈[Θ(1/log(n)),Θ(1)]. In particular, for any function f(n)=ω(1) (including f(n)=loglog n, for instance), we can produce a vertex separator of size O( OPT_c·√(log n)· f(n)) in time O(m^1+o(1)). Moreover, for an arbitrarily small constant ε=Θ(1), our algorithm also achieves the best-known approximation ratio for this problem in O(m^1+Θ(ε)) time. The algorithms are based on a semidefinite programming (SDP) relaxation of the problem, which we solve using the Matrix Multiplicative Weight Update (MMWU) framework of Arora and Kale. Our oracle for MMWU uses O(n^O(ε)polylog(n)) almost-linear time maximum-flow computations, and would be sped up if the time complexity of maximum-flow improves.