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 $c\in(0,1/2)$, where $G$ has some (unknown) $c$-balanced vertex separator of size ${\rm OPT}_c$, we give a (Monte-Carlo randomized) algorithm running in $O(n^{O(\varepsilon)}m^{1+o(1)})$ time that produces a $\Theta(1)$-balanced vertex separator of size $O({\rm OPT}_c\cdot\sqrt{(\log n)/\varepsilon})$ for any value $\varepsilon\in[\Theta(1/\log(n)),\Theta(1)]$. In particular, for any function $f(n)=\omega(1)$ (including $f(n)=\log\log n$, for instance), we can produce a vertex separator of size $O({\rm OPT}_c\cdot\sqrt{\log n}\cdot f(n))$ in time $O(m^{1+o(1)})$. Moreover, for an arbitrarily small constant $\varepsilon=\Theta(1)$, our algorithm also achieves the best-known approximation ratio for this problem in $O(m^{1+\Theta(\varepsilon)})$ 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 $O(n^{O(\varepsilon)}\text{polylog}(n))$ almost-linear time maximum-flow computations, and would be sped up if the time complexity of maximum-flow improves.