Approximating Small Balanced Vertex Separators in Almost Linear Time

Abstract

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any $\frac{2}{3}\le \alpha <1$ and any $0<\varepsilon <1-\alpha$: If G contains an $\alpha$-separator of size K, then our algorithm finds an $(\alpha +\varepsilon )$-separator of size $\mathcal O(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)$ in time $\mathcal O(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)$ w.h.p. In particular, if $K\in \mathcal O({\text {polylog}}\, n)$, then we obtain an $(\alpha +\varepsilon )$-separator of size $\mathcal O(\varepsilon ^{-1}{\text {polylog}}\, n)$ in time $\mathcal O(\varepsilon ^{-1}m\,{\text {polylog}}\, n)$ w.h.p. The presented algorithm does not require knowledge of K.

Due to space restrictions, no proofs are included in this version of the paper; the full version with a lot of additional material can be found at http://disco.ethz.ch/publications/wads2017-vertexsep.pdf.