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 $$K\in {\mathcal {O}}(\hbox {polylog } n)$$ , then we obtain an $$(\alpha +\varepsilon )$$ -separator of size $${\mathcal {O}}(\varepsilon ^{-1}\hbox { polylog } n)$$ in time $${\mathcal {O}}(\varepsilon ^{-1}m\hbox { polylog } n)$$ w.h.p. The presented algorithm does not require knowledge of K.