Balanced Cut Approximation in Random Geometric Graphs

Abstract

A random geometric graph ${\mathcal{G}}(n,r)$ is obtained by spreading n points uniformly at random in a unit square, and by associating a vertex to each point and an edge to each pair of points at Euclidian distance at most r. Such graphs are extensively used to model wireless ad-hoc networks, and in particular sensor networks. It is well known that, over a critical value of r, the graph is connected with high probability. In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges. In particular, we show that, for a sufficiently large r, any cut which separates two components of Θ(n) vertices each contains Ω(n2r3) edges with high probability. We also present a simple algorithm that, again with high probability, computes one such cut of size O(n2r3). From these two results we derive a constant expected approximation algorithm for the β-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least βn vertices each.