The minimum bisection in the planted bisection model

Abstract

In the planted bisection model a random graph G(n, p+, p-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to ±1). Any two vertices that belong to the same class are linked by an edge with probability p+ and any two that belong to different classes with probability p- < d+ that remain fixed as n → ∞, then w.h.p. the "planted" bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that d+ - d- > c√ d+ ln d+ for a certain constant c > 0.