Edge-Disjoint Paths in Expander Graphs

Abstract

Given a graph G=(V,E)and a set of $\kappa$ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of $\kappa$ paths so found is edge-disjoint. For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed. We present a polynomial time randomized algorithm for finding edge-disjoint paths in an r-regular expander graph G. We show that if G has sufficiently strong expansion properties and r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined. This is within a constant factorof best possible.