Edge disjoint paths in moderately connected graphs

We study the Edge Disjoint Paths (EDP) problem in undirected graphs: Given a graph G with n nodes and a set T of pairs of terminals, connect as many terminal pairs as possible using paths that are mutually edge disjoint. This leads to a variety of classic NP-complete problems, for which approximability is not well understood. We show a polylogarithmic approximation algorithm for the undirected EDP problem in general graphs with a moderate restriction on graph connectivity; we require the global minimum cut of G to be Omega(log(5) n). Previously, constant or polylogarithmic approximation algorithms were known for trees with parallel edges, expanders, grids and grid-like graphs, and most recently, even-degree planar graphs. These graphs either have special structure (e.g., they exclude minors) or there are large numbers of short disjoint paths. Our algorithm extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors.