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(log5 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.