Edge-Disjoint Paths in Planar Graphs

We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G=(V,E) and a set T = {s1t1, s2t2, ..., sktk} of pairs of vertices: the objective is to find the maximum number of pairs in T that can be connected via edge-disjoint paths. Our main result is a poly-logarithmic approximation for MEDP in undirected planar graphs if a congestion of 2 is allowed, that is, we allow up to 2 paths to share an edge. Prior to our work, for any constant congestion, only a polynomial-factor approximation was known for planar graphs although much stronger results are known for some special cases such as grids and grid-like graphs. We note that the natural multicommodity flow relaxation of the problem has an integrality gap of Omega(sqrt{|V|}) even on planar graphs when no congestion is allowed. Our starting point is the same relaxation and our result implies that the integrality gap shrinks to a poly-logarithmic factor once 2 paths are allowed per edge. Our result also extends to the unsplittable-flow problem and the maximum integer multicommodity flow problem. A set X subseteq V is well-linked if for each S subset V, |delta(S)| geq min{|Scap X|,|(V-S) cap X|}. The heart of our approach is to show that in any undirected planar graph, given any matching M on a well-linked set X, we can route Omega(|M|) pairs in M with a congestion of 2. Moreover, all pairs in M can be routed with constant congestion for a sufficiently large constant. This results also yields a different proof of a theorem of Klein, Plotkin, and Rao that shows an O(1) maxflow-mincut gap for uniform multicommodity flow instances in planar graphs. The framework developed in this paper applies to general graphs as well. If a certain graph theoretic conjecture is true, it will yield poly-logarithmic integrality gap for MEDP with constant congestion.