Departmental Papers (CIS)

Date of this Version

October 2004

Document Type

Conference Paper


Copyright 2004 IEEE. Reprinted from Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 2004 (FOCS 2004), pages 71-80.
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We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G = (V,E) and a set Τ = {s1t1, s2t2, . . . , sktk} of pairs of vertices: the objective is to find the maximum number of pairs in Τ that can be connected via edge-disjoint paths. Our main result is a poly-logarithmic approximation for MEDP on 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 Ω(√|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 ⊆ V is well-linked if for each S ⊂ V , |δ(S)| ≥ min{|S ∩ X|, |(V - S) ∩ 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 Ω(|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.



Date Posted: 20 February 2005