## Departmental Papers (CIS)

#### Date of this Version

October 2004

#### Document Type

Conference Paper

#### Recommended Citation

Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd, "Edge-Disjoint Paths in Planar Graphs", . October 2004.

#### Abstract

We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G = (V,E) and a set Τ = {s_{1}t_{1}, s_{2}t_{2}, . . . , s_{k}t_{k}} 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

## Comments

Copyright 2004 IEEE. Reprinted from

Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 2004 (FOCS 2004), pages 71-80.Publisher URL: http://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=29918

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