Breaking o(n1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two

  • Authors:

  • Ken-ichi Kawarabayashi;Yusuke Kobayashi

  • Affiliations:

  • National Institute of Informatics, Tokyo, Japan;University of Tokyo, Tokyo, Japan

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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Abstract

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r-approximation algorithm for this problem is a polynomial time algorithm that finds at least OPT / r edge-disjoint paths, where OPT is the maximum possible. Currently, an O(n1/2)-approximation algorithm is best known for this problem even if a congestion of two is allowed, i.e., each edge is allowed to be used in at most two of the paths. In this paper, we give a randomized O(n3/7 • poly (log n))-approximation algorithm with congestion two. This is the first result that breaks the O(n1/2)-approximation algorithm. In particular, we prove the following. 1. If we have a (randomized) polynomial time algorithm for finding Ω(OPT1/p) edge-disjoint paths for some p1, then, for some α 0, we can give a randomized O(n1/2-α)-approximation algorithm for the maximum edge-disjoint paths problem by using Rao-Zhou's algorithm. 2. Based on the well-linked set of Chekuri, Khanna, and Shepherd, we show that there is a randomized algorithm for finding Ω(OPT1/4) edge-disjoint paths connecting given terminal pairs with congestion two. Our framework for this algorithm is more general. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if the following conjecture is true. Conjecture: There is a (randomized) polynomial time algorithm for finding Ω(OPT1/p/β(n)) edge-disjoint paths connecting given terminal pairs, where β is a poly-logarithmic function.