The disjoint paths problem in quadratic time

  • Authors:

  • Ken-ichi Kawarabayashi;Yusuke Kobayashi;Bruce Reed

  • Affiliations:

  • National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan;Canada Research Chair in Graph Theory, McGill University, Montreal, Canada and Project Mascotte, INRIA, Laboratoire I3S, CNRS, Sophia-Antipolis, France

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

We consider the following well-known problem, which is called the disjoint paths problem. For a given graph G and a set of k pairs of terminals in G, the objective is to find k vertex-disjoint paths connecting given pairs of terminals or to conclude that such paths do not exist. We present an O(n^2) time algorithm for this problem for fixed k. This improves the time complexity of the seminal result by Robertson and Seymour, who gave an O(n^3) time algorithm for the disjoint paths problem for fixed k. Note that Perkovic and Reed (2000) announced in [24] (without proofs) that this problem can be solved in O(n^2) time. Our algorithm implies that there is an O(n^2) time algorithm for the k edge-disjoint paths problem, the minor containment problem, and the labeled minor containment problem. In fact, the time complexity of all the algorithms with the most expensive part depending on Robertson and Seymour@?s algorithm can be improved to O(n^2), for example, the membership testing for minor-closed class of graphs.