A nearly linear time algorithm for the half integral disjoint paths packing

  • Authors:

  • Ken-ichi Kawarabayashi;Bruce Reed

  • Affiliations:

  • National Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo, Japan;McGill University, Montreal Canada and INRIA, Laboratoire I3S, CNRS, Sophia-Antipolis, France

  • Venue:
  • Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2008

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Abstract

We consider the following problem, which is called the half integral k disjoint paths packing. Input: A graph G, k pair of vertices (s1, t1), (s2, t2),…, (sk, tk) in G (which are sometimes called terminals). Output: Paths P1,…, Pk in G such that Pi joins si and ti for i = 1,2,…, k, and in addition, each vertex is on at most two of these paths. We present an O(n log n) time algorithm for this problem for fixed k. This improves a result by Kleinberg [21] who gave an O(n3) algorithm for this problem. In fact, we also have algorithms running in O(n(1+ε)) time for any ε 0 for these problems, if k is up to o((log log n)2/5) for general graphs, up to o((log n/(log log n))1/4) for planar graphs, and up to o((log n/g/(log log n/g))1/4) for graphs on the surface, where g is the Euler genus. Furthermore, if k is fixed, then we have linear time algorithms for the planar case and for the bounded genus case. We also obtain O(n log n) algorithms for several optimization problems related to the bounded unsplittable flow problem when the number of terminal pairs is bounded. These results can all carry over to problems involving edge capacities.