The edge disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

  • Authors:

  • Ken-ichi Kawarabayashi;Yusuke Kobayashi

  • Affiliations:

  • National Institute of Informatics, Chiyoda-ku, Tokyo, Japan;University of Tokyo, Tokyo, Japan

  • Venue:
  • SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2010

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Abstract

We consider the following well-known problem, which is called the edge-disjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Output: Edge-disjoint paths P1, P2,..., Pk in G such that Pi joins si and ti for i = 1, 2,..., k. Robertson and Seymour's graph minor project gives rise to an O(m3) algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project, spanning 23 papers and at least 500 pages proof. We give a faster algorithm and a simpler proof of the correctness for the edge-disjoint paths problem for any fixed k. Our results can be summarized as follows: 1. If an input graph G is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤ 3-edge-cuts. (iii) Excluding large clique minors. 2. When an input graph G is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be non-trivially superconstant number, up to k = O((log log log n)1/2-ε) for any ε 0. Thus our hidden constant in this case is dramatically smaller than Robertson-Seymour's. In addition, if an input graph G is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((log n)1/2-ε) for any ε 0. The same thing holds for bounded genus graphs. Moreover, if an input graph is either 4-edge-connected H-minor-free or Eulerian H-minor-free for fixed graph H, k is allowed to be O((log log n)1/2-ε) for any ε 0. 3. We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, the time complexity of our algorithm is O(n2), which is faster than Robertson and Seymour's.