Journal
JOURNAL OF COMBINATORIAL THEORY SERIES B
Volume 102, Issue 2, Pages 424-435Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jctb.2011.07.004
Keywords
Disjoint paths; Quadratic time; Graph minor; Tree-width
Categories
Funding
- Japan Society for the Promotion of Science
- CC Foundation
- Kayamori Foundation
- Inoue Research Award for Young Scientists
- The research and training center for new development in mathematics, MEXT, Japan
- Grants-in-Aid for Scientific Research [22800005] Funding Source: KAKEN
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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. (C) 2011 Elsevier Inc. All rights reserved.
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