Shortest disjoint S-paths via weighted linear matroid parity

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Abstract

Mader's disjoint S-paths problem unifies two generalizations of bipartite matching: (a) non-bipartite matching and (b) disjoint s-t paths. Lovász (1980, 1981) first proposed an efficient algorithm for this problem via a reduction to matroid matching, which also unifies two generalizations of bipartite matching: (a) non-bipartite matching and (c) matroid intersection. While the weighted versions of the problems (a)-(c) in which we aim to minimize the total weight of a designated-size feasible solution are known to be solvable in polynomial time, the tractability of such a weighted version of Mader's problem has been open for a long while. In this paper, we present the first solution to this problem with the aid of a linear representation for Lovász' reduction (which leads to a reduction to linear matroid parity) due to Schrijver (2003) and polynomial-time algorithms for a weighted version of linear matroid parity announced by Iwata (2013) and by Pap (2013). Specifically, we give a reduction of the weighted version of Mader's problem to weighted linear matroid parity, which leads to an O(n5)-time algorithm for the former problem, where n denotes the number of vertices in the input graph. Our reduction technique is also applicable to a further generalized framework, packing non-zero A-paths in group-labeled graphs, introduced by Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour (2006). The extension leads to the tractability of a broader class of weighted problems not restricted to Mader's setting.

Original languageEnglish
Title of host publication27th International Symposium on Algorithms and Computation, ISAAC 2016
EditorsSeok-Hee Hong
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages63.1-63.13
ISBN (Electronic)9783959770262
DOIs
Publication statusPublished - Dec 1 2016
Event27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia
Duration: Dec 12 2016Dec 14 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume64
ISSN (Print)1868-8969

Other

Other27th International Symposium on Algorithms and Computation, ISAAC 2016
CountryAustralia
CitySydney
Period12/12/1612/14/16

All Science Journal Classification (ASJC) codes

  • Software

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  • Cite this

    Yamaguchi, Y. (2016). Shortest disjoint S-paths via weighted linear matroid parity. In S-H. Hong (Ed.), 27th International Symposium on Algorithms and Computation, ISAAC 2016 (pp. 63.1-63.13). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 64). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2016.63