License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.63
URN: urn:nbn:de:0030-drops-68325
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6832/
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Yamaguchi, Yutaro

Shortest Disjoint S-Paths Via Weighted Linear Matroid Parity

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LIPIcs-ISAAC-2016-63.pdf (0.4 MB)

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(n^5)-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.

BibTeX - Entry

@InProceedings{yamaguchi:LIPIcs:2016:6832,
  author =	{Yutaro Yamaguchi},
  title =	{{Shortest Disjoint S-Paths Via Weighted Linear Matroid Parity}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{63:1--63:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Seok-Hee Hong},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6832},
  URN =		{urn:nbn:de:0030-drops-68325},
  doi =		{10.4230/LIPIcs.ISAAC.2016.63},
  annote =	{Keywords: Mader's S-paths, packing non-zero A-paths in group-labeled graphs, linear matroid parity, weighted problems, tractability}
}

Keywords: Mader's S-paths, packing non-zero A-paths in group-labeled graphs, linear matroid parity, weighted problems, tractability
Collection: 27th International Symposium on Algorithms and Computation (ISAAC 2016)
Issue Date: 2016
Date of publication: 07.12.2016

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