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DOI: 10.4230/LIPIcs.ISAAC.2016.63
Shortest Disjoint S-Paths Via Weighted Linear Matroid Parity

Authors: Yutaro Yamaguchi

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

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.
Cite as

Yutaro Yamaguchi. Shortest Disjoint S-Paths Via Weighted Linear Matroid Parity. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{yamaguchi:LIPIcs.ISAAC.2016.63,
  author =	{Yamaguchi, Yutaro},
  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 =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.63},
  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}
}
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