### 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.

Original language | English |
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Title of host publication | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |

Editors | Seok-Hee Hong |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 63.1-63.13 |

ISBN (Electronic) | 9783959770262 |

DOIs | |

Publication status | Published - Dec 1 2016 |

Event | 27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia Duration: Dec 12 2016 → Dec 14 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 64 |

ISSN (Print) | 1868-8969 |

### Other

Other | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |
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Country | Australia |

City | Sydney |

Period | 12/12/16 → 12/14/16 |

### All Science Journal Classification (ASJC) codes

- Software

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## Cite this

*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