Packing A-paths in group-labelled graphs via linear matroid parity

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Abstract

Mader's disjoint 5-paths problem is a common generalization of non-bipartite matching and Menger's disjoint paths problems. Lovasz (1980) suggested a polynomial-time algorithm for this problem through a reduction to matroid matching. A more direct reduction to the linear matroid parity problem was given later by Schrijver (2003), which leads to faster algorithms. As a generalization of Mader's problem, Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour (2006) introduced a framework of packing non-zero A-paths in group-labelled graphs, and proved a min-max theorem. Chudnovsky, Cunningham, and Geelen (2008) provided an efficient combinatorial algorithm for this generalized problem. On the other hand, Pap (2007) introduced a framework of packing non-returning A-paths as a further genaralization. In this paper, we discuss a possible extension of Schri- jver's reduction technique to another framework introduced by Pap (2006), under the name of the subgroup model, which apparently generalizes but in fact is equivalent to packing non-returning .A-paths. We provide a necessary and sufficient condition for the groups in question to admit a reduction to the linear matroid parity problem. As a consequence, we give faster algorithms for important special cases of packing non-zero A-paths such as odd-length .4-paths. In addition, it turns out that packing non-returning A-paths admits a reduction to the linear matroid parity problem, which leads to the quite efficient solvability, if and only if the size of the input label set is at most four.

Original languageEnglish
Title of host publicationProceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
PublisherAssociation for Computing Machinery
Pages562-569
Number of pages8
ISBN (Print)9781611973389
DOIs
Publication statusPublished - 2014
Externally publishedYes
Event25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States
Duration: Jan 5 2014Jan 7 2014

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
CountryUnited States
CityPortland, OR
Period1/5/141/7/14

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Yamaguchi, Y. (2014). Packing A-paths in group-labelled graphs via linear matroid parity. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 (pp. 562-569). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611973402.42