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DOI: 10.4230/LIPIcs.MFCS.2018.12
URN: urn:nbn:de:0030-drops-95948
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9594/
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Kakimura, Naonori ; Kamiyama, Naoyuki ; Takazawa, Kenjiro

The b-Branching Problem in Digraphs

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Abstract

In this paper, we introduce the concept of b-branchings in digraphs, which is a generalization of branchings serving as a counterpart of b-matchings. Here b is a positive integer vector on the vertex set of a digraph, and a b-branching is defined as a common independent set of two matroids defined by b: an arc set is a b-branching if it has at most b(v) arcs sharing the terminal vertex v, and it is an independent set of a certain sparsity matroid defined by b. We demonstrate that b-branchings yield an appropriate generalization of branchings by extending several classical results on branchings. We first present a multi-phase greedy algorithm for finding a maximum-weight b-branching. We then prove a packing theorem extending Edmonds' disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint b-branchings. As a consequence of the packing theorem, we prove the integer decomposition property of the b-branching polytope. Finally, we deal with a further generalization in which a matroid constraint is imposed on the b(v) arcs sharing the terminal vertex v.

BibTeX - Entry

@InProceedings{kakimura_et_al:LIPIcs:2018:9594,
  author =	{Naonori Kakimura and Naoyuki Kamiyama and Kenjiro Takazawa},
  title =	{{The b-Branching Problem in Digraphs}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9594},
  URN =		{urn:nbn:de:0030-drops-95948},
  doi =		{10.4230/LIPIcs.MFCS.2018.12},
  annote =	{Keywords: Greedy Algorithm, Packing, Matroid Intersection, Sparsity Matroid, Arborescence}
}

Keywords: Greedy Algorithm, Packing, Matroid Intersection, Sparsity Matroid, Arborescence
Seminar: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 20.08.2018


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