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.MFCS.2018.12
URN: urn:nbn:de:0030-drops-95948
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9594/
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### 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 Collection: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) Issue Date: 2018 Date of publication: 27.08.2018

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