### 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 D, 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, for every vertex v of D, at most b(v) arcs entering v, and it is an independent set of a certain sparsity matroid defined by b and D. We demonstrate that b-branchings yield an appropriate generalization of branchings by extending several classic 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.

Original language | English |
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Pages (from-to) | 565-576 |

Number of pages | 12 |

Journal | Discrete Applied Mathematics |

Volume | 283 |

DOIs | |

Publication status | Published - 2020 Sep 15 |

### Keywords

- Arborescence
- Greedy algorithm
- Matroid intersection
- Packing
- Sparsity matroid

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*283*, 565-576. https://doi.org/10.1016/j.dam.2020.02.005