### 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 bbranching 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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Title of host publication | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |

Editors | Igor Potapov, James Worrell, Paul Spirakis |

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

Volume | 117 |

ISBN (Print) | 9783959770866 |

DOIs | |

Publication status | Published - 2018 Aug 1 |

Event | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 - Liverpool, United Kingdom Duration: 2018 Aug 27 → 2018 Aug 31 |

### Other

Other | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |
---|---|

Country | United Kingdom |

City | Liverpool |

Period | 18/8/27 → 18/8/31 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018*(Vol. 117). [12] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.MFCS.2018.12

**The B-branching problem in digraphs.** / Kakimura, Naonori; Kamiyama, Naoyuki; Takazawa, Kenjiro.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018.*vol. 117, 12, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, Liverpool, United Kingdom, 18/8/27. https://doi.org/10.4230/LIPIcs.MFCS.2018.12

}

TY - GEN

T1 - The B-branching problem in digraphs

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

AU - Takazawa, Kenjiro

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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 bbranching 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.

AB - 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 bbranching 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.

KW - Arborescence

KW - Greedy algorithm

KW - Matroid intersection

KW - Packing

KW - Sparsity matroid

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U2 - 10.4230/LIPIcs.MFCS.2018.12

DO - 10.4230/LIPIcs.MFCS.2018.12

M3 - Conference contribution

SN - 9783959770866

VL - 117

BT - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018

A2 - Potapov, Igor

A2 - Worrell, James

A2 - Spirakis, Paul

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -