TY - JOUR

T1 - The b-branching problem in digraphs

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

AU - Takazawa, Kenjiro

N1 - Funding Information:
The authors are thankful to the anonymous referees for helpful comments. The first author is supported by JST ERATO Grant Number JPMJER1201 , JSPS KAKENHI Grant Number JP17K00028 , Japan. The second author is supported by JST PRESTO Grant Number JPMJPR14E1 , Japan. The third author is supported by JST CREST Grant Number JPMJCR1402 , JSPS KAKENHI Grant Number JP16K16012 , Japan.
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/9/15

Y1 - 2020/9/15

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 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.

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 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.

KW - Arborescence

KW - Greedy algorithm

KW - Matroid intersection

KW - Packing

KW - Sparsity matroid

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U2 - 10.1016/j.dam.2020.02.005

DO - 10.1016/j.dam.2020.02.005

M3 - Article

AN - SCOPUS:85081238956

SN - 0166-218X

VL - 283

SP - 565

EP - 576

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -