The b-branching problem in digraphs

Naonori Kakimura, Naoyuki Kamiyama, Kenjiro Takazawa

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)565-576
Number of pages12
JournalDiscrete Applied Mathematics
Publication statusPublished - 2020 Sept 15


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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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