The b-Branching Problem in Digraphs
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.
Greedy Algorithm
Packing
Matroid Intersection
Sparsity Matroid
Arborescence
Mathematics of computing~Graph algorithms
12:1-12:15
Regular Paper
https://arxiv.org/abs/1802.02381
Naonori
Kakimura
Naonori Kakimura
Keio University, Kanagawa 223-8522, Japan
Supported by JST ERATO Grant Number JPMJER1201, JSPS KAKENHI Grant Number JP17K00028, Japan.
Naoyuki
Kamiyama
Naoyuki Kamiyama
Kyushu University and JST, PRESTO, Fukuoka 819-0395, Japan
Supported by JST PRESTO Grant Number JPMJPR14E1, Japan.
Kenjiro
Takazawa
Kenjiro Takazawa
Hosei University, Tokyo 184-8584, Japan
Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012, JP26280001, Japan.
10.4230/LIPIcs.MFCS.2018.12
N. Kakimura, N. Kamiyama, and K. Takazawa
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode