eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
12:1
12:15
10.4230/LIPIcs.MFCS.2018.12
article
The b-Branching Problem in Digraphs
Kakimura, Naonori
1
Kamiyama, Naoyuki
2
Takazawa, Kenjiro
3
Keio University, Kanagawa 223-8522, Japan
Kyushu University and JST, PRESTO, Fukuoka 819-0395, Japan
Hosei University, Tokyo 184-8584, Japan
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.12/LIPIcs.MFCS.2018.12.pdf
Greedy Algorithm
Packing
Matroid Intersection
Sparsity Matroid
Arborescence