TY - GEN

T1 - The B-branching problem in digraphs

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

AU - Takazawa, Kenjiro

N1 - Funding Information:
Supported by JST ERATO Grant Number JPMJER1201, JSPS KAKENHI Grant Number JP17K00028, Japan. 2 Supported by JST PRESTO Grant Number JPMJPR14E1, Japan. 3 Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012, JP26280001, Japan.
Publisher Copyright:
© N. Kakimura, N. Kamiyama, and K. Takazawa.

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

AN - SCOPUS:85053211348

SN - 9783959770866

T3 - Leibniz International Proceedings in Informatics, LIPIcs

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

T2 - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018

Y2 - 27 August 2018 through 31 August 2018

ER -