Abstract
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.
| Original language | English |
|---|---|
| Title of host publication | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |
| Editors | Igor Potapov, James Worrell, Paul Spirakis |
| Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
| Volume | 117 |
| ISBN (Print) | 9783959770866 |
| DOIs | |
| Publication status | Published - 2018 Aug 1 |
| Event | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 - Liverpool, United Kingdom Duration: 2018 Aug 27 → 2018 Aug 31 |
Other
| Other | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |
|---|---|
| Country | United Kingdom |
| City | Liverpool |
| Period | 18/8/27 → 18/8/31 |
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Keywords
- Arborescence
- Greedy algorithm
- Matroid intersection
- Packing
- Sparsity matroid
ASJC Scopus subject areas
- Software
Cite this
The B-branching problem in digraphs. / Kakimura, Naonori; Kamiyama, Naoyuki; Takazawa, Kenjiro.
43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018. ed. / Igor Potapov; James Worrell; Paul Spirakis. Vol. 117 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. 12.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - The B-branching problem in digraphs
AU - Kakimura, Naonori
AU - Kamiyama, Naoyuki
AU - Takazawa, Kenjiro
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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UR - http://www.scopus.com/inward/citedby.url?scp=85053211348&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.MFCS.2018.12
DO - 10.4230/LIPIcs.MFCS.2018.12
M3 - Conference contribution
SN - 9783959770866
VL - 117
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
ER -