Document

# The b-Branching Problem in Digraphs

## File

LIPIcs.MFCS.2018.12.pdf
• Filesize: 484 kB
• 15 pages

## Cite As

Naonori Kakimura, Naoyuki Kamiyama, and Kenjiro Takazawa. The b-Branching Problem in Digraphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.12

## 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 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• Greedy Algorithm
• Packing
• Matroid Intersection
• Sparsity Matroid
• Arborescence

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0