# The b-branching problem in digraphs

TL;DR: It is demonstrated that b -branchings yield an appropriate generalization of branchings by extending several classic results on branchings, and a packing theorem extending Edmonds’ disjoint branchings theorem is proved and the integer decomposition property of the b - Branching polytope is proved.

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 D , 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, for every vertex v of D , at most b ( v ) arcs entering v , and it is an independent set of a certain sparsity matroid defined by b and D . We demonstrate that b -branchings yield an appropriate generalization of branchings by extending several classic 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 .

## Summary (1 min read)

### Introduction

- Branchings have several good properties which do not hold for general matroid intersection.
- The objective of this paper is to propose a class of the matroid intersection problem which generalizes branchings and inherits those good properties of branchings.

### 3 Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012,

- JP26280001, Japan. © N. Kakimura, N. Kamiyama, and K. Takazawa; licensed under Creative Commons License CC-BY 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018).
- This offers a new direction of fundamental extensions of the classical theorems on branchings.
- The authors show that their multi-phase greedy algorithm can be extended to this generalized class.
- In Section 2, the authors review the literature of branchings and matroid intersection, including algorithmic, polyhedral, and packing results.
- The integer decomposition property of the branching polytope is a direct consequence of Theorem 2 and Corollary 4. I Corollary 5 ([1]).

### 3.1 Algorithm description

- The authors present a multi-phase greedy algorithm for finding a maximum-weight b-branching by extending the one for branchings [4, 6, 11, 23].
- The authors first show a key property of Min and Msp, which plays an important role in their algorithm.
- Its proof can be found in the full version [31].
- The authors assume that the arc weights are nonnegative, which are represented by a vector w ∈ RA+.
- It is also straightforward to see that the i-th iteration requires O(|A(i)|) time: Steps 2, 3, and 4 respectively require O(|A(i)|) time.

### 3.2 Optimality of the algorithm and totally dual integral system

- The authors first present a linear program describing MFCS 2018 Algorithm 1 Algorithm bB. Input.
- Finally, as a consequence of their packing theorem, the authors prove the integer decomposition property of the b-branching polytope.
- Its proof is described in the full version [31].
- Optimal matroid intersections, Combinatorial Structures and Their Applications, New York, 1970, Gordon and Breach, 233–234, also known as 39 E.L. Lawler.

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### Cites background or methods from "The b-branching problem in digraphs..."

...A theorem on packing disjoint b-branchings is also presented in [13], which extends Edmonds’ disjoint branchings theorem [6] and leads to the integer decomposition property of the b-branching polytope....

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...Second, the proof for the packing theorem utilizes the disjoint b-branchings theorem [13] and the supermodular coloring theorem [22], the latter of which was used in an alternative proof [22] for the packing theorem for bibranchings....

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...[13] presented a multi-phase greedy algorithm for finding a longest b-branching, which extends that for branchings [2–4, 9]....

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...In our formulation, this computation amounts to computing the minimum weight of a b-branching with prescribed indegree, which can be done by using a combinatorial algorithm for the longest b-branching [13]....

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...Finally, for the M-convex submodular flow formulation, we newly prove an exchange property of b-branchings, which extends that for branchings [23] and follows from the disjoint b-branchings theorem [13]....

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2 citations

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