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Journal Article•DOI•

The b-branching problem in digraphs

Naonori Kakimura1, Naoyuki Kamiyama2, Kenjiro Takazawa3•
Keio University1, Kyushu University2, Hosei University3
15 Sep 2020-Discrete Applied Mathematics (Elsevier)-Vol. 283, pp 565-576
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)

Jump to: [Introduction] – [3 Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012,] – [3.1 Algorithm description] and [3.2 Optimality of the algorithm and totally dual integral system]

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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Citations
More filters
Book Chapter•DOI•
Notes on Equitable Partitions into Matching Forests in Mixed Graphs and b-branchings in Digraphs

[...]

Kenjiro Takazawa1•
Hosei University1
04 May 2020
TL;DR: This paper presents two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in Digraphs, and proves that equitable partitions always exist.
Abstract: An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into k branchings, there always exists an equitable partition into k branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in digraphs. For matching forests, Kiraly and Yokoi (2018) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of the covered vertices. For b-branchings, we define an equitability based on the size of the b-branching and the indegrees of all vertices. For both matching forests and b-branchings, we prove that equitable partitions always exist.

3 citations

Posted Content•
The $b$-bibranching Problem: TDI System, Packing, and Discrete Convexity.

[...]

Kenjiro Takazawa1•
Hosei University1
09 Feb 2018-arXiv: Discrete Mathematics
TL;DR: A linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for $b-bibranchings are presented, which imply polynomial algorithms for finding a shortest bibranching.
Abstract: In this paper, we introduce the $b$-bibranching problem in digraphs, which is a common generalization of the bibranching and $b$-branching problems The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation The $b$-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa (2018), is a generalization of the branching problem admitting higher indegree, ie, each vertex $v$ can have indegree at most $b(v)$ For $b$-branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended A main contribution of this paper is to extend those previous results on bibranchings and $b$-branchings to $b$-bibranchings That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for $b$-bibranchings In particular, the linear program and M-convex submodular flow formulations respectively imply polynomial algorithms for finding a shortest $b$-bibranching

3 citations

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....

    [...]

  • ...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....

    [...]

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

    [...]

  • ...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]....

    [...]

  • ...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]....

    [...]

Posted Content•
Notes on Equitable Partition into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs

[...]

Kenjiro Takazawa
24 Mar 2020-arXiv: Combinatorics
TL;DR: This paper presents two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into $b-branchings in Digraphs, and proves that equitable partitions always exist.
Abstract: An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into $k$ branchings, there always exists an equitable partition into $k$ branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into $b$-branchings in digraphs. For matching forests, Kiraly and Yokoi (2018) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of the covered vertices. For $b$-branchings, we define an equitability based on the size of the $b$-branching and the indegrees of all vertices. For both matching forests and $b$-branchings, we prove that equitable partitions always exist.

2 citations

Journal Article•DOI•
The $b$-bibranching Problem: TDI System, Packing, and Discrete Convexity

[...]

Kenjiro Takazawa1•
Hosei University1
23 Mar 2021-Networks
TL;DR: In this article, the authors introduce the problem of finding a shortest branching in digraphs, which is a generalization of the branching and bipartite edge cover problems, and present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation.
Abstract: In this paper, we introduce the $b$-bibranching problem in digraphs, which is a common generalization of the bibranching and $b$-branching problems. The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation. The $b$-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa (2018), is a generalization of the branching problem admitting higher indegree, i.e., each vertex $v$ can have indegree at most $b(v)$. For $b$-branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended. A main contribution of this paper is to extend those previous results on bibranchings and $b$-branchings to $b$-bibranchings. That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for $b$-bibranchings. In particular, the linear program and M-convex submodular flow formulations respectively imply polynomial algorithms for finding a shortest $b$-bibranching.

2 citations

Proceedings Article•DOI•
The B-branching problem in digraphs

[...]

Naonori Kakimura1, Naoyuki Kamiyama2, Kenjiro Takazawa3•
Keio University1, Kyushu University2, Hosei University3
01 Aug 2018
TL;DR: This paper presents a multi-phase greedy algorithm for finding a maximum-weight $b-branching, proves a packing theorem extending Edmonds' disjoint branchings theorem, and provides a strongly polynomial algorithm forFinding optimal disjointed $b$-br branchings.
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 .
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Frequently Asked Questions (1)
Q1. What are the contributions in "The b-branching problem in digraphs" ?

In this paper, the authors introduce the concept of b-branchings in digraphs, which is a generalization of branchings serving as a counterpart of b-matchings. The authors demonstrate that b-branchings yield an appropriate generalization of branchings by extending several classical results on branchings. The authors first present a multi-phase greedy algorithm for finding a maximumweight b-branching. The authors 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, the authors prove the integer decomposition property of the bbranching polytope. Finally, the authors deal with a further generalization in which a matroid constraint is imposed on the b ( v ) arcs sharing the terminal vertex v. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms