The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e29" altimg="si20.svg"><mml:mi>b</mml:mi></mml:math>-branching problem in digraphs

Kakimura, Naonori; Kamiyama, Naoyuki; Takazawa, Kenjiro

doi:10.1016/j.dam.2020.02.005

Cited by 5 publications

(23 citation statements)

References 41 publications

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“…Although several papers generalizing Edmonds' theorem on packing arborescences appeared in the last decade (for recent papers with great overviews, see e.g. [18,28,38]), this problem remains widely open.…”

Section: Introductionmentioning

confidence: 99%

Complexity of packing common bases in matroids

Bérczi

Schwarcz

2020

Math. Program.

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One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of k disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases.In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into k common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when k = 2.Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when k = 2, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.

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Section: Introductionmentioning

confidence: 99%

Complexity of packing common bases in matroids

Bérczi

Schwarcz

2020

Math. Program.

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“…Let D = (V, A) be a digraph and b ∈ Z V ++ be a positive integer vector on V. An arc subset B ⊆ A is a b-branching [13] if…”

Section: B-branchingmentioning

confidence: 99%

“…What is more, b-branchings inherit several good properties of branchings. In [13], a multi-phase greedy algorithm for finding a longest b-branching and a theorem on packing disjoint b-branchings are presented. The former is an extension of that for finding a longest branching [2][3][4]9].…”

Section: B-branchingmentioning

confidence: 99%

“…A b-branching is defined as a common independent set of two matroids generalizing these two matroids: one matroid imposes that each vertex v can have indegree at most b(v); and the other matroid is a sparsity matroid defined by b (see Section 2.3 for precise description). We remark that a branching is a special case where b(v) = 1 for every vertex v. Kakimura et al [13] presented a multi-phase greedy algorithm for finding a longest b-branching, which extends that for branchings [2][3][4]9]. 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.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that a branching is a special case where b(v) = 1 for every vertex v. Kakimura et al [13] presented a multi-phase greedy algorithm for finding a longest b-branching, which extends that for branchings [2][3][4]9]. 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.…”

Section: Introductionmentioning

confidence: 99%

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The b‐bibranching problem: TDI system, packing, and discrete convexity

Takazawa

2021

Networks

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In this article, 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, 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, is a generalization of the branching problem admitting higher indegree, that is, 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 article 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.

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Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs

Takazawa¹

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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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, Kir\'{a}ly and Yokoi (2022) 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 covered vertices, which is plausible in the light of the delta-matroid structure of matching forests. While the existence of this equitable partition can be derived from a lemma in Kir\'{a}ly and Yokoi, we present its direct and simpler proof. For $b$-branchings, we define an equitability notion based on the size of the $b$-branching and the indegrees of all vertices, and prove that an equitable partition always exists. We then derive the integer decomposition property of the associated polytopes.

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The b-branching problem in digraphs

Cited by 5 publications

References 41 publications

Complexity of packing common bases in matroids

Complexity of packing common bases in matroids

The b‐bibranching problem: TDI system, packing, and discrete convexity

Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs

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