Blocking Optimal <i>k</i>-Arborescences

Bernáth, Attila; Király, Tamás

doi:10.1137/1.9781611974331.ch115

Cited by 4 publications

(9 citation statements)

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“…Our new characterization of reachability-based packing not only implies a faster algorithm for the minimum weight version of the problem but also unifies the reachability-based packing with the matroid restricted packing introduced by Frank [13] (or explicitly by Bernáth and T. Király [3]). Suppose that a matroid M v is given on ∂(v) for all v ∈ V , and let M be the directed sum of M v over v ∈ V .…”

Section: Contributionsmentioning

confidence: 96%

Packing of arborescences with matroid constraints via matroid intersection

2019

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Edmonds' arborescence packing theorem characterizes directed graphs that have arcdisjoint spanning arborescences in terms of connectivity. Later he also observed a characterization in terms of matroid intersection. Since these fundamental results, intensive research has been done for understanding and extending these results. In this paper we shall extend the second characterization to the setting of reachability-based packing of arborescences. The reachability-based packing problem was introduced by Cs. Király as a common generalization of two different extensions of the spanning arborescence packing problem, one is due to Kamiyama, Katoh, and Takizawa, and the other is due to Durand de Gevigney, Nguyen, and Szigeti. Our new characterization of the arc sets of reachability-based packing in terms of matroid intersection gives an efficient algorithm for the minimum weight reachability-based packing problem, and it also enables us to unify further arborescence packing theorems and Edmonds' matroid intersection theorem. For the proof, we also show how a new class of matroids can be defined by extending an earlier construction of matroids from intersecting submodular functions to bi-set functions based on an idea of Frank.

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Section: Contributionsmentioning

confidence: 96%

Packing of arborescences with matroid constraints via matroid intersection

2019

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“…Hereafter, for a positive integer k, we refer to the union of arc-disjoint k arborescences as a k-arborescence and that of arc-disjoint k branchings as a k-branching. Among the recent work on k-arborescences, Bernáth and Király [4] presented a theorem stating that the root vectors of the minimum-cost k-arborescences form a base polyhedron of a submodular function (Theorem 4), which extends a theorem of Frank [15] stating that the root vectors of k-arborescences form a base polyhedron. This theorem suggests a new connection of the minimum-cost k-arborescences (or k-branchings) to the theory of submodular functions [20] and discrete convex analysis [39].…”

Section: Introductionmentioning

confidence: 97%

“…Packing arborescences in digraphs, originating from the seminal work of Edmonds [12], is a classical topic in combinatorial optimization. Up to the present date, it has been actively studied and a number of generalizations have been introduced [3,4,5,10,14,21,28,30,31,35,43,47,50].…”

Section: Introductionmentioning

confidence: 99%

“…This theorem suggests a new connection of the minimum-cost k-arborescences (or k-branchings) to the theory of submodular functions [20] and discrete convex analysis [39]. However, this connection is not explored in [4]: the theorem is just obtained along the way to the main result of [4]. Indeed, while some discrete convexity of the minimum-cost branchings is analyzed in [42,48,49], to the best of our knowledge, discrete convexity of the minimum-cost packings of branchings has never been discussed in the literature.…”

Section: Introductionmentioning

confidence: 99%

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M-convexity of the minimum-cost packings of arborescences

Takazawa¹

2018

Preprint

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The aim of this paper is to reveal the discrete convexity of the minimum-cost packings of arborescences and branchings. We first prove that the minimum-cost packings of disjoint k branchings (minimum-cost k-branchings) induce an M ♮ -convex function defined on the integer vectors on the vertex set. The proof is based on a theorem on packing disjoint k-branchings, which extends Edmonds' disjoint branchings theorem and is of independent interest. We then show the M-convexity of the minimum-cost k-arborescences, which provides a short proof for a theorem of Bernáth and Király (SODA 2016) stating that the root vectors of the minimum-cost k-arborescences form a base polyhedron of a submodular function. Finally, building upon the M ♮ -convexity of k-branchings, we present a new problem of minimum-cost root location of a k-branching, and show that it can be solved in polynomial time if the opening cost function is M ♮ -convex.

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“…Király [33] generalized the result of [9] in the same direction of [32]. A matroid-restricted packing of arborescences [3,19] is another generalization concerning a matroid constraint. We remark that our packing and matroid restriction for b-branchings differ from the above matroidal extensions of packing of arborescences.…”

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confidence: 97%

The b-branching problem in digraphs

Kakimura

Kamiyama

Takazawa

2020

Discrete Applied Mathematics

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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. IntroductionSince the pioneering work of Edmonds [12,14], the importance of matroid intersection has been well appreciated. A special class of matroid intersection is branchings (or arborescences) in digraphs. Branchings have several good properties which do not hold for general matroid intersection. The objective of this paper is to propose a class of matroid intersection which generalizes branchings and inherits those good properties of branchings.One of the good properties of branchings is that a maximum-weight branching can be found by a simple combinatorial algorithm [4,6,11,23]. This algorithm is much simpler than general weighted matroid intersection algorithms, and is referred to as a "multi-phase greedy algorithm" in the textbook by Kleinberg and Tardos [34].

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Blocking Optimal k-Arborescences

Cited by 4 publications

References 9 publications

Packing of arborescences with matroid constraints via matroid intersection

Packing of arborescences with matroid constraints via matroid intersection

M-convexity of the minimum-cost packings of arborescences

The b-branching problem in digraphs

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