Abstract
Edmonds’ arborescence packing theorem characterizes directed graphs that have arc-disjoint 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.
Highlights
1.1 Edmonds’ arborescence packing theorem and its extensionsAn arborescence is a rooted directed tree in which each vertex has in-degree one except for one vertex called the root
Kobayashi [2] showed that this minimum weight packing problem can be reduced to the submodular flow problem, which leads to a polynomial-time algorithm
Our goal is to find a minimum weight arc set of an M1-based if (M1)-based packing of s-arborescences
Summary
An arborescence is a rooted directed tree in which each vertex has in-degree one except for one vertex called the root. Durand de Gevigney, Nguyen, and Szigeti [4] obtained another extension of Edmonds’ arborescence packing theorem. Tk of arborescences is said to be M-reachability-based if the root of Ti is si for every i and {si : v ∈ V (Ti )} is a base of {s j : s j ∈ P(v)} for every v ∈ V , where P(v) denotes the set of vertices from which v is reachable by a directed path in D. There exists an M-reachability-based packing of arborescences in D if and only if the multiset {si : si = v} is independent in M for every v ∈ V and. In [21], Cs. Király posed a question whether one can efficiently find a minimum weight arc set of a reachability-based packing of arborescences if a weight function is given on the arc set. Kobayashi [2] showed that this minimum weight packing problem can be reduced to the submodular flow problem, which leads to a polynomial-time algorithm
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