Arborescence Packing Research Articles

Matroid-rooted packing of arborescences

Matroid-rooted packing of arborescences

Reachability in arborescence packings

Fortier et al. proposed several research problems on packing arborescences and settled some of them. Others were later solved by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We prove that a strong version of Edmonds’ theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa’s result on packing reachability arborescences and that Durand de Gevigney, Nguyen and Szigeti’s theorem on matroid-based packing of arborescences implies Király’s result on matroid-reachability-based packing of arborescences. Further, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al.. Finally, we deal with the algorithmic aspects of the problems considered. We first obtain algorithms to find the desired packings of arborescences in all settings and then apply Edmonds’ weighted matroid intersection algorithm to also find solutions minimizing a given weight function.

Packing of maximal independent mixed arborescences

This paper studies the following packing problem: Given a mixed graph F with vertex set V, a matroid M on a set S={s1,…,sk} along with a map π:S→V, find k mutually edge and arc-disjoint mixed arborescences T1,…,Tk in F with roots π(s1),…,π(sk), such that, for any v∈V, the set {si:v∈V(Ti)} is independent and its rank reaches the theoretical maximum. This problem was mentioned by Fortier, Király, Léonard, Szigeti and Talon in [Old and new results on packing arborescences in directed hypergraphs, Discrete Appl. Math. 242 (2018), 26-33]; Matsuoka and Tanigawa gave a solution to this in [On reachability mixed arborescence packing, Discrete Optimization 32 (2019) 1-10].In this paper, we give a new characterization for above packing problem. This new characterization is simplified to the form of finding a supermodular function that should be covered by an orientation of each strong component of a matroid-based rooted mixed graph. Our proofs come along with a polynomial-time algorithm. The technique of using components opens some new ways to explore arborescence packings.

Polymatroid-based capacitated packing of branchings

Edmonds (1973) characterized the condition for the existence of a packing of spanning arborescences and also that of spanning branchings in a directed graph. Durand de Gevigney, Nguyen and Szigeti (2013) generalized the spanning arborescence packing problem to a matroid-based arborescence packing problem and gave a necessary and sufficient condition for the existence of a packing and a polynomial-time algorithm.In this paper, a generalization of this latter problem – the polymatroid-based arborescence packing problem – is considered. Two problem settings are formulated: an unsplittable version and a splittable version. The unsplittable version is shown to be strongly NP-complete. Whereas, the splittable version, which generalizes the capacitated version of the spanning arborescence packing problem, can be solved in strongly polynomial time. For convenience, we provide a strongly polynomial-time algorithm for the problem of the polymatroid-based capacitated packing of branchings for the splittable version.

Packing of arborescences with matroid constraints via matroid intersection

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.

On reachability mixed arborescence packing

As a generalization of Edmonds’ arborescence packing theorem, Kamiyama–Katoh–Takizawa (2009) provided a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root. Fortier–Király–Léonard–Szigeti–Talon (2018) asked whether the result can be extended to mixed graphs by allowing both directed arcs and undirected edges. In this paper, we solve this question by developing a polynomial-time algorithm for finding a collection of edge and arc-disjoint arborescences spanning the set of vertices reachable from each root in a given mixed graph.

On packing spanning arborescences with matroid constraint

Let us be given a rooted digraph D=(V+s,A) with a designated root vertex s. Edmonds' seminal result [Edmonds, J., Edge-disjoint branchings, in: B. Rustin (ed.) Combinatorial Algorithms, Academic Press, New York, 91–96 (1973] states that D has a packing of k spanning s-arborescences if and only if D has a packing of k (s, t)-paths for all t∈V, where a packing means arc-disjoint subgraphs.Let M be a matroid on the set of arcs leaving s. A packing of (s, t)-paths is called M-based if their arcs leaving s form a base of M while a packing of s-arborescences is called M-based if, for all t∈V, the packing of (s, t)-paths provided by the arborescences is M-based. Durand de Gevigney, Nguyen and Szigeti proved in [Durand de Gevigney, O., V.H. Nguyen, and Z. Szigeti, Matroid-based packing of arborescences, SIAM J. Discrete Math., 27(2013), 567–574] that D has an M-based packing of s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t∈V. Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds' theorem such that each s-arborescence is required to be spanning. Specifically, they conjectured that D has an M-based packing of spanning s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t∈V.We disprove this conjecture in its general form and we prove that the corresponding decision problem is NP-complete. However, we prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids.

A faster algorithm for packing branchings in digraphs

We consider the problem of finding an integral (and fractional) packing of branchings in a capacitated digraph with root-set demands. Schrijver described an algorithm that returns a packing with at most m+n3+r branchings that makes at most m(m+n3+r) calls to an oracle that basically computes a minimum cut, where n is the number of vertices, m is the number of arcs and r is the number of root-sets of the input digraph. Leston-Rey and Wakabayashi described an algorithm that returns a packing with at most m+r−1 branchings but makes a large number of oracle calls. In this work we provide an algorithm, inspired on ideas of Schrijver and in a paper of Gabow and Manu, that returns a packing with at most m+r−1 branchings and makes at most (m+r+2)n oracle calls. Moreover, for the arborescence packing problem our algorithm provides a packing with at most m−n+2 arborescences–thus improving the bound of m of Leston-Rey and Wakabayashi–and makes at most (m−n+5)n oracle calls.

Matroid-Based Packing of Arborescences

We provide the directed counterpart of a slight extension of Katoh and Tanigawa's result [SIAM J. Discrete Math., 27 (2013), pp. 155--185] on rooted-tree decompositions with matroid constraints. Our result characterizes digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds' result [Combinatorial Algorithms, Algorithmics Press, New York, 1973] on packing of spanning arborescences and implies---using a general orientation result of Frank [J. Combin. Theory Ser. B, 28 (1980), pp. 251--261]---the above result of Katoh and Tanigawa. We also give a complete description of the convex hull of the incidence vectors of the matroid-based packings of arborescences and prove that the minimum cost version of the problem can be solved in polynomial time.

Packing algorithms for arborescences (and spanning trees) in capacitated graphs

In a digraph with real-valued edge capacities, we pack the greatest number of arborescences in time O(n 3m log(n 2/m)); the packing uses at mostm distinct arborescences. Heren andm denote the number of vertices and edges in the given graph, respectively. Similar results hold for integral packing: we pack the greatest number of arborescences in time O(min{n, logN}n 2m log(n 2/)) using at mostm + n − 2 distinct arborescences; hereN denotes the largest (integral) capacity of an edge. These resuts improve the best previous bounds for capacitated digraphs. The algorithm extends to several related problems, including packing spanning trees in an undirected capacitated graph, and covering such graphs by forests. The algorithm provides a new proof of Edmonds' theorem for arborescence packing, for both integral and real capacities, based on a laminar family of sets derived from the packing. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.