Packing of mixed hyperarborescences with flexible roots via matroid intersection

Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, functions $f,g:V\rightarrow \mathbb{Z}_+$ and an integer $k$, a packing of $k$ spanning mixed hyperarborescences is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the above mentioned problem

Reachability in arborescence packings

Fortier et al. proposed several research problems on packing arborescences. Some of them were settled in that article and others were solved later 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 show how a strong version of Edmonds' theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa's result on packing reachability arborescences and how Durand de Gevigney, Nguyen and Szigeti's theorem on matroid-based packing of arborescences implies Kir\'aly's result on matroid-reachability-based packing of arborescences. Finally, 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.. In the last part of the article, 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

On packing spanning arborescences with matroid constraint

Let D = (V + s, A) be a digraph with a designated root vertex S. Edmonds’ seminal result (see J. Edmonds [4]) implies 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 [3] 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. In this paper we disprove this conjecture in its general form and we prove that the corresponding decision problem is NP-complete. We also prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids. For all the results presented in this paper, the undirected counterpart also holds