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.