Abstract

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.

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