On Maximal Independent Arborescence Packing

By generalizing the results of [N. Kamiyama, N. Katoh, and A. Takizawa, Combinatorica, 29 (2009), pp. 197--214], we solve the following problem. Given a digraph $D=(V,A)$ and a matroid on a set ${\sf{S}}=\{{\sf{s}}_1,\dots,{\sf{s}}_k\}$ along with a map $\pi:{\sf{S}}\to V$, find $k$ edge-disjoint arborescences $T_1,\dots, T_k$ with roots $\pi({\sf{s}}_1),\dots,\pi({\sf{s}}_k)$, respectively, such that, for any $v\in V$, the set $\{{\sf{s}}_i:v\in T_i\}$ is independent and its rank reaches the theoretical maximum. We also give a simplified proof for a result of [S. Fujishige, Combinatorica, 30 (2010), pp. 247--252].