The aim of this paper is to reveal the discrete convexity of the minimum-cost packings of arborescences and branchings. We first prove that the minimum-cost packings of disjoint $k$ branchings (minimum-cost $k$-branchings) induce an $\mathrm{M}^\natural$-convex function defined on the integer vectors on the vertex set. The proof is based on a theorem on packing disjoint $k$-branchings, which extends Edmonds' disjoint branchings theorem and is of independent interest. We then show the $\mathrm{M}$-convexity of the minimum-cost $k$-arborescences, which provides a short proof for a theorem of Bern\'ath and Kir\'aly (SODA 2016) stating that the root vectors of the minimum-cost $k$-arborescences form a base polyhedron of a submodular function. Finally, building upon the $\mathrm{M}^\natural$-convexity of $k$-branchings, we present a new problem of minimum-cost root location of a $k$-branching, and show that it can be solved in polynomial time if the opening cost function is $\mathrm{M}^\natural$-convex.

翻译：本文旨在揭示最小成本树形图与分支打包的离散凸性。我们首先证明，不相交$k$分支的最小成本打包（最小成本$k$分支）导出一个定义在顶点集整数向量上的$\mathrm{M}^\natural$凸函数。其证明基于一个关于打包不相交$k$分支的定理，该定理推广了Edmonds的不相交分支定理，并具有独立的理论意义。随后，我们证明了最小成本$k$树形图的$\mathrm{M}$凸性，这为Bernáth与Király（SODA 2016）的定理提供了一个简洁证明，该定理指出最小成本$k$树形图的根向量构成一个次模函数的基多面体。最后，基于$k$分支的$\mathrm{M}^\natural$凸性，我们提出了$k$分支最小成本根定位这一新问题，并证明当开设成本函数为$\mathrm{M}^\natural$凸时，该问题可在多项式时间内求解。