Covering a supermodular-like function in a mixed hypergraph

In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph F=(V, E ∪ A) having an orientation E of E such that e_E ∪ A (P) ≥ ∑X ∈ Ph(X) -b(∪ P) for every subpartition P of V, where h is an integer-valued, intersecting supermodular function on V and b a submodular function on V. As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex v belongs to exactly k of them and is the root of at least $f(v)$ and at most $g(v)$ of them.