Covering directed graphs by in-trees

Given a directed graph D = (V,A) with a set of d specified vertices S = {s₁,...,s _d }⊆ V and a function f: S → ℤ₊ where ℤ₊ denotes the set of non-negative integers, we consider the problem which asks whether there exist ∑_i=1^d f (s_i) in-trees denoted by T _i, 1, T_i, 2, ..., T_i f(s_i) for every i = 1,...,d such that T_i, 1,...,T_i, f(s_i) are rooted at s _i, each T_i,j spans vertices from which s _i is reachable and the union of all arc sets of T_i,j for i = 1,...,d and j = 1,...,f(s_i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑_i=1^d f (s_i) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.