Covering directed graphs by in-trees

Given a directed graph D = (V, A) with a set of d specified vertices S = {s(1),..., s(d)} subset of V and a function f : S -> N where N denotes the set of positive integers, we consider the problem which asks whether there exist Sigma(d)(i=1) f (s(i)) in-trees denoted by T(i,1), T(i,2),..., T(i,f(si)) for every i = 1,..., d such that T(i,1),..., T(i,f(si)) 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 Sigma(d)(i=1) 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.