Covering Directed Graphs by In-Trees

Given a directed graphD = (V,A) with a set of d specified vertices S = {s1, . . . , sd} ⊆ 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 ∑d i=1 f(si) in-trees denoted by Ti,1, Ti,2, . . . , Ti,f(si) for every i = 1, . . . , d such that Ti,1, . . . , Ti,f(si) are rooted at si, each Ti,j spans vertices from which si is reachable and the union of all arc sets of Ti,j for i = 1, . . . , d and j = 1, . . . , f(si) 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 ∑d i=1 f(si) 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 intrees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.