Covering directed graphs by in-trees

摘要

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }⊆V and a function f : S→ℕ where ℕ denotes the set of positive 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,2,¼,T_i,f(si)T_{i,1},T_{i,2},ldots,T_{i,f(s_{i})} for every i=1,…,d such that T_i,1,¼,T_i,f(si)T_{i,1},ldots,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.