Minimum cost source location problem with vertex-connectivity requirements in digraphs

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Given a digraph (or an undirected graph) G=( V, E) with a set V of vertices v with nonnegative real costs w( v), and a set E of edges and a positive integer k, we deal with the problem of finding a minimum cost subset S⊆ V such that, for each vertex v∈ V− S, there are k vertex-disjoint paths from S to v. In this paper, we show that the problem can be solved by a greedy algorithm in O( min{k, n }nm) time in a digraph (or in O( min{k, n }kn 2) time in an undirected graph), where n=| V| and m=| E|. Based on this, given a digraph and two integers k and ℓ, we also give a polynomial time algorithm for finding a minimum cost subset S⊆ V such that for each vertex v∈ V− S, there are k vertex-disjoint paths from S to v as well as ℓ vertex-disjoint paths from v to S.