Greedy Approximation for Source Location Problem with Vertex-Connectivity Requirements in Undirected Graphs

Let G = (V, E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v ∈ V has a demand d(v) ∈ Z+, and a cost c(v) ∈ R+, where Z+ and R+ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing ∑ v∈S c(v) such that there are at least d(v) pairwise vertex-disjoint paths from S to v for each vertex v ∈ V −S. It is known that the problem is not approximable within a ratio of O(ln ∑ v∈V d(v)), unless NP has an O(N log log N )-time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and d∗ = 4 holds, then the problem is NP-hard, where d∗ = max{d(v) | v ∈ V }. In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a max{d∗, 2d∗ − 6}approximate solution to the problem in O(min{d∗, √ |V |}d∗|V |2) time, while we also show that there exists an instance for which it provides no better than a (d∗ − 1)approximate solution. Especially, in the case of d∗ ≤ 4, we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to d∗ ≤ 4.