Minimum Cost Source Location Problem with Local 3-Vertex-Connectivity Requirements

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 if there exists a vertex v ∈ V with d(v) ≥ 4, then the problem is NP-hard even in the case where every vertex has a uniform cost. In this paper, we show that the problem can be solved in O(|V |4 log |V |) time if d(v) ≤ 3 holds for each vertex v ∈ V .