Minimum cost source location problem with local 3-vertex-connectivity requirements

Let G=(V,E)$G=(V,E)$ be a simple undirected graph with a set V$V$ of vertices and a set E$E$ of edges. Each vertex v∈V$v\in V$ has a demand d(v)∈_Z+$d\left(v\right)\in Z_{+}$ and a cost c(v)∈_R+$c\left(v\right)\in R_{+}$, where _Z+$Z_{+}$ and _R+$R_{+}$ 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$G$ requires finding a set S$S$ of vertices minimizing _∑v∈Sc(v)${\sum }_{v\in S}c\left(v\right)$ such that there are at least d(v)$d\left(v\right)$ pairwise vertex-disjoint paths from S$S$ to v$v$ for each vertex v∈V−S$v\in V-S$. It is known that if there exists a vertex v∈V$v\in V$ with d(v)≥4$d\left(v\right)\ge 4$, 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|⁴log²|V|)$O\left({\left|V\right|}^4{log}^2\left|V\right|\right)$ time if d(v)≤3$d\left(v\right)\le 3$ holds for each vertex v∈V$v\in V$.