The internal Steiner tree problem: Hardness and approximations

Given a graph G=(V,E) with a cost function c:E→R+ and a vertex subset R⊂V, an internal Steiner tree is a Steiner tree that contains all the vertices in R, and such that each vertex in R must be an internal vertex. The internal Steiner tree problem involves finding an internal Steiner tree T whose total cost ∑(u,v)∈E(T)c(u,v) is the minimum. In this paper, we first show that the internal Steiner tree problem is MAX SNP-hard. We then present a (2ρ+1)-approximation algorithm for solving the problem on complete graphs, where ρ is an approximation ratio for the Steiner tree problem. Currently, the best-known ρ is ln4+ϵ