On the partial terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = (V, E) with length function l : E -> R+ and two vertex subsets R subset of V and R ' subset of R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R backslash R ' belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths Sigma((u, v)epsilon T) l(u, v) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem.