Hardness and Approximation of the Selected-Leaf-Terminal Steiner Tree Problem

For a complete graph G = (V, E) with length function l : E rarr R⁺ and two vertex subsets R sub V and R´ sube R, a selected-leaf-terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R R´ belong to the leaves of this Steiner tree. The selected-leaf-terminal Steiner tree problem is to find a selected-leaf-terminal Steiner tree T whose total lengths Sigma _{(u, v)epsiT} 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