Approximating the selected-internal Steiner tree

In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G = (V, E) with a cost function c : E → R⁺ and two subsets R and R^′ satisfying R^′ ⊂ R ⊆ V, a selected-internal Steiner tree is a Steiner tree which contains (or spans) all the vertices in R such that each vertex in R^′ cannot be a leaf. The selected-internal Steiner tree problem is to find a selected-internal Steiner tree with the minimum cost. In this paper, we present a 2 ρ-approximation algorithm for the problem, where ρ is the best-known approximation ratio for the Steiner tree problem.