MAX-SNP hardness and approximation of selected-internal steiner trees

Abstract

In this paper, we consider an interesting 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 show that the problem is MAX SNP-hard even when the costs of all edges in the input graph are restricted to either 1 or 2. We also present an approximation algorithm for the problem. © Springer-Verlag Berlin Heidelberg 2006.