Bottleneck Steiner Tree with Bounded Number of Steiner Vertices

Given a complete graph G = (V, E), where each vertex is labeled either terminal or Steiner, a distance function d: E → R +, and a positive integer k, we study the problem of finding a Steiner tree T spanning all terminals and at most k Steiner vertices, such that the length of the longest edge is minimized. We first show that this problem is NP-hard and cannot be approximated within a factor 2 − ε, for any ε> 0, unless P = NP. Then, we present a polynomial-time 2-approximation algorithm for this problem