Bottleneck Steiner tree with bounded number of Steiner vertices

Given a complete graph G=(V,E)G=(V,E), where each vertex is labeled either terminal or Steiner, a distance function (i.e., a metric) d:E→R+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 of 2−ε2−ε, for any ε>0ε>0, unless P=NPP=NP. Then, we present a polynomial-time 2-approximation algorithm for this problem.