### abstract

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