Approximations for Steiner trees with minimum number of Steiner points

Given n terminals in the Euclidean plane and a positive constant, find a St einer tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the giv en positive constant. This problem is NP-hard with applications in VLSI des ign, WDM optical networks and wireless communications. In this paper, we sh ow that (a) the Steiner ratio is 1/4, that is, the minimum spanning tree yi elds a polynomial-time approximation with performance ratio exactly 4, (b) there exists a polynomial-time approximation with performance ratio 3, and (c) there exists a polynomial-time approximation scheme under certain condi tions.