An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem

Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find a bottleneck Steiner tree of P with at most k Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP-hard and cannot be approximated within factor √ 2, unless P = NP . Then, we present a polynomial-time approximation algorithm with performance ratio 2.