Bottleneck bichromatic full Steiner trees

Abstract

Given two sets of points in the plane, Q of n (terminal) points and S of m (Steiner) points, where each of Q and S contains bichromatic points (red and blue points), a full bichromatic Steiner tree is a Steiner tree in which all points of Q are leaves and each edge of the tree is bichromatic, i.e., connects a red and a blue point. In the bottleneck bichromatic full Steiner tree (BBFST) problem, the goal is to compute a bichromatic full Steiner tree T, such that the length of the longest edge in T is minimized. In the k-BBFST problem, the goal is to find a bichromatic full Steiner tree T with at most k≤m Steiner points from S, such that the length of the longest edge in T is minimized.In this paper, we first present an O((n+m)log⁡m) time algorithm that solves the BBFST problem. Then, we show that k-BBFST problem is NP-hard and cannot be approximated within a factor of 5 in polynomial time, unless P=NP. Finally, we give a polynomial-time 9-approximation algorithm for the k-BBFST problem.