An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem

TL;DR

This paper addresses the Euclidean bottleneck Steiner tree problem, proposing an approximation algorithm with a performance ratio of 2, after establishing its NP-hardness and approximation limits.

Contribution

It introduces a polynomial-time 2-approximation algorithm for the Euclidean bottleneck Steiner tree problem, a problem previously shown to be NP-hard with limited approximability.

Findings

The problem is NP-hard.

Cannot be approximated within factor √2 unless P=NP.

A polynomial-time algorithm with a performance ratio of 2 is provided.

Abstract

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 $k \le m$, 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 $\sqrt{2}$, unless $P=NP$. Then, we present a polynomial-time approximation algorithm with performance ratio 2.