On the Longest Spanning Tree with Neighborhoods

TL;DR

This paper introduces an approximation algorithm for a geometric network design problem involving selecting points in neighborhoods to maximize the longest spanning tree length, achieving a ratio of 0.511, slightly better than 1/2.

Contribution

The paper presents the first approximation algorithm with a ratio above 0.5 for the problem of maximizing the longest spanning tree in neighborhoods.

Findings

Approximation ratio achieved is 0.511.

The problem is likely NP-hard even in the plane.

Open question remains on NP-hardness proof.

Abstract

We study a maximization problem for geometric network design. Given a set of $n$ compact neighborhoods in $\mathbb{R}^d$, select a point in each neighborhood, so that the longest spanning tree on these points (as vertices) has maximum length. Here we give an approximation algorithm with ratio $0.511$, which represents the first, albeit small, improvement beyond $1/2$. While we suspect that the problem is NP-hard already in the plane, this issue remains open.