Long plane trees

Cabello, Sergio (author), Hoffmann, Michael (author), Klost, Katharina (author), Mulzer, Wolfgang (author), Tkadlec, Josef (author)

In the longest plane spanning tree problem, we are given a finite planar point set ${\mathcal P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for ${\mathcal P}$ with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous results have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates $\text{OPT}$, the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically two to four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least $0.546\cdot\text{OPT}$. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter $d \ge 3$, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most $d$ (compared to a longest plane tree without constraints).

• Language: English
• Keywords: computational geometry, geometric network design, spanning trees, plane straight-line graphs, approximation algorithms
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2026
• Year: 2026
• Number of pages: Str. 5:1-5:40
• Numbering: Vol. 22, iss. 1, art. 5
• PID: 20.500.12556/RUL-177721
• UDC: 519.17:004
• ISSN on article: 1549-6325
• DOI: 10.1145/3765740
• COBISS.SI-ID: 263386883
• Publication date in RUL: 05.01.2026

Record is a part of a journal

• Title: ACM transactions on algorithms
• Publisher: Association for Computing Machinery
• ISSN: 1549-6325
• COBISS.SI-ID: 14508889

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.