Long Plane Trees

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 \(\mathrm{OPT}\) , 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 \(0.546\cdot\mathrm{OPT}\) . 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 \(d\geq 3\) , 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).