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 efforts have focused on olynomial-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 three or 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 $d \geq 3$, 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).

翻译：在最长平面生成树问题中，给定一个有限平面点集 $\mathcal{P}$，我们的目标是找到一棵平面（即无交叉）生成树，使其总欧几里得边长最大。尽管经过了二十多年的研究，该问题是否为NP困难仍未解决。因此，先前的研究集中在多项式时间算法上，这些算法能生成总边长近似于$\text{OPT}$（最大可能长度）的平面树。这些算法中的近似树均具有较小的无权直径，通常为三或四。一个自然的问题是：这是最长平面生成树的共同特征，还是特定近似算法的人为产物？我们通过三个结果来阐明近似保证与近似树无权直径之间的相互作用。首先，我们描述了一种多项式时间算法，用于构造直径至多为四且总边长至少为$0.546 \cdot \text{OPT}$的平面树。这相较现有技术有显著改进。其次，我们证明可以在多项式时间内找到直径至多为三的最长平面树。第三，对于任何候选直径 $d \geq 3$，我们给出了直径至多为 $d$的最长平面树（与无约束的最长平面树相比）所能达到的近似因子的上界。