In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)} n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach can be generalized to the problem with $\ell_p$ metric for any rational $1 \le p \le \infty$, or even other metrics on $\mathbb{R}^2$.

翻译：在欧几里得瓶颈斯坦纳树问题中，输入包含平面$\mathbb{R}^2$中的一组$n$个点（称为终端）和一个参数$k$，目标是计算一棵覆盖所有终端且最多包含$k$个$\mathbb{R}^2$中的点作为斯坦纳点的斯坦纳树，使得该树的最大边长最小化，其中树边的长度定义为两端点之间的欧几里得距离。该问题已被广泛研究，并已知为NP难问题。本文给出了欧几里得瓶颈斯坦纳树的$k^{O(k)} n^{O(1)}$时间算法，证明了该问题是固定参数可解的。这解决了Bae等人[Algorithmica, 2011]明确提出的一个公开问题，他们曾证明该问题的$\ell_1$和$\ell_{\infty}$变体是FPT的。我们的方法可推广到任意有理数$1 \le p \le \infty$的$\ell_p$度量问题，甚至$\mathbb{R}^2$上的其他度量。