The Euclidean Steiner Minimal Tree problem takes as input a set $\mathcal P$ of points in the Euclidean plane and finds the minimum length network interconnecting all the points of $\mathcal P$. In this paper, in continuation to the works of Du et al. and Weng et al., we study Euclidean Steiner Minimal Tree when $\mathcal P$ is formed by the vertices of a pair of regular, concentric and parallel $n$-gons. We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for $\mathcal P$. We also consider point sets $\mathcal P$ of size $n$ where the number of input points not on the convex hull of $\mathcal P$ is $f(n) \leq n$. We give an exact algorithm with running time $2^{\mathcal{O}(f(n)\log n)}$ for such input point sets $\mathcal P$. Note that when $f(n) = \mathcal{O}(\frac{n}{\log n})$, our algorithm runs in single-exponential time, and when $f(n) = o(n)$ the running time is $2^{o(n\log n)}$ which is better than the known algorithm stated in Hwang et al. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P=NP, as shown by Garey et al. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets, as given by Scott Provan. In this paper, we show that if the number of input points in $\mathcal P$ not belonging to the convex hull of $\mathcal P$ is $\mathcal{O}(\log n)$, then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any $\epsilon \in (0,1]$, when there are $\Omega(n^{\epsilon})$ points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P=NP.

翻译：欧几里得斯坦纳最小树问题的输入是欧几里得平面上的点集 $\mathcal P$，目标是找到连接 $\mathcal P$ 中所有点的最短长度网络。本文延续杜等人和翁等人的工作，研究了当 $\mathcal P$ 由一对规则、共心且平行的 $n$ 边形顶点构成时的欧几里得斯坦纳最小树。我们重点关注两个多边形相距不太近的情形。在此类情形下，我们证明了欧几里得斯坦纳最小树可在多项式时间内求解，并描述了 $\mathcal P$ 的欧几里得斯坦纳最小树的显式结构。我们还考虑大小为 $n$ 的点集 $\mathcal P$，其中不在 $\mathcal P$ 凸包上的输入点数量为 $f(n) \leq n$。对于此类输入点集 $\mathcal P$，我们给出一个运行时间为 $2^{\mathcal{O}(f(n)\log n)}$ 的精确算法。注意，当 $f(n) = \mathcal{O}(\frac{n}{\log n})$ 时，我们的算法以单指数时间运行；当 $f(n) = o(n)$ 时，运行时间为 $2^{o(n\log n)}$，优于黄等人所述已知算法。根据加里等人的结果，除非 P=NP，否则欧几里得斯坦纳最小树不存在 FPTAS。另一方面，斯科特·普罗文给出了凸点集上欧几里得斯坦纳最小树的 FPTAS。本文中，我们证明：若 $\mathcal P$ 中不属于其凸包点的输入点数量为 $\mathcal{O}(\log n)$，则欧几里得斯坦纳最小树存在 FPTAS。对比之下，我们证明：对于任意 $\epsilon \in (0,1]$，当不属于输入集凸包的点数量为 $\Omega(n^{\epsilon})$ 时，除非 P=NP，否则欧几里得斯坦纳最小树不存在任何 FPTAS。