In the Euclidean Steiner Tree problem, we are given as input a set of points (called terminals) in the $\ell_2$-metric space and the goal is to find the minimum-cost tree connecting them. Additional points (called Steiner points) from the space can be introduced as nodes in the solution. The seminal works of Arora [JACM'98] and Mitchell [SICOMP'99] provide a Polynomial Time Approximation Scheme (PTAS) for solving the Euclidean Steiner Tree problem in fixed dimensions. However, the problem remains poorly understood in higher dimensions (such as when the dimension is logarithmic in the number of terminals) and ruling out a PTAS for the problem in high dimensions is a notoriously long standing open problem (for example, see Trevisan [SICOMP'00]). Moreover, the explicit construction of optimal Steiner trees remains unknown for almost all well-studied high-dimensional point configurations. Furthermore, a vast majority the state-of-the-art structural results on (high-dimensional) Euclidean Steiner trees were established in the 1960s, with no noteworthy update in over half a century. In this paper, we revisit high-dimensional Euclidean Steiner trees, proving new structural results. We also establish a link between the computational hardness of the Euclidean Steiner Tree problem and understanding the optimal Steiner trees of regular simplices (and simplicial complexes), proposing several conjectures and showing that some of them suffice to resolve the status of the inapproximability of the Euclidean Steiner Tree problem. Motivated by this connection, we investigate optimal Steiner trees of regular simplices, proving new structural properties of their optimal Steiner trees, revisiting an old conjecture of Smith [Algorithmica'92] about their optimal topology, and providing the first explicit, general construction of candidate optimal Steiner trees for that topology.

翻译：在欧几里得斯坦纳树问题中，输入是$\ell_2$度量空间中的一组点（称为终端），目标是找到连接它们的最小成本树。允许引入空间中的额外点（称为斯坦纳点）作为解的节点。Arora [JACM'98] 和 Mitchell [SICOMP'99] 的开创性工作提供了在固定维度下求解欧几里得斯坦纳树问题的多项式时间近似方案 (PTAS)。然而，该问题在更高维度（例如，当维度与终端数量的对数成正比时）仍未被充分理解，而排除高维情况下该问题存在 PTAS 的可能性是一个长期悬而未决的难题（例如，参见 Trevisan [SICOMP'00]）。此外，对于几乎所有被深入研究的 高维点构型，其最优斯坦纳树的显式构造仍属未知。并且，关于（高维）欧几里得斯坦纳树的绝大部分前沿结构性结果均确立于20世纪60年代，半个多世纪以来未有显著更新。在本文中，我们重新审视高维欧几里得斯坦纳树，证明了新的结构性质。我们还建立了欧几里得斯坦纳树问题的计算困难性 与理解正单形（及单纯复形）的最优斯坦纳树之间的联系，提出了若干猜想，并表明其中一些猜想足以解决欧几里得斯坦纳树问题不可近似性的状态。受此联系启发，我们研究了正单形的最优斯坦纳树，证明了其最优斯坦纳树的新结构性质，重新审视了 Smith [Algorithmica'92] 关于其最优拓扑的一个旧猜想，并首次给出了针对该拓扑的候选最优斯坦纳树的一般性显式构造。