We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.

翻译：我们提出了一种在$d$维双曲空间中TSP问题的近似方案，该方案在Gap-ETH下对$\varepsilon$的依赖达到最优。对于任意固定的维度$d\geq 2$和任意$\varepsilon>0$，我们的随机算法在$2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$时间内给出一个$(1+\varepsilon)$-近似解。我们还为双曲Steiner树问题提供了一个具有相同运行时间的算法。我们的算法是一种基于随机平移层次分解的Arora式动态规划。然而，为实现所需的大尺度结构，我们引入了一种称为混合双曲四叉树的新层次分解，这与Kisfaludi-Bak和Van Wordragen（JoCG'25）最近提出的双曲四叉树有显著差异。此外，我们采用了一种新的非均匀门户点放置方法，并且我们的结构定理运用了一种新的加权交叉分析。我们相信这些技术可以为进一步发展弯曲空间中的几何优化奠定基础。