AI 中文总结
本文研究双曲平面与闭合双曲曲面上的非交叉生成树问题，提出带Steiner点的稀疏$(1+\varepsilon)$-生成树构造，给出厚薄分解与颈部分解快速算法，并推导得双曲曲面TSP的EPTAS。
AI 中文摘要
我们研究双曲平面或闭合双曲曲面上点集的生成树（spanner），约束条件是生成树的边不允许交叉，这是非交叉欧几里得生成树的自然推广，因此得到的生成图嵌入在双曲平面或双曲曲面上。作为核心贡献，我们证明在允许使用Steiner点（斯坦纳点）的情况下，这类问题存在稀疏的$(1+\varepsilon)$-生成树：\n- 在双曲平面上，我们得到边数为$\mathcal{O}(n / \varepsilon^2)$的非交叉Steiner $(1+\varepsilon)$-生成树；\n- 在亏格为$g$的双曲曲面上，我们得到边数为$\mathcal{O}(n / \varepsilon^{3/2} + g/\varepsilon^2)$的非交叉Steiner $(1+\varepsilon)$-生成树，或是边数为$\mathcal{O}(n / \sqrt{\varepsilon} + g/\varepsilon)$、允许边交叉的生成树。\n特别地，我们的曲面上生成树的稀疏度与$g$呈线性依赖关系，而非更容易实现的指数依赖；其中$n/\varepsilon^{3/2}$和$n/\sqrt{\varepsilon}$项分别匹配了当前平面和可交叉Steiner生成树的最优欧几里得结果。\n结合我们的非交叉生成树与现有轻量生成树、无minor TSP相关文献中的技术，我们得到了双曲曲面上TSP的EPTAS（高效多项式时间近似方案）。\n我们的曲面构造依赖于厚薄分解（thick-thin decomposition）——这是研究双曲曲面的标准工具。对于凸双曲多边形，我们提出了类似的颈部分解（neck decomposition）。我们给出的算法可在$\mathcal{O}(g^4\log g)$时间内计算亏格为$g$的曲面的厚薄分解，在$\mathcal{O}(n)$时间内计算$n$顶点多边形的颈部分解。
英文摘要
We consider spanners for point sets lying in the hyperbolic plane or on a closed hyperbolic surface with the restriction that spanner edges are not allowed to cross. This is a natural generalization of non-crossing Euclidean spanners. Thus, the resulting spanner graphs are embedded in the hyperbolic plane or on the hyperbolic surface. As our main contribution, we show that there are sparse $(1+\varepsilon)$-spanners for these problems when we are allowed to use Steiner points: - on the hyperbolic plane we get a non-crossing Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^2)$ edges, - on hyperbolic surfaces of genus $g$ we get a Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^{3/2} + g/\varepsilon^2)$ non-crossing edges, or with $\mathcal{O}(n / \sqrt{\varepsilon} + g/\varepsilon)$ edges that are allowed to cross. In particular, our spanners on surfaces have sparsity with linear dependence on $g$, rather than the easier-to-attain exponential dependence, and the terms $n/\varepsilon^{3/2}$ and $n/\sqrt{\varepsilon}$ match the current best Euclidean results for plane and crossing Steiner spanners, respectively. As a corollary of our non-crossing spanner and techniques from the existing literature on light spanners and minor-free TSP, we get an EPTAS for TSP on hyperbolic surfaces. Our surface constructions rely on the thick-thin decomposition, a standard tool for studying hyperbolic surfaces. For convex hyperbolic polygons, we introduce an analogous neck decomposition. We give algorithms that compute the thick-thin decomposition of a genus-$g$ surface in $\mathcal{O}(g^4\log g)$ time and the neck decomposition of an $n$-vertex polygon in $\mathcal{O}(n)$ time.
