We consider intersection graphs of disks of radius $r$ in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of $r$, where very small $r$ corresponds to an almost-Euclidean setting and $r \in \Omega(\log n)$ corresponds to a firmly hyperbolic setting. We observe that larger values of $r$ create simpler graph classes, at least in terms of separators and the computational complexity of the \textsc{Independent Set} problem. First, we show that intersection graphs of disks of radius $r$ in the hyperbolic plane can be separated with $\mathcal{O}((1+1/r)\log n)$ cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set $S$ of $n$ points with pairwise distance at least $2r$ in the hyperbolic plane the corresponding Delaunay complex has outerplanarity $1+\mathcal{O}(\frac{\log n}{r})$, which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that \textsc{Independent Set} can be solved in $n^{\mathcal{O}(1+\frac{\log n}{r})}$ time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, \textsc{Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of $r\in \Omega(\log n)$. Finally, in the case when the disks have ply (depth) at most $\ell$, we give a PTAS for \textsc{Maximum Independent Set} that has only quasi-polynomial dependence on $1/\varepsilon$ and $\ell$. Our PTAS is a further generalization of our exact algorithm.

翻译：我们考虑双曲平面中半径为$r$的圆盘的相交图。与欧几里得设定不同，这些图类随$r$值的变化而不同：极小的$r$对应近乎欧几里得的情形，而$r \in \Omega(\log n)$则对应严格双曲的情形。我们观察到，更大的$r$值会产生更简单的图类，至少在分离器与\textsc{独立集}问题的计算复杂度方面如此。首先，我们证明双曲平面中半径为$r$的圆盘的相交图可以用$\mathcal{O}((1+1/r)\log n)$个团进行平衡分离。我们的第二个结构洞见涉及双曲平面中的Delaunay复形，这可能具有独立的研究价值。我们证明，对于双曲平面中任意两两距离至少为$2r$的$n$个点集$S$，对应的Delaunay复形的外平面度为$1+\mathcal{O}(\frac{\log n}{r})$，这意味着此类Delaunay复形的平衡分离器与树宽具有类似上界。利用此外平面度（及树宽）界，我们证明\textsc{独立集}可在$n^{\mathcal{O}(1+\frac{\log n}{r})}$时间内求解。该算法基于对某个未知球面切割分解（其构造依赖于问题解）进行动态规划。所得算法是Kisfaludi-Bak（SODA 2020）结果的深远推广，且在指数时间假设下是紧的。特别地，在$r\in \Omega(\log n)$的严格双曲设定中，\textsc{独立集}可在多项式时间内求解。最后，当圆盘的层数（深度）至多为$\ell$时，我们给出了\textsc{最大独立集}的PTAS，该方案仅对$1/\varepsilon$和$\ell$具有拟多项式依赖。我们的PTAS是精确算法的进一步推广。