Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

TL;DR

This work studies shortest paths in regular hyperbolic tilings $G_{p,q}$ with $\frac{1}{p}+\frac{1}{q}<\frac{1}{2}$, introducing geodesic convex hulls and minimal isometric closures of terminal sets. It develops line-based, boundary-driven techniques to compute shortest paths and isometric closures in near-linear time, showing the convex hull has $\mathcal{O}(N)$ vertices and a treewidth bounded by $\max\{12, \mathcal{O}(\log\frac{n}{p+q})\}$. The paper then proves that optimal Steiner trees and Subset TSP tours can be realized within a minimal isometric closure and provides an $\mathcal{O}(N \log N)$-time algorithmic framework, with an additional polynomial dependence on $n/(p+q)$. These results culminate in near-linear algorithms for key NP-hard problems on hyperbolic tilings by leveraging small treewidth and outerplanarity properties, offering significant improvements over the planar case. Overall, the work advances efficient geometric and graph-theoretic methods on hyperbolic graphs and highlights the algorithmic potential of hyperbolic geometry in combinatorial optimization.

Abstract

Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$ terminals, we can compute a so-called isometric closure (closely related to the geodesic convex hull) of the terminals in near-linear time, using a classic geometric convex hull algorithm as a black box. We show that the size of the convex hull is $O(N)$ where $N$ is the total length of the paths to the terminals from a fixed origin. Furthermore, we prove that the geodesic convex hull of a set of $n$ terminals has treewidth only $\max(12,O(\log\frac{n}{p + q}))$, a bound independent of the distance of the points involved. As a consequence, we obtain algorithms for subset TSP and Steiner tree with running time $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.

Figures (5)

• Figure 1: The regular tiling in spherical, Euclidean and hyperbolic geometry where $p=3$ and $q=5,6$, and $7$, respectively.
• Figure 2: The convex hull (gray) and a minimal isometric closure (black) of a set of terminals (red) in the grid graph.
• Figure 3: The subgraph $G_\ell$ (blue and green) intersected by a line $\ell$ (dashed) and the sequence $S_\ell$ of vertices and edges (green) intersected by $\ell$. An additional layer of nearby tiles is depicted (grey).
• Figure 4: The path $P_{xy}$ (red) cannot be shorter than $Q_{x,y}$ (blue). Only the solid lines represent edges of $G$.
• Figure 5: Optimal Steiner tree $T$ (black) can be made to use only edges in $G_K$ (blue) by replacing subtrees $R_i$ (dotted black) with boundary walks (red).

Theorems & Definitions (31)

• Lemma 1: Informal, weaker version of \ref{['lemma:shortestPath']}(i)
• Lemma 2: Informal, weaker version of \ref{['lemma:extensionByIntersection']}
• Theorem 3
• Theorem 4
• Lemma 5
• Definition 6: Subgraph intersected by a line
• Lemma 7
• Corollary 8
• Lemma 8
• Definition 9: Sequence of edges/vertices intersected by $\ell$, $S_{\ell}$