Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

TL;DR

This paper investigates shortest paths in hyperbolic tilings, establishing bounds on convex hull size and treewidth, and providing efficient algorithms for TSP and Steiner tree problems in these graphs.

Contribution

It introduces near-linear algorithms for computing isometric closures and bounds on convex hull size and treewidth in hyperbolic tilings, enabling efficient solutions to TSP and Steiner tree problems.

Findings

Convex hull size is linear in total path length.

Treewidth of geodesic convex hull is bounded by a logarithmic function.

Algorithms for TSP and Steiner tree run in near-linear time.

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 $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.