Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

2025-10-29

AI Summary

This paper investigates the structure and algorithms for computing geodesic convex hulls (i.e., isometric closures) of terminal point sets in regular hyperbolic tessellations. For regular hyperbolic tilings, we present the first near-linear-time algorithm—running in (O(N log N) + ext{poly}(n/(p+q)) cdot N) time—to compute the isometric closure of (n) terminals, where (N) denotes the total length of shortest paths from terminals to the origin. Our key theoretical contribution is the first proof that the treewidth of this convex hull depends only logarithmically on (n/(p+q)), independent of inter-terminal distances—breaking prior bounds reliant on diameter. Methodologically, we integrate classical geometric convex hull techniques with the inherent symmetry and shortest-path structure of hyperbolic graphs. As direct corollaries, we obtain efficient algorithms for the subset Traveling Salesman Problem and the Steiner Tree Problem on such tessellations, establishing a new paradigm for combinatorial optimization in hyperbolic networks.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $max(12,O(logfrac{n}{p + q}))$, 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 $O(N log N) + mathrm{poly}(frac{n}{p + q}) cdot N$.

Problem

• Studying shortest paths structure in regular hyperbolic tilings
• Computing isometric closure of terminals in near-linear time
• Analyzing treewidth of geodesic convex hulls in hyperbolic graphs

Innovation

• Computes isometric closure using geometric convex hull
• Proves convex hull size is linear in path length
• Shows geodesic convex hull has logarithmic treewidth