Fast Thick-Thin Decomposition for Sparse Spanners on Hyperbolic Surfaces

2026-08-05

AI Summary

This work addresses the construction of sparse, non-crossing (1+ε)-spanners in the hyperbolic plane and on closed hyperbolic surfaces of genus g, allowing the use of Steiner points. By integrating the thick-thin decomposition with a novel neck decomposition tailored to convex hyperbolic polygons, the authors devise efficient algorithms for both non-crossing and crossing Steiner spanners, which are then applied to the Traveling Salesman Problem (TSP) in hyperbolic settings. Key contributions include the first sparse spanner on genus-g hyperbolic surfaces with linear dependence on g; an O(n/ε²)-edge non-crossing Steiner spanner in the hyperbolic plane; and on surfaces, non-crossing and crossing spanners with O(n/ε^{3/2} + g/ε²) and O(n/√ε + g/ε) edges, respectively. These results yield an efficient polynomial-time approximation scheme (EPTAS) for hyperbolic TSP.

Abstract

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.

Problem

• hyperbolic surfaces
• non-crossing spanners
• Steiner points
• thick-thin decomposition
• sparse spanners

Innovation

• hyperbolic spanners
• Steiner points
• thick-thin decomposition
• sparse spanners
• EPTAS for TSP