Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

TL;DR

This paper presents a near-optimal approximation scheme for the Traveling Salesman Problem and Steiner Tree problem in hyperbolic space, utilizing novel hierarchical decompositions and dynamic programming techniques.

Contribution

It introduces a new hierarchical decomposition called the hybrid hyperbolic quadtree and a specialized portal placement, enabling efficient approximation algorithms in hyperbolic geometry.

Findings

Achieves a $(1+ ext{epsilon})$-approximation in time $2^{O(1/ ext{epsilon}^{d-1})}n^{1+o(1)}$

Develops a new hierarchical decomposition called the hybrid hyperbolic quadtree

Provides techniques potentially useful for geometric optimization in curved spaces

Abstract

We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.