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

2026-03-10

AI Summary

This work addresses the design of efficient approximation algorithms for the Traveling Salesman Problem (TSP) and the Steiner Tree problem in high-dimensional hyperbolic space. Building upon an Arora-style dynamic programming framework, the authors introduce several key techniques—including a hybrid hyperbolic quadtree hierarchy, randomized shifting decomposition, non-uniform portal placement, and weighted crossing analysis—to overcome limitations of existing approaches in hyperbolic geometric optimization. For both problems in $d$-dimensional hyperbolic space, the proposed algorithms achieve a $(1+\varepsilon)$-approximation with running time $2^{O(1/\varepsilon^{d-1})} n^{1+o(1)}$. Notably, under the Gap-ETH assumption, the dependence on $\varepsilon$ is provably optimal.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Hyperbolic TSP
• Steiner Tree
• Gap-ETH
• Approximation Algorithms
• Geometric Optimization

Innovation

• hybrid hyperbolic quadtree
• non-uniform portal placement
• weighted crossing analysis
• hyperbolic TSP
• Gap-ETH-tight algorithm