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

Authors Sándor Kisfaludi-Bak , Saeed Odak , Satyam Singh , Geert van Wordragen

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Hyperbolic traveling salesman problem
  • TSP
  • Hyperbolic Steiner tree problem
  • Approximation scheme
  • Banyan
  • Hyperbolic geometry

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Abstract

We give an approximation scheme for the TSP in d-dimensional hyperbolic space that has optimal dependence on ε under Gap-ETH. For any fixed dimension d ≥ 2 and for any ε > 0 our randomized algorithm gives a (1+ε)-approximation in time 2^O(1/ε^{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.

Cite As Get BibTex

Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen. Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.64

Author Details

Sándor Kisfaludi-Bak
  • Department of Computer Science, Aalto University, Espoo, Finland
Saeed Odak
  • Department of Computer Science, Aalto University, Espoo, Finland
Satyam Singh
  • Department of Computer Science, Aalto University, Espoo, Finland
Geert van Wordragen
  • Department of Computer Science, Aalto University, Espoo, Finland

Funding

This work was supported by the Research Council of Finland, Grant 363444.

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