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

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

TL;DR

The hybrid hyperbolic quadtree is introduced to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25).

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.

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

TL;DR

The hybrid hyperbolic quadtree is introduced to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25).

Abstract

We give an approximation scheme for the TSP in -dimensional hyperbolic space that has optimal dependence on under Gap-ETH. For any fixed dimension and for any our randomized algorithm gives a -approximation in time . 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.
Paper Structure (31 sections, 16 theorems, 45 equations, 6 figures)

This paper contains 31 sections, 16 theorems, 45 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Overview of our techniques, comparison with earlier work
  4. The techniques of Krauthgamer and Lee KrauthgamerL06.
  5. Overcoming exponential expansion.
  6. The number of crossings, weighted portal analysis.
  7. The dynamic programming: a banyan for hyperbolic Steiner tree.
  8. Organization of the paper
  9. Preliminaries
  10. Poincaré upper half-space model.
  11. Hyperbolic quadtree of Kisfaludi-BakW24.
  12. Building blocks of the technique of Arora Arora98.
  13. Building blocks of the technique of Kisfaludi-Bak, Nederlof and Węgrzycki EucTSPJACM.
  14. Hybrid Hyperbolic Quadtree and Portals
  15. Bounding box.
  16. ...and 16 more sections

Key Result

Theorem 1

For any fixed $d \geq 2$ and any $\varepsilon>0$, there is a randomized $(1 + \varepsilon)$-approximation for TSP and Steiner tree in $d$-dimensional hyperbolic space in $2^{O(1/\varepsilon^{d-1})}n(\log n)^{2d(d-1)}$ time.

Figures (6)

  • Figure 1: (i) The binary tiling of $\mathbb{H}^2$ with isometric tiles. (ii) The cells of various positive levels of the corresponding quadtree illustrated on the binary tree $T$. (iii) Cells of the new hybrid tree of various positive levels illustrated on $T$.
  • Figure 2: Illustration of (i) uniform (naïve) portal placement and (ii) non-uniform portal placement, on a $\mathsf{Side}$ facet of a 3-dimensional cell of the hybrid tree. For simplicity, Figure (ii) uses a scaling factor of $2$, although the true scaling factor is $2^{1-1/d}$. The empty circles represent the portals.
  • Figure 3: Illustration of the hyperbolic quadtree.
  • Figure 4: The compressed hybrid tree. (i) A point set and the cells of its compressed hybrid tree (A compressed cell is depicted in dashed lines). (ii) The tree structure of the hybrid tree.
  • Figure 5: (i) The geodesic $s$ has a crossing with the hyperplane $H_1$, but not with the hyperplane $H_2$: although $s$ intersects $H_2$ twice, its endpoints lie on the same side. (ii) Two adjacent negative-level hybrid tree cells $C_1$ and $C_2$ share a common facet $F$. Extending $F$ upward by a factor of $4$ yields $F'$, which is intersected by any geodesic $s$ joining points $a\in C_1$ and $b\in C_2$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem 1
  • Corollary 1
  • Corollary 2: Embedding the lower bound construction of EucTSPJACM
  • Theorem 3: Patching Lemma Arora98
  • Theorem 4: Arora’s Structure Theorem Arora98
  • Definition 5: Euclidean $r$-simple geometric graph
  • Definition 6: Euclidean $r$-simplification
  • Theorem 7: Kisfaludi-Bak, Nederlof and Węgrzycki Structure Theorem EucTSPJACM
  • Lemma 7
  • proof
  • ...and 28 more