# Hyperbolic intersection graphs and (quasi)-polynomial time

## TL;DR

This paper explores the complexity of classical graph problems in hyperbolic space, showing that many can be solved more efficiently than in Euclidean space, with some problems reaching polynomial time in hyperbolic settings.

## Contribution

The authors introduce new algorithms and bounds for graph problems in hyperbolic space, demonstrating dimension reduction effects and establishing ETH-based optimality results.

## Key findings

- Algorithms for Independent Set, Dominating Set, Steiner Tree, Hamiltonian Cycle in hyperbolic space are quasi-polynomial or polynomial.
- Treewidth bounds in hyperbolic disk graphs enable efficient algorithms for several problems.
- Matching lower bounds under ETH indicate the complexity of problems in hyperbolic geometry.

## Abstract

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.