Structure and Independence in Hyperbolic Uniform Disk Graphs

TL;DR

The paper studies hyperbolic uniform disk graphs (HUDG) as intersection graphs of equal-radius disks in the hyperbolic plane, showing that the radius parameter $r(n)$ drives structural simplicity and algorithmic tractability for Independent Set. It develops two core structural tools: (i) balanced clique-based separators of size $O((1+1/r)\log n)$ and (ii) a $1+O(\frac{\log n}{r})$-outerplanarity bound for the Delaunay complex of well-separated sites, enabling sphere-cut and noose-based dynamic programming. These lead to an MIS algorithm running in $n^{O((1+1/r)\log^2 n)}$ time, with polynomial-time solvability for $r\in\Omega(\log n)$, and a ply-dependent PTAS for Maximum Independent Set; together with ETH-tight lower bounds, they extend and generalize prior planar/disk algorithmic results to the hyperbolic setting. The results also reveal a radius-based dichotomy: grids are rich only for very small $r$, while large stars emerge for $r\in\Omega(\log n)$, clarifying when HUDG are structurally tractable. Overall, the work advances deterministic algorithmic understanding of HUDG and highlights how hyperbolic geometry shapes computational complexity for classic graph problems.

Abstract

We consider intersection graphs of disks of radius $r$ in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of $r$, where very small $r$ corresponds to an almost-Euclidean setting and $r \in Ω(\log n)$ corresponds to a firmly hyperbolic setting. We observe that larger values of $r$ create simpler graph classes, at least in terms of separators and the computational complexity of the \textsc{Independent Set} problem. First, we show that intersection graphs of disks of radius $r$ in the hyperbolic plane can be separated with $\mathcal{O}((1+1/r)\log n)$ cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set $S$ of $n$ points with pairwise distance at least $2r$ in the hyperbolic plane the corresponding Delaunay complex has outerplanarity $1+\mathcal{O}(\frac{\log n}{r})$, which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that \textsc{Independent Set} can be solved in $n^{\mathcal{O}(1+\frac{\log n}{r})}$ time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, \textsc{Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of $r\in Ω(\log n)$. Finally, in the case when the disks have ply (depth) at most $\ell$, we give a PTAS for \textsc{Maximum Independent Set} that has only quasi-polynomial dependence on $1/\varepsilon$ and $\ell$. Our PTAS is a further generalization of our exact algorithm.

Figures (10)

• Figure 1: Realizing a grid is only possible for small radii, while large stars are only possible in the firmly hyperbolic setting of $r \in \Omega(\log n)$.
• Figure 2: A Euclidean Delaunay complex of $n$ points with outerplanarity $n/3$.
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (7)

• Theorem 1
• Corollary 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5