Structure and Independence in Hyperbolic Uniform Disk Graphs

2024-07-12

arXiv.org

AI Summary

This paper investigates the structural properties and independent set problem on disk intersection graphs of radius-$r$ disks in the hyperbolic plane. As $r$ increases, the graph structure simplifies: balanced clique separators shrink to size $O((1+1/r)log n)$, and both treewidth and layered treewidth decrease significantly. We establish the first balanced separator bound for hyperbolic disk graphs and reveal the outerplanarity of their Delaunay complexes. For $r = Omega(log n)$, we prove that the maximum independent set can be computed exactly in polynomial time. Furthermore, we present the first quasi-polynomial PTAS for hyperbolic settings: for graphs with layered treewidth bounded by $ell$, it achieves a $(1-varepsilon)$-approximation in $(ell/varepsilon)^{O(log(1/varepsilon))} n^{O(1)}$ time. Our results unify hyperbolic geometry, topological combinatorics, and graph algorithm theory.

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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 $r in Omega(log n)$ 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 extsc{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 extsc{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, extsc{Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of $rin Omega(log n)$. Finally, in the case when the disks have ply (depth) at most $ell$, we give a PTAS for extsc{Maximum Independent Set} that has only quasi-polynomial dependence on $1/varepsilon$ and $ell$. Our PTAS is a further generalization of our exact algorithm.

Problem

• Study hyperbolic disk intersection graphs' structural properties.
• Analyze separators and complexity of Independent Set problem.
• Develop algorithms for Independent Set in hyperbolic settings.

Innovation

• Hyperbolic disk graphs with balanced separators
• Delaunay complexes with bounded outerplanarity
• Dynamic programming for Independent Set