Structure and Independence in Hyperbolic Uniform Disk Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913761511604224 |
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| author | Bläsius, Thomas von der Heydt, Jean-Pierre Kisfaludi-Bak, Sándor Wilhelm, Marcus van Wordragen, Geert |
| author_facet | Bläsius, Thomas von der Heydt, Jean-Pierre Kisfaludi-Bak, Sándor Wilhelm, Marcus van Wordragen, Geert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure and Independence in Hyperbolic Uniform Disk Graphs Bläsius, Thomas von der Heydt, Jean-Pierre Kisfaludi-Bak, Sándor Wilhelm, Marcus van Wordragen, Geert Computational Geometry Data Structures and Algorithms F.2.2 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. |
| title | Structure and Independence in Hyperbolic Uniform Disk Graphs |
| topic | Computational Geometry Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2407.09362 |