Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs
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arXiv
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2024
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| author | Koana, Tomohiro Purohit, Nidhi Simonov, Kirill |
| author_facet | Koana, Tomohiro Purohit, Nidhi Simonov, Kirill |
| contents | In Clique Cover, given a graph $G$ and an integer $k$, the task is to partition the vertices of $G$ into $k$ cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster.
Many classical NP-hard problems are known to admit $2^{O(n^{(1 - 1/d)})}$-time algorithms on unit ball graphs in $\mathbb{R}^d$ [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in $\mathbb{R}^3$, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021].
In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a $2^{O(\sqrt{n})}$-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a $2^{o(n)}$-time algorithm on unit ball graphs in dimension $5$, unless the ETH fails. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs Koana, Tomohiro Purohit, Nidhi Simonov, Kirill Data Structures and Algorithms Computational Geometry In Clique Cover, given a graph $G$ and an integer $k$, the task is to partition the vertices of $G$ into $k$ cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster. Many classical NP-hard problems are known to admit $2^{O(n^{(1 - 1/d)})}$-time algorithms on unit ball graphs in $\mathbb{R}^d$ [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in $\mathbb{R}^3$, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021]. In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a $2^{O(\sqrt{n})}$-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a $2^{o(n)}$-time algorithm on unit ball graphs in dimension $5$, unless the ETH fails. |
| title | Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/2410.03609 |