Computing Maximum Cliques in Unit Disk Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915361369096192 |
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| author | Tkachenko, Anastasiia Wang, Haitao |
| author_facet | Tkachenko, Anastasiia Wang, Haitao |
| contents | Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum clique in $G(P)$. The previously best algorithm for the problem runs in $O(n^{7/3+o(1)})$ time. We show that the problem can be solved in $O(n \log n + n K^{4/3+o(1)})$ time, where $K$ is the maximum clique size. The algorithm is faster than the previous one when $K=o(n)$. In addition, if $P$ is in convex position, we give a randomized algorithm that runs in $O(n^{15/7+o(1)})= O(n^{2.143})$ worst-case time and the algorithm can compute a maximum clique with high probability. For points in convex position, one special case we solve is when a point in the maximum clique is given; we present an $O(n^2\log n)$ time (deterministic) algorithm for this special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Maximum Cliques in Unit Disk Graphs Tkachenko, Anastasiia Wang, Haitao Computational Geometry Data Structures and Algorithms Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum clique in $G(P)$. The previously best algorithm for the problem runs in $O(n^{7/3+o(1)})$ time. We show that the problem can be solved in $O(n \log n + n K^{4/3+o(1)})$ time, where $K$ is the maximum clique size. The algorithm is faster than the previous one when $K=o(n)$. In addition, if $P$ is in convex position, we give a randomized algorithm that runs in $O(n^{15/7+o(1)})= O(n^{2.143})$ worst-case time and the algorithm can compute a maximum clique with high probability. For points in convex position, one special case we solve is when a point in the maximum clique is given; we present an $O(n^2\log n)$ time (deterministic) algorithm for this special case. |
| title | Computing Maximum Cliques in Unit Disk Graphs |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.21926 |