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# Finding a Maximum Clique in a Disk Graph

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## Cite As

Jared Espenant, J. Mark Keil, and Debajyoti Mondal. Finding a Maximum Clique in a Disk Graph. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.30

## Abstract

A disk graph is an intersection graph of disks in the Euclidean plane, where the disks correspond to the vertices of the graph and a pair of vertices are adjacent if and only if their corresponding disks intersect. The problem of determining the time complexity of computing a maximum clique in a disk graph is a long-standing open question that has been very well studied in the literature. The problem is known to be open even when the radii of all the disks are in the interval [1,(1+ε)], where ε > 0. If all the disks are unit disks then there exists an O(n³log n)-time algorithm to compute a maximum clique, which is the best-known running time for over a decade. Although the problem of computing a maximum clique in a disk graph remains open, it is known to be APX-hard for the intersection graphs of many other convex objects such as intersection graphs of ellipses, triangles, and a combination of unit disks and axis-parallel rectangles. Here we obtain the following results. - We give an algorithm to compute a maximum clique in a unit disk graph in O(n^2.5 log n)-time, which improves the previously best known running time of O(n³log n) [Eppstein '09]. - We extend a widely used "co-2-subdivision approach" to prove that computing a maximum clique in a combination of unit disks and axis-parallel rectangles is NP-hard to approximate within 4448/4449 ≈ 0.9997. The use of a "co-2-subdivision approach" was previously thought to be unlikely in this setting [Bonnet et al. '20]. Our result improves the previously known inapproximability factor of 7633010347/7633010348 ≈ 0.9999. - We show that the parameter minimum lens width of the disk arrangement may be used to make progress in the case when disk radii are in [1,(1+ε)]. For example, if the minimum lens width is at least 0.265 and ε ≤ 0.0001, which still allows for non-Helly triples in the arrangement, then one can find a maximum clique in polynomial time.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Maximum clique
• Disk graph
• Time complexity
• APX-hardness

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