# QPTAS and subexponential algorithm for maximum clique on disk graphs

Bonnet, Edouard, Giannopoulos, Panos, Kim, Eun Jung, Rzążewski, Paweł and Sikora, Florian (2018) QPTAS and subexponential algorithm for maximum clique on disk graphs. 34th International Symposium on Computational Geometry (SoCG 2018). In: 34th International Symposium on Computational Geometry, 11-14 Jun 2018, Budapest, Hungary. ISBN 9783959770668. ISSN 1868-8969 [Conference or Workshop Item] (doi:10.4230/LIPIcs.SoCG.2018.12)

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A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.