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QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

Giannopoulos, P., Bonnet, E., Kim, E. J., Rzazewski, P. and Sikora, F. (2018). QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs. Paper presented at the 34th International Symposium on Computational Geometry (SoCG 2018), 11 - 14 July 2018, Budapest, Hungary.

Abstract

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 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^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, 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-epsilon}}, unless the Exponential Time Hypothesis fails.

Publication Type: Conference or Workshop Item (Paper)
Additional Information: © Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, and Florian Sikora; licensed under Creative Commons License CC-BY
Publisher Keywords: disk graph, maximum clique, computational complexity
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Mathematics, Computer Science & Engineering > Computer Science
URI: http://openaccess.city.ac.uk/id/eprint/21507
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