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arXiv · 1712.05010

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

Édouard Bonnet·Panos Giannopoulos·Eun Jung Kim·Paweł Rzążewski·Florian Sikora
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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 \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 approximation which is not attainable even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.

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Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora. 2017-12-13. QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs. https://arxiv.org/abs/1712.05010

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