QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Pawel Rzazewski, Florian Sikora, Marc Herbstritt
2018 International Symposium on Computational Geometry  
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
more » ... most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2Õ (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−ε , unless the Exponential Time Hypothesis fails. ACM Subject Classification Theory of computation → Computational geometry
doi:10.4230/lipics.socg.2018.12 dblp:conf/compgeom/BonnetG0RS18 fatcat:67msd4qudbasro5p2neguvcy5m