QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

Edouard Bonnet (1) , Panos Giannopoulos (2) , Eun Jung Kim (3) , Paweł Rzążewski (4) , Florian Sikora (3)

1 MC2 - Modèles de calcul, Complexité, Combinatoire
2 Middlesex University [London]
3 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
4 Faculty of Mathematics and Information Science [Warszawa]

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 graph 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.

Mots clés: computational complexity; maximum clique; disk graph; intersection graph

Domaines: Informatique [cs]; Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]

Edouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora. QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs. SoCG 2018, Jun 2018, Budapest, Hungary. ⟨10.4230/LIPIcs⟩. ⟨hal-01991635⟩