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EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

Bonamy, M., Bonnet, É., Bousquet, N. , Charbit, P., Giannopoulos, P. ORCID: 0000-0002-6261-1961, Kim, E. J., Rzazewski, P., Sikora, F. & Thomasse, S. (2021). EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs. Journal of the ACM, 68(2), 9. doi: 10.1145/3433160


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 Cliqe 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 that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails.

Publication Type: Article
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Publisher Keywords: disk graph, maximum clique, computational complexity, approximation, algorithms, subexponential algorithms
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Science & Technology > Computer Science
School of Science & Technology > Computer Science > giCentre
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