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 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 $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} 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 \textsc{Max Clique} on disk and unit ball graphs. \textsc{Max Clique} on unit ball graphs is equivalent to finding, given a collection of points in $\mathbb R^3$, a maximum subset of points with diameter at most some fixed value. In stark contrast, \textsc{Maximum Clique} 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 $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.

翻译：磁盘图( unit) 是磁盘中封闭( unit) 磁盘的交叉图。 近30年前, 在单盘图[ Clark, Colbourn, Johnson; Discrete Mathemats'90] 中找到了优雅的多元时间算法。 从那时起, 磁盘和单位图中一直存在一个令人感兴趣的问题。 我们显示两个奇数周期的脱节从来不是磁盘图或单位( 3维) 球图的补充。 从这个事实和现有结果, 我们得出简单的 QPTAS 和在时间图中运行的亚化算法( Clokbilde) 。 磁盘中, 磁盘和直线性平面图中的一些直线性数字, 以正数中的一种正数表示。 以正数为正数的正数的正数和正数, 以正数的正数为正数的正数 。