We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes $\mathcal I$ follow the same road. They show that, for every graph $G$ of a large-enough class $\mathcal C$, the complement of an even subdivision of $G$ belongs to the intersection class $\mathcal I$. Then they conclude invoking the hardness of Maximum Independent Set on the class $\mathcal C$, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.

中文翻译：

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类盘交图中的最大团

我们研究了平面中凸对象相交图中最大集团的复杂性。在算法方面，我们将单位圆盘的多项式时间算法 [Clark '90，Raghavan 和 Spinrad '03] 扩展到任何固定凸集的平移。我们还概括了磁盘的有效多项式时间近似方案 (EPTAS) 和次指数算法 [Bonnet 等人。'18，博纳米等人。'18]到固定中心对称凸集的同位词。关于该主题的主要悬而未决的问题是磁盘图中最大集团的复杂性。不知道这个问题是否是 NP-hard。我们观察到，到目前为止，所有最大集团在交叉图类 $\mathcal I$ 中的硬度证明都遵循相同的路径。他们表明，对于足够大的类 $\mathcal C$ 的每个图 $G$，$G$ 的偶数细分的补集属于交类 $\mathcal I$。然后他们得出结论，在类 $\mathcal C$ 上调用最大独立集的硬度，以及偶数细分保持该硬度的事实。然而，有强有力的证据表明这种方法不能用于磁盘图 [Bonnet 等人。'18]。我们建议了一种新方法，基于我们称为最大间隔置换避免的问题，我们证明它不太可能具有次指数时间近似方案。我们将该硬度转移到对象的交集图中的最大集团，这些对象可以是半平面（或单位圆盘）或轴平行矩形。该问题不适用于以前的方法。