Maximum Clique in Disk-Like Intersection Graphs

Abstract

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 I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude invoking the hardness of Maximum Independent Set on the class 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.

Computing a maximum clique in geometric superclasses of disk graphs

“…Relatedly, it would be interesting to generalise the existence of an EPTAS for maximum clique to superclasses of disk graphs. This was achieved with Bonnet and Miltzow for intersection graphs of homothets of a fixed bounded centrally symmetric convex set [6]. In this paper, we aim at generalising further to intersection graphs of convex pseudo-disks, for which we conjecture the existence of an EPTAS, and give partial results towards proving it.…”

“…In this paper, we give a polynomial time algorithm for solving maximum clique in Π 2 : the intersection graphs class of unit disks and 2-pancakes. This is to put in contrast with the fact that computing a maximum clique in intersection graphs of unit disks and axis-parallel rectangles (instead of 2-pancakes) is NP-hard and even APX-hard, as shown together with Bonnet and Miltzow [6], even though maximum clique can be solved in polynomial time in axis-parallel rectangle graphs [11].…”

Computing a Maximum Clique in Geometric Superclasses of Disk Graphs

“…Relatedly, it would be interesting to generalise the existence of an EPTAS for maximum clique to superclasses of disk graphs. This was achieved with Bonnet and Miltzow for intersection graphs of homothets of a fixed bounded centrally symmetric convex set (Bonnet et al 2020). In this paper, we aim at generalising further to intersection graphs of convex pseudo-disks, for which we conjecture the existence of an EPTAS, and give partial results towards proving it.…”

“…In this paper, we give a polynomial time algorithm for solving maximum clique in Π 2 : the intersection graphs class of unit disks and 2pancakes. This is to put in contrast with the fact that computing a maximum clique in intersection graphs of unit disks and axis-parallel rectangles (instead of 2-pancakes) is NP-hard and even APX-hard, as shown together with Bonnet and Miltzow (2020), even though maximum clique can be solved in polynomial time in axis-parallel rectangle graphs (Imai and Asano 1983).…”