Homothetic polygons and beyond: Maximal cliques in intersection graphs

Brimkov, Valentin E.; Junosza-Szaniawski, Konstanty; Kafer, Sean; Kratochvı́l, Jan; Pergel, Martin; Rzążewski, Paweł; Szczepankiewicz, Matthew; Terhaar, Joshua

doi:10.1016/j.dam.2018.03.046

Cited by 2 publications

(4 citation statements)

References 19 publications

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“…Proof. The argument is similar to the one used by Brimkov et al [20], which was in turn inspired by the construction by Kratochvíl and Matoušek [47]. Consider the graph G in Figure 13, containing what we will henceforth refer to as black, gray, red, blue, and white vertices.…”

Section: Filled Ellipses and Filled Trianglesmentioning

confidence: 92%

“…It is known that P hom graphs, i.e., intersection graphs of homothetic copies of a fixed polygon P, are pseudodisk intersection graphs [4]. As shown by Brimkov et al, for every convex k-gon P, a P hom graph with n vertices has at most n k maximal cliques [20]. This clearly implies that Maximum Cliqe, but also Cliqe p-Partition for fixed p is polynomially solvable in P hom graphs.…”

Section: Homothets Of a Convex Polygonmentioning

confidence: 96%

“…This clearly implies that Maximum Clique, but also Clique p-Partition for fixed p is polynomially solvable in P hom graphs. Actually, the bound on the maximum number of maximal cliques from [20] holds for a more general class of graphs, called k DIR -CONV, which admit a intersection representation by convex polygons, whose every side is parallel to one of k directions.…”

Section: Homothets Of a Convex Polygonmentioning

confidence: 99%

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

Bonamy

Bonnet

Bousquet

et al. 2021

J. ACM

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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 M AXIMUM C LIQUE 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 Õ( n 2/3 ) for M AXIMUM C LIQUE 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 M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE 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 that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

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Section: Filled Ellipses and Filled Trianglesmentioning

confidence: 92%

Section: Homothets Of a Convex Polygonmentioning

confidence: 96%

Section: Homothets Of a Convex Polygonmentioning

confidence: 99%

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

Bonamy

Bonnet

Bousquet

et al. 2021

J. ACM

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“…We recall that Maximum Clique can be solved in polynomial-time in unit disk graphs [20,52] and in axis-parallel rectangle intersection graphs [14]. Now if the objects can be unit disks and axis-parallel rectangles, we show that even a SUBEXPAS is unlikely.…”

Section: Max Interval Permutation Avoidancementioning

confidence: 87%

Maximum Clique in Disk-Like Intersection Graphs

Bonnet,

Grelier,

Miltzow

2020

Preprint

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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.

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Homothetic polygons and beyond: Maximal cliques in intersection graphs

Cited by 2 publications

References 19 publications

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

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

Maximum Clique in Disk-Like Intersection Graphs

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