On Forbidden Induced Subgraphs for Unit Disk Graphs

Atminas, Aistis; Zamaraev, Viktor

doi:10.1007/s00454-018-9968-1

Cited by 8 publications

(12 citation statements)

References 18 publications

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“…We only show the following part of Theorem 3.1 to emphasize that, rather unexpectedly, parity plays a crucial role in disk graphs of co-degree at most 2. It is also amusing that the complement of any odd cycle is a unit disk graph while the complement of any even cycle of length at least 8 is not [10]. Here, the situation is somewhat reversed: complements of even cycles are easier to represent than complements of odd cycles.…”

Section: The Disjoint Union Of Cycles With At Most One Odd Is Co-diskmentioning

confidence: 99%

See 1 more Smart Citation

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: The Disjoint Union Of Cycles With At Most One Odd Is Co-diskmentioning

confidence: 99%

“…This representation of C 2s+1 can now be put on top of complements of even cycles. We identify the small region (point) where the disk D 1 intersects the disks of even index (in complements of [10]. Unfortunately, we cannot use this representation.…”

mentioning

confidence: 99%

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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show abstract

“…As shown in Theorem 4 below, it was recently proved that there exist infinitely many minimal forbidden induced subgraphs for the class of unit disk graphs (UDGs) or infinitely many minimal non-UDGs (Atminas and Zamaraev 2018). Readers may refer to the proof of Theorem 4 in (Atminas and Zamaraev 2018).…”

Section: Background On Unit Disk Graphsmentioning

confidence: 99%

“…A graph is a unit disk graph, if its vertices can be represented as points in a 2D Euclidean space such that there is an edge between two vertices iff the distance between their corresponding points is at most m, where m is a positive constant (Atminas and Zamaraev 2018). Unit disk graphs are useful in many applications, for example wireless networks, and have been studied actively in recent years (Breu and Kirkpatrick 1998;McDiarmid and Müller 2013;da Fonseca et al 2015).…”

Section: Background On Unit Disk Graphsmentioning

confidence: 99%

Qualitative Spatial Logic over 2D Euclidean Spaces Is Not Finitely Axiomatisable

Alechina

2019

AAAI

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Several qualitative spatial logics used in reasoning about geospatial data have a sound and complete axiomatisation over metric spaces. It has been open whether the same axiomatisation is also sound and complete for 2D Euclidean spaces. We answer this question negatively by showing that the axiomatisations presented in (Du et al. 2013; Du and Alechina 2016) are not complete for 2D Euclidean spaces and, moreover, the logics are not finitely axiomatisable.

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“…Atminas and Zamaraev [7] showed that the complement of K 2 + C s is not a unit disk graph when s is odd (where K 2 is an edge and C s is a cycle on s vertices). Is this obstruction enough to obtain an alternative polynomial-time algorithm for Maximum Clique on unit disk graphs?…”

Section: Problem 3 Is There a Ptas For Maximum Independent Set On Grmentioning

confidence: 99%

EPTAS for Max Clique on Disks and Unit Balls

Bonamy

Bonnet

Bousquet

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We propose a polynomial-time algorithm which takes as input a finite set of points of R 3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane.Almost three decades ago, an elegant polynomial-time algorithm was found for 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. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzążewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VCdimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.

show abstract

On Forbidden Induced Subgraphs for Unit Disk Graphs

Cited by 8 publications

References 18 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

Qualitative Spatial Logic over 2D Euclidean Spaces Is Not Finitely Axiomatisable

EPTAS for Max Clique on Disks and Unit Balls

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