Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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 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 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 2Õ(n2/3) for 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 MAX CLIQUE on disk and unit ball graphs. MAX CLIQUE on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, 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 that cannot be attained even in time 2n1−ɛ, unless the Exponential Time Hypothesis fails. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the ACM (JACM) Association for Computing Machinery

Loading next page...
 
/lp/association-for-computing-machinery/eptas-and-subexponential-algorithm-for-maximum-clique-on-disk-and-unit-YUiDYSh9Ga

References (74)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2021 ACM
ISSN
0004-5411
eISSN
1557-735X
DOI
10.1145/3433160
Publisher site
See Article on Publisher Site

Abstract

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 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 2Õ(n2/3) for 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 MAX CLIQUE on disk and unit ball graphs. MAX CLIQUE on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, 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 that cannot be attained even in time 2n1−ɛ, unless the Exponential Time Hypothesis fails.

Journal

Journal of the ACM (JACM)Association for Computing Machinery

Published: Jan 6, 2021

Keywords: Disk graph

There are no references for this article.