research-article

New existence proofs ε-nets

Authors:

Evangelia Pyrga,

Saurabh RayAuthors Info & Claims

SCG '08: Proceedings of the twenty-fourth annual symposium on Computational geometry

Pages 199 - 207

https://doi.org/10.1145/1377676.1377708

Published: 09 June 2008 Publication History

Abstract

We describe a new technique for proving the existence of small μ-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o(1/μ log 1/μ) upper bounds which the standard VC-dimension theory does not allow. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of O(1/μ) size μ-nets for them. This includes the geometric hypergraph in which the vertex set is a set of points in the plane and the hyperedges are defined by a set of pseudo-disks. This result was not known previously. We also get a very short proof for O(1/μ) size μ-nets for half-spaces in R³.

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https://doi.org/10.5817/CZ.MUNI.EUROCOMB23-107
Elbassioni K(2023)A Bicriteria Approximation Algorithm for the Minimum Hitting Set Problem in Measurable Range SpacesOperations Research Letters10.1016/j.orl.2023.07.005Online publication date: Aug-2023
https://doi.org/10.1016/j.orl.2023.07.005
Alon NJartoux BKeller CSmorodinsky SYuditsky Y(2022)The $$\varepsilon $$-t-Net ProblemDiscrete & Computational Geometry10.1007/s00454-022-00376-x68:2(618-644)Online publication date: 7-Mar-2022
https://doi.org/10.1007/s00454-022-00376-x
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Index Terms

New existence proofs ε-nets
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '08: Proceedings of the twenty-fourth annual symposium on Computational geometry

June 2008

304 pages

ISBN:9781605580715

DOI:10.1145/1377676

Program Chair:
Monique Teillaud
INRIA Sophia Antipolis - Méditerranée, France

Copyright © 2008 ACM.

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Published: 09 June 2008

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SoCG08: 24th Annual Symposium on Computational Geometry

June 9 - 11, 2008

MD, College Park, USA

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Cited By

Raman RSingh K(2023)On Hypergraph Supports.European Conference on Combinatorics, Graph Theory and Applications10.5817/CZ.MUNI.EUROCOMB23-107Online publication date: 28-Aug-2023
https://doi.org/10.5817/CZ.MUNI.EUROCOMB23-107
Elbassioni K(2023)A Bicriteria Approximation Algorithm for the Minimum Hitting Set Problem in Measurable Range SpacesOperations Research Letters10.1016/j.orl.2023.07.005Online publication date: Aug-2023
https://doi.org/10.1016/j.orl.2023.07.005
Alon NJartoux BKeller CSmorodinsky SYuditsky Y(2022)The $$\varepsilon $$-t-Net ProblemDiscrete & Computational Geometry10.1007/s00454-022-00376-x68:2(618-644)Online publication date: 7-Mar-2022
https://doi.org/10.1007/s00454-022-00376-x
Bansal NBatra JMarx D(2021)Non-uniform geometric set cover and scheduling on multiple machinesProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458243(3011-3021)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458243
Huang LLi JShi Q(2020)Approximation algorithms for the connected sensor cover problemTheoretical Computer Science10.1016/j.tcs.2020.01.020Online publication date: Jan-2020
https://doi.org/10.1016/j.tcs.2020.01.020
Bishnu ADesai SGhosh AMishra GPaul S(2020)Existence of planar support for geometric hypergraphs using elementary techniquesDiscrete Mathematics10.1016/j.disc.2020.111853343:6(111853)Online publication date: Jun-2020
https://doi.org/10.1016/j.disc.2020.111853
Raman RRay S(2020)Constructing Planar Support for Non-Piercing RegionsDiscrete & Computational Geometry10.1007/s00454-020-00216-wOnline publication date: 22-Jun-2020
https://doi.org/10.1007/s00454-020-00216-w
Ashok PBasu Roy AGovindarajan S(2019)Local search strikes again: PTAS for variants of geometric covering and packingJournal of Combinatorial Optimization10.1007/s10878-019-00432-yOnline publication date: 21-Jun-2019
https://doi.org/10.1007/s10878-019-00432-y
Agarwal PPan J(2019)Near-Linear Algorithms for Geometric Hitting Sets and Set CoversDiscrete & Computational Geometry10.1007/s00454-019-00099-6Online publication date: 9-May-2019
https://doi.org/10.1007/s00454-019-00099-6
Kobylkin KDryakhlova I(2019)Approximation Algorithms for Piercing Special Families of Hippodromes: An Extended AbstractMathematical Optimization Theory and Operations Research10.1007/978-3-030-22629-9_40(565-580)Online publication date: 12-Jun-2019
https://doi.org/10.1007/978-3-030-22629-9_40
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