### Abstract

We describe a new technique for proving the existence of small e-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving O(1/ε log 1/ε) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of ( _{ε}^{1}) size ε-nets for them. This includes the geometric hypergraph in wdiich 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 the existence of O( _{ε}^{1}/_{ε}) size ε-nets for halfspaces in Double strok R sign.

Original language | English (US) |
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Title of host publication | Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08 |

Pages | 199-207 |

Number of pages | 9 |

DOIs | |

State | Published - Dec 12 2008 |

Event | 24th Annual Symposium on Computational Geometry, SCG'08 - College Park, MD, United States Duration: Jun 9 2008 → Jun 11 2008 |

### Other

Other | 24th Annual Symposium on Computational Geometry, SCG'08 |
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Country | United States |

City | College Park, MD |

Period | 6/9/08 → 6/11/08 |

### Fingerprint

### Keywords

- Discrete geometry
- Hitting sets
- Hypergraph transversals
- Strong ε-nets

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08*(pp. 199-207) https://doi.org/10.1145/1377676.1377708

**New existence proofs for ε-nets.** / Pyrga, Evangelia; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08.*pp. 199-207, 24th Annual Symposium on Computational Geometry, SCG'08, College Park, MD, United States, 6/9/08. https://doi.org/10.1145/1377676.1377708

}

TY - GEN

T1 - New existence proofs for ε-nets

AU - Pyrga, Evangelia

AU - Ray, Saurabh

PY - 2008/12/12

Y1 - 2008/12/12

N2 - We describe a new technique for proving the existence of small e-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving O(1/ε log 1/ε) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of ( ε1) size ε-nets for them. This includes the geometric hypergraph in wdiich 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 the existence of O( ε1/ε) size ε-nets for halfspaces in Double strok R sign.

AB - We describe a new technique for proving the existence of small e-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving O(1/ε log 1/ε) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of ( ε1) size ε-nets for them. This includes the geometric hypergraph in wdiich 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 the existence of O( ε1/ε) size ε-nets for halfspaces in Double strok R sign.

KW - Discrete geometry

KW - Hitting sets

KW - Hypergraph transversals

KW - Strong ε-nets

UR - http://www.scopus.com/inward/record.url?scp=57349089195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57349089195&partnerID=8YFLogxK

U2 - 10.1145/1377676.1377708

DO - 10.1145/1377676.1377708

M3 - Conference contribution

AN - SCOPUS:57349089195

SN - 9781605580715

SP - 199

EP - 207

BT - Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08

ER -