### Abstract

We show the existence of e-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R^{3} and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of size O (1/εe log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

Original language | English (US) |
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Title of host publication | STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing |

Pages | 639-648 |

Number of pages | 10 |

DOIs | |

State | Published - 2009 |

Event | 41st Annual ACM Symposium on Theory of Computing, STOC '09 - Bethesda, MD, United States Duration: May 31 2009 → Jun 2 2009 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Other

Other | 41st Annual ACM Symposium on Theory of Computing, STOC '09 |
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Country | United States |

City | Bethesda, MD |

Period | 5/31/09 → 6/2/09 |

### Keywords

- E-nets
- Geometric range spaces
- Hitting set
- Randomized algorithms
- Set cover

### ASJC Scopus subject areas

- Software

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## Cite this

*STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing*(pp. 639-648). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/1536414.1536501