### Abstract

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ℝ
^{3} and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.

Original language | English (US) |
---|---|

Pages (from-to) | 883-895 |

Number of pages | 13 |

Journal | Discrete and Computational Geometry |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - Sep 16 2010 |

### Fingerprint

### Keywords

- Approximation algorithms
- Epsilon nets
- Greedy algorithms
- Hitting sets
- Local search
- Transversals

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete and Computational Geometry*,

*44*(4), 883-895. https://doi.org/10.1007/s00454-010-9285-9

**Improved results on geometric hitting set problems.** / Mustafa, Nabil H.; Ray, Saurabh.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 44, no. 4, pp. 883-895. https://doi.org/10.1007/s00454-010-9285-9

}

TY - JOUR

T1 - Improved results on geometric hitting set problems

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2010/9/16

Y1 - 2010/9/16

N2 - We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ℝ 3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.

AB - We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ℝ 3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.

KW - Approximation algorithms

KW - Epsilon nets

KW - Greedy algorithms

KW - Hitting sets

KW - Local search

KW - Transversals

UR - http://www.scopus.com/inward/record.url?scp=77957980032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957980032&partnerID=8YFLogxK

U2 - 10.1007/s00454-010-9285-9

DO - 10.1007/s00454-010-9285-9

M3 - Article

AN - SCOPUS:77957980032

VL - 44

SP - 883

EP - 895

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -