### 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 halfspaces in ℝ3 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) |
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Title of host publication | Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 |

Pages | 17-22 |

Number of pages | 6 |

DOIs | |

State | Published - Dec 4 2009 |

Event | 25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark Duration: Jun 8 2009 → Jun 10 2009 |

### Other

Other | 25th Annual Symposium on Computational Geometry, SCG'09 |
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Country | Denmark |

City | Aarhus |

Period | 6/8/09 → 6/10/09 |

### Fingerprint

### Keywords

- Approximation algorithm
- Epsilon nets
- Hitting sets
- Local search

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09*(pp. 17-22) https://doi.org/10.1145/1542362.1542367

**PTAS for geometric hitting set problems via local search.** / Mustafa, Nabil H.; Ray, Saurabh.

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

*Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09.*pp. 17-22, 25th Annual Symposium on Computational Geometry, SCG'09, Aarhus, Denmark, 6/8/09. https://doi.org/10.1145/1542362.1542367

}

TY - GEN

T1 - PTAS for geometric hitting set problems via local search

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2009/12/4

Y1 - 2009/12/4

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 halfspaces in ℝ3 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 halfspaces in ℝ3 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 algorithm

KW - Epsilon nets

KW - Hitting sets

KW - Local search

UR - http://www.scopus.com/inward/record.url?scp=70849098296&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70849098296&partnerID=8YFLogxK

U2 - 10.1145/1542362.1542367

DO - 10.1145/1542362.1542367

M3 - Conference contribution

SN - 9781605585017

SP - 17

EP - 22

BT - Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09

ER -