## Abstract

Let D be a set of disks of arbitrary radii in the plane, and let P be a set of points. We study the following three problems: (i) Assuming P contains the set of center points of disks in D, find a minimum-cardinality subset ^{P} of P (if exists), such that each disk in D is pierced by at least h points of ^{P}, where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming P is such that for each D∈D there exists a point in P whose distance from D's center is at most αr(D), where r(D) is D's radius and 0≤α<1 is a given constant, find a minimum-cardinality subset ^{P} of P, such that each disk in D is pierced by at least one point of ^{P}. We call this problem minimum discrete piercing with cores. (iii) Assuming P is the set of center points of disks in D, and that each D∈D covers at most l points of P, where l is a constant, find a minimum-cardinality subset ^{D} of D, such that each point of P is covered by at least one disk of ^{D}. We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178-189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+)-approximation for minimum discrete piercing (i.e., for arbitrary P).

Original language | English |
---|---|

Pages (from-to) | 209-218 |

Number of pages | 10 |

Journal | Computational Geometry: Theory and Applications |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - 1 Apr 2008 |

## Keywords

- Approximation algorithms
- Covering
- Discrete piercing
- Geometric optimization
- Wireless networks

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics