### Abstract

This paper deals with vector covering problems in ddimensional space. The input to a vector covering problem consists of a set X of d-dimensional vectors in [0, 1] d. The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability. For the on-fine version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This result contradicts a statement of Csirik and Freak (1990) in [5] where it is claimed that for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. For the off-fine version, we derive polynomial time approximation algorithms with worst case guarantee Ω (1/log d). For d = 2, we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the area of compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d ≥ 2.

Original language | English (US) |
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Title of host publication | Algorithms - ESA 1996 - 4th Annual European Symposium, Proceedings |

Editors | Josep Diaz, Maria Serna |

Publisher | Springer Verlag |

Pages | 406-418 |

Number of pages | 13 |

ISBN (Print) | 3540616802, 9783540616801 |

DOIs | |

State | Published - Jan 1 1996 |

Externally published | Yes |

Event | 4th European Symposium on Algorithms, ESA 1996 - Barcelona, Spain Duration: Sep 25 1996 → Sep 27 1996 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1136 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th European Symposium on Algorithms, ESA 1996 |
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Country | Spain |

City | Barcelona |

Period | 9/25/96 → 9/27/96 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Keywords

- Approximation algorithm
- Covering problem
- On-line algorithm
- Packing problem
- Worst case ratio

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## Cite this

*Algorithms - ESA 1996 - 4th Annual European Symposium, Proceedings*(pp. 406-418). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1136). Springer Verlag. https://doi.org/10.1007/3-540-61680-2_71