### 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 |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 406-418 |

Number of pages | 13 |

Volume | 1136 |

ISBN (Print) | 3540616802, 9783540616801 |

DOIs | |

Publication status | Published - 1996 |

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) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

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

City | Barcelona |

Period | 9/25/96 → 9/27/96 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 1136, 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

**On-line and off-line approximation algorithms for vector covering problems.** / Alon, Noga; Csirik, János; Sevastianov, Sergey V.; Vestjens, Arjen P A; Woeginger, Gerhard J.

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

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 1136, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1136, Springer Verlag, pp. 406-418, 4th European Symposium on Algorithms, ESA 1996, Barcelona, Spain, 9/25/96. https://doi.org/10.1007/3-540-61680-2_71

}

TY - GEN

T1 - On-line and off-line approximation algorithms for vector covering problems

AU - Alon, Noga

AU - Csirik, János

AU - Sevastianov, Sergey V.

AU - Vestjens, Arjen P A

AU - Woeginger, Gerhard J.

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

KW - Approximation algorithm

KW - Covering problem

KW - On-line algorithm

KW - Packing problem

KW - Worst case ratio

UR - http://www.scopus.com/inward/record.url?scp=0344750918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0344750918&partnerID=8YFLogxK

U2 - 10.1007/3-540-61680-2_71

DO - 10.1007/3-540-61680-2_71

M3 - Conference contribution

SN - 3540616802

SN - 9783540616801

VL - 1136

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 406

EP - 418

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -