### Abstract

This paper deals with vector covering problems in d-dimensional 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-line 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 Frenk in [5] where it is claimed that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(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 |
---|---|

Pages (from-to) | 104-118 |

Number of pages | 15 |

Journal | Algorithmica (New York) |

Volume | 21 |

Issue number | 1 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Safety, Risk, Reliability and Quality
- Applied Mathematics

### Cite this

*Algorithmica (New York)*,

*21*(1), 104-118.

**On-line and off-line approximation algorithms for vector covering problems.** / Alon, N.; Azar, Y.; Csirik, J.; Epstein, L.; Sevastianov, S. V.; Vestjens, A. P A; Woeginger, G. J.

Research output: Contribution to journal › Article

*Algorithmica (New York)*, vol. 21, no. 1, pp. 104-118.

}

TY - JOUR

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

AU - Alon, N.

AU - Azar, Y.

AU - Csirik, J.

AU - Epstein, L.

AU - Sevastianov, S. V.

AU - Vestjens, A. P A

AU - Woeginger, G. J.

PY - 1998

Y1 - 1998

N2 - This paper deals with vector covering problems in d-dimensional 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-line 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 Frenk in [5] where it is claimed that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(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 d-dimensional 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-line 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 Frenk in [5] where it is claimed that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(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 - Competitive analysis

KW - Covering problem

KW - On-line algorithm

KW - Packing problem

KW - Worst case ratio

UR - http://www.scopus.com/inward/record.url?scp=0013454612&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013454612&partnerID=8YFLogxK

M3 - Article

VL - 21

SP - 104

EP - 118

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -