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

N. Alon, Y. Azar, J. Csirik, L. Epstein, S. V. Sevastianov, A. P A Vestjens, G. J. Woeginger

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 languageEnglish
Pages (from-to)104-118
Number of pages15
JournalAlgorithmica (New York)
Volume21
Issue number1
Publication statusPublished - 1998

Fingerprint

Covering Problem
Approximation algorithms
Approximation Algorithms
Line
Approximability
Summation
Polynomial-time Algorithm
NP-complete problem
Partition
Polynomials
Zero

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

Alon, N., Azar, Y., Csirik, J., Epstein, L., Sevastianov, S. V., Vestjens, A. P. A., & Woeginger, G. J. (1998). On-line and off-line approximation algorithms for vector covering problems. 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.

In: Algorithmica (New York), Vol. 21, No. 1, 1998, p. 104-118.

Research output: Contribution to journalArticle

Alon, N, Azar, Y, Csirik, J, Epstein, L, Sevastianov, SV, Vestjens, APA & Woeginger, GJ 1998, 'On-line and off-line approximation algorithms for vector covering problems', Algorithmica (New York), vol. 21, no. 1, pp. 104-118.
Alon N, Azar Y, Csirik J, Epstein L, Sevastianov SV, Vestjens APA et al. On-line and off-line approximation algorithms for vector covering problems. Algorithmica (New York). 1998;21(1):104-118.
Alon, N. ; Azar, Y. ; Csirik, J. ; Epstein, L. ; Sevastianov, S. V. ; Vestjens, A. P A ; Woeginger, G. J. / On-line and off-line approximation algorithms for vector covering problems. In: Algorithmica (New York). 1998 ; Vol. 21, No. 1. pp. 104-118.
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