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 Scopus citations

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 (US)
Pages (from-to)104-118
Number of pages15
JournalAlgorithmica (New York)
Volume21
Issue number1
DOIs
StatePublished - Jan 1 1998
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

Keywords

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

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  • 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. https://doi.org/10.1007/PL00009203