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

## 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/(2*d*) in *d* ≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk (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-line 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 areaof compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/*d* for every *d* ≥ 2.

## Keywords

Approximation algorithm worst case ratio packing problem covering problem on-line algorithm## Preview

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