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

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 greater than or equal to 2 dimensions. This result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d greater than or equal to 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d greater than or equal to 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 Theta(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 greater than or equal to 2.