 
Summary: Online and Offline Approximation Algorithms
for Vector Covering Problems
Noga Alon
Yossi Azar
J´anos Csirik
Leah Epstein §
Sergey V. Sevastianov ¶
Arjen P.A. Vestjens
Gerhard J. Woeginger
Abstract
This paper deals with vector covering problems in ddimensional space. The input to
a vector covering problem consists of a set X of ddimensional 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
NPcomplete, and we are mainly interested in its online and offline approximability.
For the online 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 (1990) in [5] where it is claimed that for d 2, no online algorithm can have a
worst case ratio better than zero. Moreover, we prove that for d 2, no online algorithm can
