Summary: Online and Offline Approximation Algorithms
for Vector Covering Problems
Noga Alon \Lambda Yossi Azar y J' anos Csirik z Leah Epstein x
Sergey V. Sevastianov -- Arjen P.A. Vestjens k
Gerhard J. Woeginger \Lambda\Lambda
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  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
have worst case ratio better than 2=(2d + 1). For the offline 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 offline 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 offline approximation algorithms with worst case ratio 1=d for every d – 2.