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
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
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.

Source: Azar, Yossi - School of Computer Science, Tel Aviv University
Epstein, Leah - Department of Mathematics, University of Haifa

Collections: Computer Technologies and Information Sciences; Mathematics