 
Summary: The Online Set Cover Problem
Noga Alon Baruch Awerbuch y Yossi Azar z Niv Buchbinder x Joseph (SeĈ) Naor {
Abstract
Let X = f1; 2; : : : ; ng be a ground set of n elements, and let S be a family of subsets of X ,
jSj = m, with a positive cost c S associated with each S 2 S.
Consider the following online version of the set cover problem, described as a game between
an algorithm and an adversary. An adversary gives elements to the algorithm from X oneby
one. Once a new element is given, the algorithm has to cover it by some set of S containing
it. We assume that the elements of X and the members of S are known in advance to the
algorithm, however, the set X 0 X of elements given by the adversary is not known in advance
to the algorithm. (In general, X 0 may be a strict subset of X.) The objective is to minimize
the total cost of the sets chosen by the algorithm. Let C denote the family of sets in S that
the algorithm chooses. At the end of the game the adversary also produces (oline) a family
of sets COPT that covers X 0 . The performance of the algorithm is the ratio between the cost of
C and the cost of COPT . The maximum ratio, taken over all input sequences, is the competitive
ratio of the algorithm.
We present an O(log m log n) competitive deterministic algorithm for the problem, and es
tablish a nearly
matching
