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Summary: The Online Set Cover Problem
(Extended Abstract)
Noga Alon
Baruch Awerbuch
Yossi Azar
Niv Buchbinder §
Joseph (Seffi) Naor ¶
Abstract
Let X = {1, 2, . . ., n} be a ground set of n elements, and let S be a family of subsets of X,
|S| = m, with a positive cost cS associated with each S 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 one-by-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 X of elements given by the adversary is not known in advance to the
algorithm. (In general, X 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 (off-line) a family of sets COP T
that covers X . The performance of the algorithm is the ratio between the cost of C and the
cost of COP T . The maximum ratio, taken over all input sequences, is the competitive ratio of the
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