The Online Set Cover Problem

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_s associated with each s \in 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' \subset 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 C_OPT that covers X'. The performance of the algorithm is the ratio between the cost of C and the cost of C_OPT . The maximum ra io, 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 establish a nearly matching \Omega ( log n logm / log logm+log log n) lower bound for all interesting values of m and n. The techniques used are motivated by similar techniques developed in computational learning theory for online prediction (e.g., the WINNOW algorithm) together with a novel way of converting the fractional solution they supply into a deterministic online algorithm.