Approximating the online set multicover problems via randomized winnowing

In this paper, we consider the weighted online set k-multicover problem. In this problem, we have a universe V of elements, a family S of subsets of V with a positive real cost for every [email protected]?S, and a ''coverage factor'' (positive integer) k. A subset {i"0,i"1,...}@?V of elements are presented online in an arbitrary order. When each element i"p is presented, we are also told the collection of all (at least k) sets S"i"""[email protected]?S and their costs to which i"p belongs and we need to select additional sets from S"i"""p if necessary such that our collection of selected sets contains at leastk sets that contain the element i"p. The goal is to minimize the total cost of the selected sets. In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of [N. Littlestone, Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm, Machine Learning 2 (1988) 285-318]. This algorithm generalizes and improves some earlier results in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, A general approach to online network optimization problems, in: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, 2004, pp. 570-579; N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100-105]. We also discuss lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100-105].

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