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 an universe V of elements, a family S of subsets of V with a positive real cost for every S G ε S, and a "coverage factor" (positive integer) k. A subset {i_o, i₁...} ⊆ 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_ip ⊆ S and their costs in which _p belongs and we need to select additional sets from S_ip if necessary such that our collection of selected sets contains at least k sets that contain the element i_p. The goal is to minimize the total cost of the selected sets¹. In this paper, we describe a new randomized algorithm for the online multicover problem based on the randomized winnowing approach of [11]. This algorithm generalizes and improves some earlier results in [1]. We also discuss lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [1].