Approximating the Online Set Multicover Problems via Randomized Winnowing

摘要

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 ∈ S, and a "coverage factor" (positive integer) k. A subset {i_0,i_1,...} is contained in 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 Si_p is contained in S and their costs in which i_p belongs and we need to select additional sets from Si_p 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 sets. 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].