  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 set in S and a "coverage factor" (positive
integer) k. A subset of elements are presented online in an arbitrary order.
When each element, say i, is presented, we are also told the collection of all
(at least k) sets and their costs to which i belongs and we need to select
additional sets from these sets containing i, if necessary, such that our
collection of selected sets contains at least k sets that contain the element
i. The goal is to minimize the total cost of the selected sets (our algorithm
and competitive ratio bounds can be extended to the case when a set can be
selected at most a pre-specified number of times instead of just once; we do
not report these extensions for simplicity and also because they have no
relevance to the biological applications that motivated our work). In this
paper, we describe a new randomized algorithm for the online multicover problem
based on a randomized version of the winnowing approach of Littlestone. This
algorithm generalizes and improves some earlier results by N. Alon, B.
Awerbuch, Y. Azar, N. Buchbinder, and J. Naor. We also discuss lower bounds on
competitive ratios for deterministic algorithms for general $k$.
