Approximating the Online Set Multicover Problems via Randomized Winnowing

Abstract

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 S ∈ 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 ip ⊆ S and their costs to which i 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 sets 1 . In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of [15]. This algorithm generalizes and improves some earlier results in [1,2]. We also discuss lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [2].

Computational Complexities of Combinatorial Problems With Applications to Reverse Engineering of Biological Networks

“…An appropriate on-line variation of the set-multicover problem, as outlined in the reference [3], is also appropriate for the reverse engineering problems as will be mentioned in the next section.…”

“…In [3] we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of [12]. The winnowing algorithm has two scaling factors: a multiplicative scaling factor µ cS that depends on the particular set S containing i and another additive scaling factor…”

“…We also discussed in [3] lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [1] 4…”