Approximating the online set multicover problems via randomized winnowing

Authors:
Piotr Berman;Bhaskar DasGupta
Affiliations:
Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA;Computer Science Department, University of Illinois at Chicago, Chicago, IL
Venue:
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
Year:
2005

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Abstract

In this paper, we consider the weighted online set k- multicover problem. In this problem, we have an universe V of elements, a family $\mathcal{S}$ of subsets of V with a positive real cost for every $S \in \mathcal{S}$, and a “coverage factor” (positive integer) k. A subset {i0, i1,...}⊆V of elements are presented online in an arbitrary order. When each element ip is presented, we are also told the collection of all (at least k) sets $\mathcal{S}_{i_p} \subseteq \mathcal{S}$ and their costs in which ip belongs and we need to select additional sets from $\mathcal{S}_{i_p}$ if necessary such that our collection of selected sets contains at leastk sets that contain the element ip. 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].