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Online Budgeted Maximum Coverage

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Abstract

We study the Online Budgeted Maximum Coverage problem. Subsets of a weighted ground set U arrive one by one, where each set has a cost. The online algorithm has to select a collection of sets, under the constraint that their cost is at most a given budget. Upon arrival of a set the algorithm must decide whether to accept or to irrevocably reject the arriving set, and it may also irrevocably drop previously accepted sets. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We give a deterministic \(\frac{4}{1-r}\)-competitive algorithm where r is the maximum ratio between the cost of a set and the total budget, and show that the competitive ratio of any deterministic online algorithm is \(\Omega (\frac{1}{1-r})\). We further give a randomized O(1)-competitive algorithm. We also give a deterministic \(O(\Delta )\)-competitive algorithm, where \(\Delta \) is the maximum weight of a set and a modified version of it with competitive ratio of \(O(\min \{\Delta ,\sqrt{w(U)}\})\) for the case that the total weight of the elements, w(U), is known in advance. A matching lower bound of \(\Omega (\min \{\Delta ,\sqrt{w(U)}\})\) is given. Finally, our results, including the lower bounds, apply also to Removable Online Knapsack.

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Notes

  1. A function f is called submodular if \(f(T) + f(T') \ge f(T \cup T') + f(T \cap T')\) for every two sets T and \(T'\) in the domain of f.

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Acknowledgements

We would like to thank an anonymous referee for improving the lower bound to \(\Omega (1/(1-r))\). (The lower bound that appeared in the conference version [36] was \(\Omega (1/\sqrt{1-r})\).) We also thank the referee for a suggestion that lead to an improvement in the constant randomized competitive ratio.

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Authors and Affiliations

  1. Bar-Ilan University, Ramat Gan, 52900, Israel

    Dror Rawitz

  2. CNRS and Université de Paris, Paris, France

    Adi Rosén

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  1. Dror Rawitz
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Correspondence to Dror Rawitz.

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A preliminary version of this paper appeared in the proceedings of ESA 2016 [36].

Dror Rawitz: Supported in part by a Grant from the Israeli Ministry of Science, Technology, and Space, Israel (French-Israeli Project Maimonide No. 3-10996) and by the Israel Science Foundation (Grant No. 497/14). Adi Rosén: Research supported in part by ANR Project NeTOC, and by a French-Israeli Grant PHC Maimonide 31768XL.

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Rawitz, D., Rosén, A. Online Budgeted Maximum Coverage. Algorithmica 83, 2989–3014 (2021). https://doi.org/10.1007/s00453-021-00850-7

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