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Online Bin Covering with Advice

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Abstract

The bin covering problem asks for covering a maximum number of bins with an online sequence of n items of different sizes in the range (0, 1]; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide asymptotically tight bounds of \(\Theta (n \log {\textsc {Opt}})\) on the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size \(o(\log \log n)\) has a competitive ratio of at most 0.5. In other words, advice of size \(o(\log \log n)\) is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size \(O(\log \log n)\) is sufficient to achieve an asymptotic competitive ratio of \(0.5\bar{3}\) which is strictly better than the best ratio 0.5 attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of advice bits is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.

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Notes

  1. There is another problem, also sometimes referred to as “dual bin packing”, which asks for maximizing the number of items packed into a fixed number of bins; for the advice complexity of that dual bin packing problem, see [10, 22].

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Acknowledgements

We thank anonymous referees for their valuable comments.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Joan Boyar, Lene M. Favrholdt & Kim S. Larsen

  2. Department of Computer Science, E2-EITC, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Shahin Kamali

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  1. Joan Boyar
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  4. Kim S. Larsen
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Correspondence to Kim S. Larsen.

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Joan Boyar, Lene M. Favrholdt, Kim S. Larsen were supported in part by the Danish Council for Independent Research, Natural Sciences, Grant DFF-1323-00247. A preliminary version of this paper appeared in the 16th International Algorithms and Data Structures Symposium (WADS), volume 11646 of Lecture Notes in Computer Science, Springer 2019.

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Boyar, J., Favrholdt, L.M., Kamali, S. et al. Online Bin Covering with Advice. Algorithmica 83, 795–821 (2021). https://doi.org/10.1007/s00453-020-00728-0

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