Abstract
We study several variants of bin covering and design lower bounds on the asymptotic competitive ratio of online algorithms for these problems. Our main result is for vector covering with \(d \ge 2\) dimensions, for which our new lower bound is \(d+1\), improving over the previously known lower bound of \(d+\frac{1}{2}\), which was proved more than twenty years ago by Alon et al. Two special cases of vector covering are considered as well. We prove an improved lower bound of approximately 2.8228 for the asymptotic competitive ratio of the bin covering with cardinality constraints problem, and we also study vector covering with small components and show tight bounds of d for it. Finally, we define three models for one-dimensional black and white covering and show that no online algorithms of finite asymptotic competitive ratios can be designed for them.
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J. Balogh was supported by the European Union, co-financed by the European Social Fund (project “Integrated program for training new generation of scientists in the fields of computer science”, No. EFOP-3.6.3-VEKOP-16-2017-00002). L. Epstein and A. Levin were partially supported by a Grant from GIF—the German-Israeli Foundation for Scientific Research and Development (Grant No. I-1366-407.6/2016).
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Balogh, J., Epstein, L. & Levin, A. Lower bounds for online bin covering-type problems. J Sched 22, 487–497 (2019). https://doi.org/10.1007/s10951-018-0590-0
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DOI: https://doi.org/10.1007/s10951-018-0590-0