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 Θ(nlogOPT)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (n \log {\textsc {Opt}})$$\end{document} on the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size o(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(\log \log n)$$\end{document} has a competitive ratio of at most 0.5. In other words, advice of size o(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(\log \log n)$$\end{document} 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(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log \log n)$$\end{document} is sufficient to achieve an asymptotic competitive ratio of 0.53¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.5\bar{3}$$\end{document} 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.

中文翻译：

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在线垃圾桶覆盖与建议

我们展示了任何建议大小为 o(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(\log \log n)$$\end{document} 的竞争比率最多为 0.5。换句话说，建议大小为 o(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \ usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(\log \log n)$$\end{document} 对于提高 0.5 的竞争比没有用，可以通过在线实现没有建议的算法。这个结果突出了建议模型中的装箱问题和装箱问题之间的区别：对于装箱问题，有几种具有恒定大小建议的算法优于没有建议的在线算法。此外，我们展示了大小为 O(loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} 的建议\usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log \log n)$$\end{document} 足以达到 0.53¯\documentclass 的渐近竞争比率[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt } \begin{文档}$$0。5\bar{3}$$\end{document} 严格优于纯在线算法可获得的最佳比率 0.5。引入和分析该算法所涉及的技术细节与装箱问题的现有结果有很大不同，并证实了这两个问题的不同性质。最后，我们表明，对于在线箱覆盖问题，要实现任何优于 15/16 的竞争比率，需要线性数量的建议位。