Lower bounds for online bin covering-type problems

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≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 2$$\end{document} dimensions, for which our new lower bound is d+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+1$$\end{document}, improving over the previously known lower bound of d+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+\frac{1}{2}$$\end{document}, 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.