We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ratio of 1+ϵ\documentclass[12pt]{minimal}
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\begin{document}$$1 + \epsilon $$\end{document} for the number of bins, a relatively simple argument proves a lower bound of Ω(1/ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ({1}/{\epsilon })$$\end{document} for the migration factor. We establish a nearly matching upper bound of O(1/ϵ4log1/ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$O({1}/{\epsilon }^4 \log {1}/{\epsilon })$$\end{document} using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items nand in 1/ϵ\documentclass[12pt]{minimal}
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\begin{document}$${1}/{\epsilon }$$\end{document}. The previous best trade-off was for an asymptotic competitive ratio of 5/4\documentclass[12pt]{minimal}
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\begin{document}$${5}/{4}$$\end{document} for the bins (rather than 1+ϵ\documentclass[12pt]{minimal}
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\begin{document}$$1+\epsilon $$\end{document}) and needed an amortized number of O(logn)\documentclass[12pt]{minimal}
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\begin{document}$$O(\log n)$$\end{document} repackings (while in our scheme the number of repackings is independent of n and non-amortized).
