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On the monotonicity of certain bin packing algorithms
This paper examines the monotonicity of the approximation bin packing algorithms Worst-Fit (WF), Worst-Fit Decreasing (WFD), Best-Fit (BF), Best-Fit Decreasing (BFD), and Next-Fit-k (NF-k). Let X and Y be two sets of items such that the set X can be derived from the set Y by possibly deleting some members of Y or by reducing the size of some members of Y. If an algorithm never uses more bins to pack X than it uses to pack Y we say that algorithm is monotonic. It is shown that NF and NF-2 are monotonic. It was already known that First-Fit and First-Fit Decreasing were non-monotonic and we give examples which show BF, BFD, WF, and WFD also suffer from this anomaly. One may consider First-Fit as the limiting case of NF-k. We notice that NF-1 is monotonic while First-Fit is not, suggesting there exists some critical k for which NF-k' is monotonic, for k' k. We establish that this is indeed the case and determine that critical k. An upper bound on the non-monotonicity of selected algorithms is also provided
Online Bin Stretching with Three Bins
Online Bin Stretching is a semi-online variant of bin packing in which the
algorithm has to use the same number of bins as an optimal packing, but is
allowed to slightly overpack the bins. The goal is to minimize the amount of
overpacking, i.e., the maximum size packed into any bin.
We give an algorithm for Online Bin Stretching with a stretching factor of
for three bins. Additionally, we present a lower bound of for Online Bin Stretching on three bins and a lower bound of
for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference
paper. Conference paper split into two journal submissions; see
arXiv:1601.0811
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An average-case analysis of bin packing with uniformly distributed item sizes
We analyze the one-dimensional bin-packing problem under the assumption that bins have unit capacity, and that items to be packed are drawn from a uniform distribution on [0,1]. Building on some recent work by Frederickson, we give an algorithm which uses n/2+0(n^½) bins on the average to pack n items. (Knodel has achieved a similar result.) The analysis involves the use of a certain 1-dimensional random walk. We then show that even an optimum packing under this distribution uses n/2+0(n^1/2) bins on the average, so our algorithm is asymptotically optimal, up to constant factors on the amount of wasted space. Finally, following Frederickson, we show that two well-known greedy bin-packing algorithms use no more bins than our algorithm; thus their behavior is also in asymptotically optimal in this sense
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