4 research outputs found
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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
Anomalous behavior in bin packing algorithms
AbstractIn the classical bin packing problem one is given a list of items and asked to pack them into the fewest possible unit-sized bins. Given two lists, L1 and L2, where L2 is derived from L1 by deleting some elements of L1 and/or reducing the size of some elements of L1, one might hope that an approximation algorithm would use no more bins to pack L2 than it uses to pack L1. Johnson and Graham have given examples showing that First-Fit and First-Fit Decreasing can actually use more bins to pack L2 than L1. Graham has also studied this type of behavior among multiprocessor scheduling algorithms. In the present paper we extend this study of anomalous behavior to a broad class of approximation algorithms for bin packing. To do this we introduce a technique which allows one to characterize the monotonic/anomalous behavior of any algorithm in a large, natural class. We then derive upper and lower bounds on the anomalous behavior of the algorithms which are anomalous and provide conditions under which a normally nonmonotonic algorithm becomes monotonic
Recommended from our members
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