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Best Fit Bin Packing with Random Order Revisited
arXiv - CS - Data Structures and Algorithms Pub Date : 2020-12-01 , DOI: arxiv-2012.00511
Susanne Albers, Arindam Khan, Leon Ladewig

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon's result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

中文翻译:

再造最佳随机包装箱

“最佳拟合”是解决垃圾箱包装问题的一种著名的在线算法,其中必须将一维物品的集合包装到最小数量的单位大小的垃圾箱中。在一项开创性的工作中,Kenyon [SODA 1996]引入了(渐近)随机阶数比作为在线算法的一种替代性能指标。在这里,对手指定了物品,但是到达顺序是随机地统一绘制的。对于最佳拟合的随机顺序比率,Kenyon的结果分别确定了1.08和1.5的下限和上限。尽管这种分析模型在在线算法领域变得越来越流行,但在Kenyon的研究结果之后,Best Fit算法并未取得任何进展。我们研究了最佳拟合的随机顺序比率,并通过建立改进的1.10下限来缩小长期存在的差距。对于所有项目都大于1/3的情况,我们表明随机顺序比率迅速收敛至1.25。如此大的项目的存在决定了一般情况下Best Fit的性能。此外,这种情况与完全在线模型中的经典最大基数匹配问题密切相关。作为副产品,我们证明在这种情况下,Best Fit满足单调性,这与通常情况不同。另外,我们开始研究这个问题的绝对随机序数比。与渐近比率相反,即使可以装入少量箱柜的情况,绝对比率也必须保持不变。我们显示最佳拟合的绝对随机顺序比率至少为1.3。对于所有项目都大于1/3的情况,我们得出21/16和1.2的上限和下限,
更新日期:2020-12-02
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