Abstract
Bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k≥2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem and prove a lower bound of 2 on the overall asymptotic competitive ratio. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Last, we show a lower bound strictly larger than 2 on the asymptotic competitive ratio of the online 2-dimensional vector packing problem, where no such lower bound was known even for fixed high dimensions.
Highlights
Bin packing with cardinality constraints (CCBP, called cardinality constrained bin packing) is a well-known variant of bin packing [18, 19, 17, 9, 10, 11, 15]
Items of indices 1, 2, . . . , n, where item i has a size si ∈
CCBP is a special case of vector packing (VP) [14]
Summary
Bin packing with cardinality constraints (CCBP, called cardinality constrained bin packing) is a well-known variant of bin packing [18, 19, 17, 9, 10, 11, 15]. We conclude this work by establishing a lower bound strictly larger than 2 on the competitive ratio of 2-dimensional VP, and we show here for the first time that the 2-dimensional VP is provably harder for online algorithms than its special case of CCBP. 10:4 Online Bin Packing with Cardinality Constraints Resolved case that the competitive ratio for some specific algorithms for CCBP is larger by 1 with comparison to that of the corresponding algorithms for standard bin packing [18, 16, 20, 10] This is not the case with respect to the results shown in this paper
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