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.

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.

We revisit the classic online bin packing problem. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called "bins" of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class of items to be packed in multiple kinds of bins, and not only into their own bins. We combine this with the approach of packing of sufficiently big items according to their exact sizes. Finally, we simplify the analysis of such algorithms, allowing the analysis to be based on the most standard weight functions. This simplified analysis allows us to study the algorithm which we defined based on all these ideas. This leads us to the design and analysis of the first algorithm of asymptotic competitive ratio strictly below 1.58, specifically, we break this barrier and provide an algorithm AH (Advanced Harmonic) whose asymptotic competitive ratio does not exceed 1.5783.

Cardinality constrained bin packing or 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 in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes the long standing open problem of finding the value of the best possible overall asymptotic competitive ratio, since an algorithm of an absolute competitive ratio 2 for any fixed value of k is known. 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. Thus, our lower bound inputs do not follow the common practice for online bin packing problems of having a known in advance input consisting of batches for which the algorithm needs to be competitive on every prefix of the input. Last, we show a lower bound strictly larger than 2 on the asymptotic competitive ratio of the online 2-dimensional vector packing problem, and thus provide for the first time a lower bound larger than 2 on the asymptotic competitive ratio for the vector packing problem in any fixed dimension.

We present an online bin packing algorithm with absolute competitive ratio 5/3, which is optimal.

In 1996 Ivkovič and Lloyd [A fundamental restriction on fully dynamic maintenance of bin packing, Inform. Process. Lett., 59 (1996), pp. 229–232] gave the lower bound $\frac{4}{3}$ on the asymptotic worst-case ratio for so-called fully dynamic bin packing algorithms, where the number of repackable items in each step is restricted by a constant. In this paper we improve this result to about $1.3871$. We present our proof for a semionline case of the classical bin packing, but it works for fully dynamic bin packing as well. We prove the lower bound by analyzing and solving a specific optimization problem. The bound can be expressed exactly using the Lambert W function.