We consider a known variant of bin packing called {\it cardinality constrained bin packing}, also called {\it bin packing with cardinality constraints} (BPCC). In this problem, there is a parameter k\geq 2, and items of rational sizes in [0,1] are to be packed into bins, such that no bin has more than k items or total size larger than 1. The goal is to minimize the number of bins. A recently introduced concept, called the price of clustering, deals with inputs that are presented in a way that they are split into clusters. Thus, an item has two attributes which are its size and its cluster. The goal is to measure the relation between an optimal solution that cannot combine items of different clusters into bins, and an optimal solution that can combine items of different clusters arbitrarily. Usually the number of clusters may be large, while clusters are relatively small, though not trivially small. Such problems are related to greedy bin packing algorithms, and to batched bin packing, which is similar to the price of clustering, but there is a constant number of large clusters. We analyze the price of clustering for BPCC, including the parametric case with bounded item sizes. We discuss several greedy algorithms for this problem that were not studied in the past, and comment on batched bin packing.

中文翻译：

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基数约束装箱的几种分析方法

我们考虑一种已知的装箱变体，称为 {\it 基数约束装箱}，也称为 {\it 带基数约束的装箱}（BPCC）。在这个问题中，有一个参数 k\geq 2，并且 [0,1] 中的有理大小的项目将被打包到 bin 中，使得没有 bin 的项目超过 k 或总大小大于 1。目标是以最小化 bin 的数量。最近引入的一个概念称为聚类的价格，它处理以将它们拆分为聚类的方式呈现的输入。因此，一个项目有两个属性，即它的大小和它的集群。目标是衡量一个不能将不同簇的项组合成 bin 的最优解与一个可以任意组合不同簇的项的最优解之间的关系。通常集群的数量可能很大，虽然集群相对较小，但并非微不足道。此类问题与贪心装箱算法和批量装箱有关，这与聚类的代价相似，但大簇的数量是恒定的。我们分析了 BPCC 聚类的代价，包括项目大小有界的参数情况。我们讨论了过去没有研究过的针对这个问题的几种贪心算法，并对批量装箱进行了评论。