We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannotexceed 1.

In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors into bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to an optimal clustered solution, under certain constraints on the coloring provided with the input. We show close bounds on the worst-case ratio between these two costs, called the price of clustering, improving and simplifying previous results. Specifically, we show that the price of clustering does not exceed 1.93667, improving over the previous upper bound of 1.951, and that it is at least 1.93558, improving over the previous lower bound of 1.93344.

In online bin packing with delays, items are presented over time. Items may wait to be packed, and an algorithm can create a new bin at any time, packing a subset of already existing unpacked items into it, under the condition that the bin is valid. A created bin cannot be used again in the future, and all items have to be packed into bins eventually. The objective is to minimize the number of used bins plus the sum of waiting costs of all items, called delays. We build on previous work and modify a simple phase-based algorithm. We combine the modification with a careful analysis to improve the previously known competitive ratio from 3.951 to below 3.1551.

我们继续研究最近引入的两个装箱类型问题，称为带聚类的装箱和有延迟的在线装箱。一个 bin 打包输入由大小不大于 1 的项目组成，目标是将它们分区或打包到 bin 中，其中每个有效 bin 的项目总大小不能超过 1。

在使用聚类的装箱中，物品也有与其相关联的颜色。全局最优解决方案可以将不同颜色的项目组合到 bin 中，而聚类解决方案只能打包单色 bin。目标是在对输入提供的着色的某些限制下，将全局最优解与最优聚类解进行比较。我们展示了这两个成本之间最坏情况比率的接近边界，称为聚类、改进和简化先前结果的价格。具体来说，我们表明聚类的价格不超过 1.93667，比之前的上限 1.951 有所提高，并且至少为 1.93558，比之前的下限 1.93344 有所提高。

在具有延迟的在线装箱中，物品会随着时间的推移而呈现。物品可能会等待被打包，并且算法可以随时创建一个新的 bin，在 bin 有效的情况下，将已经存在的未包装物品的子集打包到其中。创建的 bin 以后不能再次使用，最终所有物品都必须装入 bin 中。目标是最小化使用的垃圾箱数量加上所有物品的等待成本之和，称为延迟。我们以之前的工作为基础，修改了一个简单的基于相位的算法。我们将修改与仔细分析相结合，将之前已知的竞争比率从 3.951 提高到 3.1551 以下。