We consider the Bin Packing problem with a partition matroid constraint. The input is a set of items of sizes in $(0,1]$, and a partition matroid over the items. The goal is to pack all items in a minimum number of unit-size bins, such that each bin forms an independent set in the matroid. The problem is a generalization of both Group Bin Packing and Bin Packing with Cardinality Constraints. Bin Packing with Partition Matroid naturally arises in resource allocation to ensure fault tolerance and security, as well as in harvesting computing capacity. Our main result is a polynomial-time algorithm that packs the items in $OPT + o(OPT)$ bins, where OPT is the minimum number of bins required for packing the given instance. This matches the best known result for the classic Bin Packing problem up to the function hidden by o(OPT). As special cases, our result improves upon the existing APTAS for Group Bin Packing and generalizes the AFTPAS for Bin Packing with Cardinality Constraints. Our approach is based on rounding a solution for a configuration-LP formulation of the problem. The rounding takes a novel point of view of prototypes in which items are interpreted as placeholders for other items and applies fractional grouping to modify a fractional solution (prototype) into one having nice integrality properties.
翻译:我们用分割型类固醇来考虑本包装问题。 输入是一组大小的物品, 以$( 0. 1) 美元计算, 并且用一个分隔型的类固醇来计算。 目标是将所有物品包装在最小数量的单位大小的垃圾桶中, 这样每个垃圾桶就组成一个独立的机器人。 问题在于将“ 集体包装” 和“ 附带红心限制” 两者都普遍化。 与分割型的包装一起的纸质包装自然会在资源分配中产生, 以确保对错误的容忍度和安全性, 以及在回收计算能力中产生。 我们的主要结果就是将物品包装在$OPT + o( OPT) $( OPT) 的组合时间算法, 将所有物品包装在最小数量的垃圾箱中包装, 将每个垃圾桶的最小数量包装成一个最小的垃圾箱。 这是典型的宾包装问题的最大已知结果 。 作为特殊情况, 我们的结果是改进了现有的“ APTAS” 用于 Bin 包装和“ 本包装限制” 的AFTASAS 。 我们的方法是基于将一个解决方案的解决方案用于将一个配置- Lproal 版本的特性转换成一个配置式的特性的特性, 。