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：我们考虑了带有划分拟阵约束的装箱问题。输入为一组大小在$(0,1]$内的物品，以及物品上的一个划分拟阵。目标是将所有物品装入数量最少的单位容量箱子中，使得每个箱子在拟阵中构成一个独立集。该问题是群组装箱问题和容量约束装箱问题的推广。带划分拟阵的装箱问题自然产生于资源分配中的容错与安全保障，以及计算能力收集等场景。我们的主要成果是一个多项式时间算法，该算法可将物品装入$OPT + o(OPT)$个箱子中，其中$OPT$是给定实例所需的最小箱子数。这一结果在$o(OPT)$所隐藏的函数项内匹配了经典装箱问题已知的最佳结果。作为特例，我们的结果改进了群组装箱问题的现有APTAS，并推广了容量约束装箱问题的AFTPAS。该方法基于对问题的配置线性规划解的舍入处理。舍入采用了一种新颖的原型视角，其中物品被解释为其他物品的占位符，并通过分数分组将分数解（原型）修改为具有良好整数性质的解。