We consider the following generalization of the bin packing problem. We are given a set of items each of which is associated with a rational size in the interval [0,1], and a monotone non-decreasing non-negative cost function f defined over the cardinalities of the subsets of items. A feasible solution is a partition of the set of items into bins subject to the constraint that the total size of items in every bin is at most 1. Unlike bin packing, the goal function is to minimize the total cost of the bins where the cost of a bin is the value of f applied on the cardinality of the subset of items packed into the bin. We present an APTAS for this strongly NP-hard problem. We also provide a complete complexity classification of the problem with respect to the choice of f.

翻译：我们考虑装箱问题的以下推广形式。给定一组物品，每个物品关联一个位于区间[0,1]内的有理数尺寸，以及一个定义在物品子集基数上的单调非递减非负成本函数f。可行解是将物品集合划分到若干箱子中，且满足每个箱子内物品总尺寸不超过1的约束条件。与标准装箱问题不同，目标函数是最小化所有箱子的总成本，其中单个箱子的成本是将函数f应用于该箱所装物品子集的基数所得的值。针对这一强NP难问题，我们提出了一个近似多项式时间近似方案。同时，我们根据函数f的选择给出了该问题的完整计算复杂性分类。