In this work, we study the square min-sum bin packing problem (SMSBPP), where a list of square items has to be packed into indexed square bins of dimensions $1 \times 1$ with no overlap between the areas of the items. The bins are indexed and the cost of packing each item is equal to the index of the bin in which it is placed in. The objective is to minimize the total cost of packing all items, which is equivalent to minimizing the average cost of items. The problem has applications in minimizing the average time of logistic operations such as cutting stock and delivery of products. We prove that classic algorithms for two-dimensional bin packing that order items in non-increasing order of size, such as Next Fit Decreasing Height or Any Fit Decreasing Height heuristics, can have an arbitrarily bad performance for SMSBPP. We, then, present a $\frac{53}{22}$-approximation and a PTAS for the problem.

翻译：本文研究平方最小和装箱问题（SMSBPP），其中一系列正方形物品需装入尺寸为$1 \times 1$的索引化正方形箱子中，且物品面积不得重叠。箱子按索引编号，每个物品的装箱成本等于其所在箱子的索引编号。目标是最小化所有物品的总成本，等价于最小化物品的平均成本。该问题在物流操作（如切割库存和产品配送）中具有平均时间最小化的应用场景。我们证明，针对二维装箱问题的经典算法（如按尺寸非递增顺序排序物品的Next Fit Decreasing Height或Any Fit Decreasing Height启发式算法）在SMSBPP中可能产生任意差的性能表现。随后，我们提出该问题的$\frac{53}{22}$-近似算法和多项式时间近似方案（PTAS）。