Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into the minimum number of bins such that the sum of the item sizes in each bin does not exceed the capacity. We define a new variant, k-times bin-packing (kBP), in which the goal is to pack the items so that each item appears exactly k times in k different bins. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. The fair electricity division problem motivates the study of kBP. The goal is to allocate the available supply among households using some fairness criteria, such as the egalitarian principle. We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve kBP and apply them to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time. We study another variant of the egalitarian allocation problem, in which the goal is to maximize the minimum number of watts allocated to a household. For this variant, we prove an impossibility result: there does not exist such a k that depends only on the number of agents. This impossibility result motivates us to develop four different heuristic algorithms to solve the egalitarian allocation of watts problem. We evaluate the heuristics by summing the minimum watts allocated to any household in each hour, yielding a fairness metric that reflects the lowest watt allocation across all hours. A higher total minimum of watts indicates a more equitable distribution. Thus, we establish new benchmarks for fair allocation of watts.

翻译：给定不同大小的物品和固定的箱容量，装箱问题旨在将这些物品装入最少数量的箱子，使得每个箱子中物品大小之和不超过容量。我们定义了一个新变体，即 $k$ 次装箱问题（kBP），其目标是将物品打包，使得每个物品恰好出现在 $k$ 个不同的箱子中各一次。我们将现有的装箱近似算法推广到求解 kBP，并分析了它们的性能比。公平电力分配问题推动了 kBP 的研究，旨在根据某些公平性准则（如平等主义原则）在住户之间分配可用供应量。我们证明，每个电力分配问题都可以通过某个有限 $k$ 的 $k$ 次装箱求解，该 $k$ 仅取决于住户数量。我们实现了首次适应和首次适应递减装箱算法的推广版本，以求解 kBP，并将其应用于实际电力需求数据。我们表明，在连接时间的平等分配方面，我们的推广优于现有的该问题启发式解法。我们研究了平等分配问题的另一个变体，其目标是最大化分配给单个住户的最小瓦特数。对于该变体，我们证明了一个不可能性结果：不存在仅依赖于代理数量的 $k$。这一不可能性结果促使我们开发了四种不同的启发式算法来解决瓦特数的平等分配问题。我们通过求每小时分配给任何住户的最小瓦特数之和来评估这些启发式算法，从而得到一个反映所有小时中最低瓦特分配的公平性度量。总最小瓦特数越高表示分配越公平。因此，我们为瓦特数的公平分配建立了新的基准。