AI 中文总结
本文研究不可分割混合甘露的时间公平分配问题，确定了多个易处理设置，给出了对应规则或算法，同时证明了部分相关问题的NP难性。
AI 中文摘要
我们研究不可分割混合甘露的时间公平分配问题。物品随时间到达且必须不可撤销地分配；一件物品对部分智能体而言是益品，对另一些智能体而言是负担，对其余智能体而言是中性物品。我们要求每一轮后的累计分配满足无嫉妒至多一件物品（TEF1）。尽管判定TEF1分配是否存在即使对于益品也是NP难问题，但我们确定了几个易处理设置。首先，对于至多k种物品类型，在线循环规则在每一件物品到达后保证EF⌈k/2⌉，因此每一个至多两种类型的实例都允许在线TEF1分配；此外，当智能体数量和类型数量固定时，TEF1的存在性可在多项式时间内判定。其次，在智能体特定缩放后的一致性条件下，若缩放因子在物品到达前已知，在线规则会在每一件物品到达后生成满足EF1且帕累托最优的分配。第三，对于具有共同排序的两部分到达序列，我们给出的规则在每一件物品到达后满足EF1。第四，当智能体数量固定且价值为有界整数时，我们给出判定TEF1存在性的精确伪多项式算法。最后，对于益品，每一个TEF1分配在每一轮后为每个智能体提供至少其最大最小份额的1/n；该因子即使对于相同价值和两轮也是紧的。判定精确时间最大最小份额分配是否存在对于益品和负担都是NP难问题，即使在价值相同、两个智能体和两轮的情况下也是如此。
英文摘要
We study temporal fair division of indivisible mixed manna. Items arrive over time and must be allocated irrevocably; an item may be a good for some agents, a chore for others, and neutral for the rest. We require the cumulative allocation after every round to be envy-free up to one item (TEF1). Although deciding whether a TEF1 allocation exists is NP-hard even for goods, we identify several tractable settings. First, with at most $k$ item types, an online cyclic rule guarantees EF$\lceil k/2\rceil$ after every item arrival. Thus, every instance with at most two types admits an online TEF1 allocation; moreover, when the numbers of agents and types are fixed, TEF1 existence can be decided in polynomial time. Second, under agreement after agent-specific scaling, provided that the scaling factors are known before arrivals begin, an online rule produces an allocation that is EF1 and Pareto optimal after every item arrival. Third, for a two-part arrival sequence with common rankings, we give a rule that is EF1 after every item arrival. Fourth, when the number of agents is fixed and values are bounded integers, we give an exact pseudo-polynomial algorithm for deciding TEF1 existence. Finally, for goods, every TEF1 allocation gives each agent at least $1/n$ of her maximin share after every round; this factor is tight even for identical valuations and two rounds. Deciding whether an exact temporal maximin-share allocation exists is NP-hard for both goods and chores, even with identical valuations, two agents, and two rounds.
