发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究顺序分配不可分割物品时最小化累积最大嫉妒的问题，证明其强NP完全性并给出动态规划与贪心算法，后者对两个智能体达到3/2近似。
AI 中文摘要
我们研究时间公平分配问题，其中不可分割的物品按顺序到达，并且必须不可撤销地分配。与通常的在线模型不同，我们假设估值和未来到达情况事先已知，并询问不公平性在此过程中如何演变。我们引入了“累积最大嫉妒”：所有轮次中每轮最大成对嫉妒之和。等价地，这是最差嫉妒曲线下的面积，它同时捕捉了嫉妒的幅度和持续时间。对于固定的到达顺序，我们证明了相应的决策问题是强NP完全的，并且即使估值相同，即使估值是二元的，除非P=NP，否则最小化该目标不存在常数因子近似。我们用动态规划补充了这些困难结果，该动态规划对于常数数量的智能体给出伪多项式时间可解性，在进一步受限的设置中给出多项式时间算法，并且在相同整数估值下对固定$n$给出FPTAS。然后我们研究了一种排序变体，其中算法可以选择到达顺序。即使对于两个估值相同的智能体，该变体仍然是NP完全的；然而，一个简单的贪心算法对于$n=2$个智能体实现了$3/2$近似，对于任意数量的智能体实现了$n/(n-1)$近似，并且给出了取决于任何物品最大值的加性保证。
英文摘要
We study temporal fair division with indivisible goods that arrive sequentially and must be allocated irrevocably. In contrast to the usual online model, we assume that valuations and future arrivals are known in advance, and ask how unfairness evolves during the process. We introduce \emph{cumulative maximum envy}: the sum, over all rounds, of the maximum pairwise envy at that round. Equivalently, this is the area under the worst-envy curve, and it captures both the magnitude and the duration of envy. For a fixed arrival order, we show that the corresponding decision problem is strongly NP-complete and that minimizing this objective admits no constant-factor approximation unless P = NP, even under identical valuations and even under binary valuations. We complement these hardness results with a dynamic program that gives pseudopolynomial-time solvability for a constant number of agents, polynomial-time algorithms in further restricted settings, and an FPTAS for fixed $n$ under identical integer valuations. We then study a sequencing variant where the algorithm may choose the arrival order. This variant remains NP-complete even for two agents with identical valuations; however, a simple greedy algorithm achieves a $3/2$-approximation for $n=2$ agents, an $n/(n-1)$-approximation for any number of agents, and an additive guarantee depending on the maximum value of any good.
Comments40 pages. Preliminary version of this paper was presented at the 19th SAGT (Sept. 2026)
