We study temporal fair division, where agents receive goods over multiple rounds and cumulative fairness is required. We investigate Temporal Envy-Freeness Up to One Good (TEF1) and Up to Any Good (TEFX), its approximation $α$-TEFX, and Temporal Maximin Share (TMMS). Motivated by known impossibilities in standard settings, we consider the model in various restricted settings and extend it by introducing scheduling. Our main contributions draw the boundary between possibility and impossibility. First, regarding temporal fair division without scheduling, we prove that while constant-factor $α$-TEFX is impossible in general, a $1/2$-approximation is achievable for generalized binary valuations and identical days with two agents. Second, regarding temporal fair division with scheduling, we demonstrate that a scheduling buffer of size at least $n/2$ enables TEF1 for identical days. However, we establish that TEFX and TMMS remain largely impossible even with scheduling or restricted domains. These results highlight the inherent difficulty of strict temporal fairness and quantify the trade-offs required to achieve approximation guarantees.

翻译：我们研究时间公平分配问题，其中智能体在多个轮次中获得物品且需要满足累积公平性。我们探讨了时间意义下的至多一个物品无嫉妒性（TEF1）、任意物品无嫉妒性（TEFX）及其近似α-TEFX，以及时间意义下的最大最小份额（TMMS）。受标准场景中已知的不可能性结果启发，我们考虑多种受限场景下的模型，并通过引入调度机制进行扩展。我们的主要贡献在于划定了可能性与不可能性的边界。首先，在无调度的时间公平分配中，我们证明尽管常数因子α-TEFX在一般情况下不可实现，但对于广义二元估值且天数相同的两智能体场景，1/2近似是可达的。其次，在具有调度的时间公平分配中，我们证明规模至少为n/2的调度缓冲区可在天数相同时实现TEF1。然而，我们证实即使采用调度或受限域，TEFX和TMMS在多数情况下仍不可实现。这些结果揭示了严格时间公平性的固有难度，并量化了实现近似保证所需付出的权衡代价。