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：本文研究时序公平分配问题，其中智能体在多轮次中接收物品，并要求满足累积公平性。我们探讨时序单物品无嫉妒性（TEF1）、时序任意物品无嫉妒性（TEFX）及其近似形式$α$-TEFX，以及时序最大最小份额（TMMS）。受标准设定中已知不可能性结果的启发，我们在多种受限设定中考察该模型，并通过引入调度机制对其进行扩展。我们的主要贡献划定了可能性与不可能性之间的边界。首先，对于无调度的时序公平分配，我们证明虽然常数因子$α$-TEFX在一般情况下不可实现，但在广义二元估值且具有相同天数双智能体场景下可达成$1/2$近似。其次，对于带调度的时序公平分配，我们证明当调度缓冲区规模不小于$n/2$时，可在相同天数条件下实现TEF1。然而，即使引入调度机制或限制定义域，TEFX与TMMS在多数情况下仍不可实现。这些结果揭示了严格时序公平性的内在困难，并量化了达成近似保证所需的权衡关系。