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：我们研究时序公平分配问题：智能体在多轮次中获取物品，需满足累积公平性要求。本文系统考察了时序嫉妒无嫉妒性（TEF1）与任意物品嫉妒无嫉妒性（TEFX）及其近似形式α-TEFX，同时研究了时序最大最小份额（TMMS）。鉴于标准设置中已知的不可能性结论，我们在多种受限设置下考察该模型，并通过引入调度机制对其进行扩展。主要贡献在于界定可能性与不可能性的边界：首先，针对无调度时序公平分配，我们证明常数因子α-TEFX存在一般性不可能，但针对广义二分估值与双智能体相同时段场景，可达到1/2近似比；其次，针对含调度时序公平分配，我们证明至少需要n/2规模的调度缓冲池才能实现相同时段下的TEF1，但即便引入调度或限制域，TEFX与TMMS仍普遍不可实现。这些结果揭示了严格时序公平的固有难度，并量化了实现近似保证所需的权衡代价。