We study fair division of indivisible items under a variable input setting, where the set of agents or items may change over time. Starting from an arbitrary allocation, the goal is to restore envy-freeness up to one item (EF1) through item transfers while causing as little disruption as possible. We formalize this via `valid transfers' and introduce the EF1-Restoration problem. We give efficient algorithms for EF1-Restoration when agents have identical monotone valuations and the items are either all goods or all chores. In contrast, even for identical additive valuations, we prove that optimizing the number of valid transfers is NP-hard. For the stronger notion of EFX, we show that deciding whether EFX-Restoration admits any positive solution is weakly NP-hard for identical additive valuations. We also show that, unlike the pure goods and pure chores cases, EF1-Restoration may be impossible for mixed manna. For additive binary valuations, we prove that deciding whether EF1-Restoration is possible is NP-hard, and so is finding the minimum number of valid transfers when restoration is possible. We complement these hardness results with a polynomial-time algorithm for the subclass of additive binary valuations defined using multigraphs, introduced by Christodoulou et al. (EC 2023), when allocations are required to be orientations. Finally, for monotone binary valuations, we prove that deciding whether EF1-Restoration is possible is PSPACE-complete. Together, our results give a broad complexity landscape for restoring EF1 under variable inputs across several natural valuation classes.

翻译：我们研究了可变输入设置下不可分割物品的公平分配问题，其中代理或物品集合可能随时间变化。从任意初始分配出发，目标是通过物品转移恢复至多一个物品的无嫉妒性（EF1），同时尽可能减少干扰。我们通过"有效转移"形式化这一问题，并提出了EF1恢复问题。当代理具有相同的单调估值且物品均为商品或均为杂务时，我们给出了EF1恢复的高效算法。相反，即使对于相同的可加估值，我们证明了优化有效转移数量是NP难的。对于更强的EFX概念，我们证明了判断EFX恢复是否存在任何正解在相同可加估值下是弱NP难的。我们还表明，与纯商品和纯杂务情形不同，混合物品情形下EF1恢复可能不可行。对于可加二元估值，我们证明了判断EF1恢复是否可行是NP难的，且当恢复可行时寻找最少有效转移数量也是NP难的。我们通过克里斯托杜卢等人（EC 2023）引入的基于多重图的子类（要求分配为定向分配）的多项式时间算法，补充了这些困难性结果。最后，对于单调二元估值，我们证明了判断EF1恢复是否可行是PSPACE完全的。综上，我们的结果在多种自然估值类别下揭示了可变输入中恢复EF1的广泛复杂性景观。