In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.

翻译：在不可分割商品的分配中，一个重要的公平性概念是至多一个物品的无嫉妒性（EF1）。我们通过研究从一个EF1分配能否通过一系列交换到达另一个EF1分配的问题，且每个中间分配也必须满足EF1，来开创公平分配中可行性问题的研究。我们表明，即使在两个代理人的情况下，两个EF1分配也可能无法相互到达，并且判断其可行性在一般情况下是PSPACE完全的。另一方面，我们证明，对于具有相同或二元效用的两个代理人，以及任意数量具有相同二元效用的代理人，可行性是有保障的。我们还研究了判断是否存在交换次数最优的EF1交换序列的复杂性。