  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.
