  Rent division is the well-studied problem of fairly assigning rooms and
dividing rent among a set of roommates within a single apartment. A shortcoming
of existing solutions is that renters are assumed to be considering apartments
in isolation, whereas in reality, renters can choose among multiple apartments.
In this paper, we generalize the rent division problem to the multi-apartment
setting, where the goal is to both fairly choose an apartment among a set of
alternatives and fairly assign rooms and rents within the chosen apartment. Our
main contribution is a generalization of envy-freeness called rearrangeable
envy-freeness. We show that a solution satisfying rearrangeable envy-freeness
is guaranteed to exist and that it is possible to optimize over all
rearrangeable envy-free solutions in polynomial time. We also define an even
stronger fairness notion called universal envy-freeness and study its existence
when values are drawn randomly.
