Fair Allocation of Items in Multiple Regions

We initiate the study of fair allocation with the set of divisible or indivisible items distributed in multiple regions. The key requirement is that each agent can only obtain items from one region. In this work, we consider two kinds of fairness concepts: envy-based notions including envy-freeness (EF) and envy-freeness up to one/any item (EF1/EFX), and share-based notions including proportionality (PROP) and proportionality up to one/any item (PROP1/PROPX). On the negative side, we show NP-hardness and inapproximability results about the aforementioned fairness notions. On the positive side, we propose several algorithms to compute the partial allocations that satisfy envy-based notions and allocations that approximate the above fairness notions.