Fair division under cardinality constraints

We consider the problem of fairly allocating indivisible goods, among agents, under cardinality constraints and additive valuations. In this setting, we are given a partition of the entire set of goods-i.e., the goods are categorized-and a limit is specified on the number of goods that can be allocated from each category to any agent. The objective here is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. This problem naturally captures a number of resource-allocation applications, and is a generalization of the well-studied (unconstrained) fair division problem. The two central notions of fairness, in the context of fair division of indivisible goods, are envy freeness up to one good (EF1) and (approximate) maximin share guarantee (MMS). We show that the existence and algorithmic guarantees established for these solution concepts in the unconstrained setting can essentially be achieved under cardinality constraints. Furthermore, focusing on the case wherein all the agents have the same additive valuation, we establish that EF1 allocations exist even under matroid constraints.