We study fair division of goods under the broad class of generalized
assignment constraints. In this constraint framework, the sizes and values of
the goods are agent-specific, and one needs to allocate the goods among the
agents fairly while further ensuring that each agent receives a bundle of total
size at most the corresponding budget of the agent. Since, in such a constraint
setting, it may not always be feasible to partition all the goods among the
agents, we conform -- as in recent works -- to the construct of charity to
designate the set of unassigned goods. For this allocation framework, we obtain
existential and computational guarantees for envy-free (appropriately defined)
allocation of divisible and indivisible goods, respectively, among agents with
individual, additive valuations for the goods.
We deem allocations to be fair by evaluating envy only with respect to
feasible subsets. In particular, an allocation is said to be feasibly envy-free
(FEF) iff each agent prefers its bundle over every (budget) feasible subset
within any other agent's bundle (and within the charity). The current work
establishes that, for divisible goods, FEF allocations are guaranteed to exist
and can be computed efficiently under generalized assignment constraints.
In the context of indivisible goods, FEF allocations do not necessarily
exist, and hence, we consider the fairness notion of feasible envy-freeness up
to any good (FEFx). We show that, under generalized assignment constraints, an
FEFx allocation of indivisible goods always exists. In fact, our FEFx result
resolves open problems posed in prior works. Further, for indivisible goods and
under generalized assignment constraints, we provide a pseudo-polynomial time
algorithm for computing FEFx allocations, and a fully polynomial-time
approximation scheme (FPTAS) for computing approximate FEFx allocations.Comment: 29 page