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.

翻译：我们研究广义分配约束这一广泛类别下物品的公平分配问题。在此约束框架中，物品的大小和价值均具有主体特异性，需要在公平分配物品的同时，确保每个主体获得的物品组合总大小不超过其对应的预算上限。由于在此类约束设定下，将所有物品全部分配给主体可能不可行，我们遵循近期研究的做法，引入“慈善”机制来指定未分配物品的集合。针对该分配框架，我们分别针对可分割与不可分割物品，在主体具有个体化加性估值的情况下，获得了无嫉妒性（适当定义）分配的存在性保证与计算保证。我们通过仅评估对可行子集的嫉妒程度来判定分配的公平性。具体而言，当且仅当每个主体都更偏好自己的组合而非其他主体组合（以及慈善组合）中的任意（预算）可行子集时，该分配被称为可行无嫉妒分配（FEF）。本研究证明，对于可分割物品，在广义分配约束下，FEF分配必然存在且可高效计算。对于不可分割物品，FEF分配不一定存在，因此我们考虑“任意物品下可行无嫉妒性”（FEFx）这一公平性概念。我们证明，在广义分配约束下，不可分割物品的FEFx分配始终存在。事实上，我们的FEFx结果解决了先前研究中提出的开放问题。此外，针对不可分割物品及广义分配约束，我们提出了计算FEFx分配的伪多项式时间算法，以及计算近似FEFx分配的完全多项式时间近似方案（FPTAS）。