We study fair division of divisible goods under generalized assignment constraints. Here, each good has an agent-specific value and size, and every agent has a budget constraint that limits the total size of the goods she can receive. Since it may not always be feasible to assign all goods to the agents while respecting the budget constraints, we use the construct of charity to accommodate the unassigned goods. In this constrained setting with charity, we obtain several new existential and computational results for feasible envy-freeness (FEF); this fairness notion requires that agents are envy-free, considering only budget-feasible subsets. First, we simplify and extend known existential results for FEF allocations. Then, we show that the space of FEF allocations has a non-convex structure. Next, using a fixed-point argument, we establish a novel guarantee that FEF can always be achieved with Pareto-optimality. Furthermore, we give an alternative proof of the fact that one cannot additionally obtain truthfulness in this context: There does not exist a mechanism that is simultaneously truthful, fair, and Pareto-optimal. On the positive side, we show that truthfulness is compatible with each of FEF and Pareto-optimality, individually.

翻译：本文研究广义分配约束下的可分割物品公平分配问题。在此框架中，每个物品对每个智能体具有特定的价值与尺寸，且每个智能体存在预算约束以限制其可接收物品的总尺寸。由于在满足预算约束的前提下将全部物品分配给智能体并非总是可行，我们引入慈善机构这一构造以容纳未分配物品。在此带慈善机构的约束环境中，我们针对可行无嫉妒性（FEF）获得了若干新的存在性与计算性结果；该公平性概念要求智能体仅考虑预算可行子集时保持无嫉妒状态。首先，我们简化并扩展了已知的FEF分配存在性结果。其次，我们证明FEF分配空间具有非凸结构。接着，通过不动点论证，我们建立了FEF总能与帕累托最优性同时实现的新保证。此外，我们提供了该背景下无法同时获得真实性的替代证明：不存在同时满足真实性、公平性与帕累托最优性的机制。从积极角度看，我们证明了真实性可分别与FEF及帕累托最优性兼容。