In many applications such as rationing medical care and supplies, university admissions, and the assignment of public housing, the decision of who receives an allocation can be justified by various normative criteria. Such settings have motivated the following priority-respecting allocation problem: several categories, each with a quota of interchangeable items, wish to allocate the items among a set of agents. Each category has a list of eligible agents and a priority ordering over these agents; agents may be eligible in multiple categories. The goal is to select a valid allocation: one that respects quotas, eligibility, and priorities and ensures Pareto efficiency. We provide an algorithmic characterization of all valid allocations, exhibiting a bijection between sets of agents who can be allocated and maximum-weight matchings under carefully chosen rank-based weights. While prior work provides a polynomial-time algorithm to locate a valid allocation, our characterization admits a simpler algorithm that enables two wide-reaching extensions: 1. Selecting valid allocations that satisfy additional criteria: Via three examples -- inclusion/exclusion of some chosen agent; agent-side Pareto efficiency vs. welfare maximization; and fairness from the perspective of allocated vs. unallocated agents -- we show that finding priority-respecting allocations subject to some secondary constraint straddles a complexity knife-edge; in each example, one problem variant can be solved efficiently, while its variant is NP-hard. 2. Efficiency-envy tradeoffs in dynamic allocation: In settings where allocations must be made to T agents arriving sequentially via some stochastic process, we show that while insisting on zero priority violations leads to an Omega(T) loss in efficiency, one can design allocation policies ensuring that the sum of the efficiency loss and priority violations in hindsight is O(1).

翻译：在医疗资源配给、大学招生及公共住房分配等众多应用场景中，分配决策往往需要依据多种规范性标准进行合理性论证。此类场景催生了以下尊重优先级的分配问题：设有若干类别，每类拥有可互换物品的配额，需将这些物品分配给一组代理人。每个类别持有合格代理人名单及其优先级排序；代理人可能同时符合多个类别的资格。目标在于选择有效分配方案——即满足配额限制、资格条件与优先级要求，并确保帕累托效率的分配。我们提出所有有效分配的算法化特征刻画，揭示了可被分配的代理人集合与基于精心设计的分级权重下的最大权匹配之间存在双射关系。虽然既有文献已提供多项式时间算法用于寻找有效分配，但我们的特征刻画催生了一种更简洁的算法，该算法支持两大重要扩展：1. 满足附加条件的有效分配选择：通过三个示例（指定代理人的包含/排除、代理人侧帕累托效率与福利最大化的权衡、从已分配与未分配代理人视角的公平性）证明，在次级约束下寻找尊重优先级的分配方案正处于计算复杂度的刀锋边缘——每个示例中，问题的一个变体可高效求解，而其对应变体则为NP难问题。2. 动态分配中的效率-嫉妒权衡：在需通过随机过程依次向T个到达代理人进行分配的动态场景中，我们证明，虽然坚持零优先级违反将导致Omega(T)的效率损失，但可设计分配策略使得事后效率损失与优先级违反之和为O(1)。