带优先级与配额进行分配：算法、复杂性与动态
Allocating with Priorities and Quotas: Algorithms, Complexity, and Dynamics
AI总结：
本文研究带优先级和配额的资源分配问题，通过基于排名的最大权重匹配刻画所有有效分配，并据此给出更简单算法，进而揭示附加约束下的复杂性分界，以及在动态顺序分配中实现效率损失与优先级违反之和为常数的最优权衡。
摘要：
在医疗资源与物资配给、大学录取、公共住房分配等许多应用中，谁应获得分配的决定可以通过各种规范性标准来加以论证。这类场景引出了如下尊重优先级的分配问题：存在若干类别，每个类别都拥有一定配额的可互换物品，并希望将这些物品分配给一组智能体。每个类别都有一份合格智能体名单以及对这些智能体的优先级排序；智能体可能在多个类别中都具有资格。目标是选择一个有效分配：即尊重配额、资格和优先级，并确保帕累托效率的分配。我们给出了所有有效分配的算法刻画，展示了可被分配的智能体集合与在精心选择的基于排名的权重下的最大权重匹配之间存在一一对应关系。尽管已有工作提供了一个多项式时间算法来寻找有效分配，但我们的刻画给出了一种更简单的算法，并支持两个影响广泛的扩展：1. 选择满足附加标准的有效分配：通过三个例子——包含/排除某个被选中的智能体；智能体侧的帕累托效率与福利最大化；以及从被分配智能体与未被分配智能体视角看的公平性——我们表明，在某个次要约束下寻找尊重优先级的分配横跨一个复杂性刀刃；在每个例子中，一个问题变体可以高效求解，而它的另一个变体则是NP困难的。2. 动态分配中的效率-嫉妒权衡：在必须通过某种随机过程向顺序到达的T个智能体进行分配的设置中，我们表明，虽然坚持零优先级违反会导致效率损失为Omega(T)，但可以设计分配策略，确保事后效率损失与优先级违反之和为O(1)。
英文摘要：
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).
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