在线公平分配下的资格约束

中文摘要 AI 辅助

我们研究了在资格约束下不可分割物品的公平分配问题：每个物品只能分配给符合条件的代理子集。该模型与调度中的受限分配相吻合，其中代理作为机器，物品作为作业，并涵盖了工作或资源必须在能力不同的各方之间分配的场景。资格约束使得通常的公平性基准不再适用，因为代理不应被与它们永远无法获得的物品进行比较。因此，我们采用了考虑资格的比例性和无嫉妒性版本，其中代理的比例份额将每个物品在其符合条件的代理之间平均分配，而无嫉妒比较则忽略代理不符合条件的物品。我们的主要发现是比例性在杂务和商品之间的鲜明分离。我们关注相同估值：在任意估值下，没有在线算法能保证对杂务实现甚至$o(m)$近似。对于杂务，Prop1在离线情况下可实现，而精确比例性和任意物品比例性则不可实现。对于在线杂务，确定性算法的Prop1最优近似因子为$\Theta(\sqrt m)$，下界甚至对单位大小作业也成立。对于商品，Prop1在离线情况下同样可实现，但在线紧界降至$\Theta(\log m)$。对于基于嫉妒的公平性，情况则不同。在考虑资格的无嫉妒概念下，商品和杂务是等价的，而概念的选择是决定性的。强EF1即使在离线情况下也可能不存在，且无法在线近似。相比之下，弱EF1可以在线维持，而弱EFX则区分了离线和在线情况。这些结果共同对受限分配中的在线比例性和基于嫉妒的公平性进行了系统研究，并展示了在资格约束下哪些公平性保证仍然可实现。

英文摘要

We study fair division of indivisible items under eligibility constraints: each item may be assigned only to a subset of eligible agents. This model coincides with restricted assignment in scheduling, with agents as machines and items as jobs, and captures settings where work or resources must be split among parties with different capabilities. Eligibility makes the usual fairness benchmarks inadequate, as agents should not be measured against items they could never receive. We therefore use eligibility-aware versions of proportionality and envy-freeness, where an agent's proportional share splits each item equally among its eligible agents, and envy comparisons discount items the agents are not eligible for. Our main finding is a sharp chore--goods separation for proportionality. We focus on identical valuations: under arbitrary valuations, no online algorithm can guarantee even an $o(m)$-approximation for chores. For chores, Prop1 is achievable offline, whereas exact proportionality and proportionality up to any item are not. For online chores, the optimal approximation factor for Prop1 for deterministic algorithms is $Θ(\sqrt m)$, with the lower bound holding even for unit-size jobs. For goods, Prop1 is again achievable offline, but the tight online bound drops to $Θ(\log m)$. For envy-based fairness, the picture is different. Goods and chores are equivalent under eligibility-aware envy notions, while the choice of notion is decisive. Strong EF1 may fail to exist even offline and cannot be approximated online. In contrast, weak EF1 can be maintained online, while weak EFX separates offline from online. Together, these results give a systematic study of online proportional and envy-based fairness in restricted assignment and show which fairness guarantees remain achievable under eligibility constraints.

发表机构

• University of Bremen（不来梅大学）
• Max Planck Institute for Informatics（马克斯·普朗克 informatics 研究所）

机构由 AI 辅助整理，请以论文原文为准。