Inspired by real-world applications such as the assignment of pupils to schools or the allocation of social housing, the one-sided matching problem studies how a set of agents can be assigned to a set of objects when the agents have preferences over the objects, but not vice versa. For fairness reasons, most mechanisms use randomness, and therefore result in a probabilistic assignment. We study the problem of decomposing these probabilistic assignments into a weighted sum of ex-post (Pareto-)efficient matchings, while maximizing the worst-case number of assigned agents. This decomposition preserves all the assignments’ desirable properties, most notably strategy-proofness. Next to discussing the complexity of the problem, we obtain tight lower and upper bounds on the optimal worst-case number of assigned agents. Moreover, we propose two alternative column generation frameworks for the introduced problem, which prove to be capable of finding decompositions with the theoretically best possible worst-case number of assigned agents, both for randomly generated data, and for real-world school choice data from the Belgian cities Antwerp and Ghent. Lastly, the proposed column generation frameworks are inherently flexible, and can therefore also be applied to settings where other ex-post criteria are desirable, or to find decompositions that satisfy other worst-case measures.

受现实世界应用的启发，例如分配学生到学校或分配社会住房，单边匹配问题研究了当代理对对象有偏好时如何将一组代理分配给一组对象，但反之亦然。出于公平原因，大多数机制都使用随机性，因此会导致概率分配。我们研究了将这些概率分配分解为事后（帕累托）有效匹配的加权和的问题，同时最大化最坏情况下分配代理的数量。这种分解保留了所有分配的理想属性，最显着的是策略证明。接下来讨论问题的复杂性，我们获得了分配代理的最佳最坏情况数量的严格下限和上限。而且，我们为引入的问题提出了两个替代的列生成框架，证明能够找到具有理论上最佳可能最坏情况分配代理数的分解，既适用于随机生成的数据，也适用于来自比利时的真实学校选择数据城市安特卫普和根特。最后，所提出的列生成框架本质上是灵活的，因此也可以应用于需要其他事后标准的设置，或者找到满足其他最坏情况措施的分解。