The seminal work of Roth, Sönmez, & Ünver shows that the Edmonds-Gallai structure theorem for non-bipartite matching can be leveraged to yield a randomized algorithm to match patient-donor pairs in kidney exchange with extraordinarily strong properties. This breakthrough led to randomized polynomial-time algorithms to find a maximum-cardinality matching maximizing individual fairness objectives--measured by the probability that nodes are matched--such as Nash social welfare. But the exchanges allowed in practice go beyond cardinality matching, generalizing to weighted variants and allowing structures such as paths and 3-cycles. We show that strongly polynomial algorithms guaranteeing the same fairness properties can be obtained in weighted settings for matching and 2-paths. While even maximum cardinality coverage with cycles and paths of length at least three is NP-hard, we provide a general result showing that any optimization subroutine (for whichever structure is allowed) can be bootstrapped using a polynomial number of calls to yield a mechanism that has analogous fairness properties to those obtained for matching. We complement these theoretical results with computational results, both on well-studied synthetic data-sets and on samples drawn from real data, that demonstrate the striking advantages of adding fairness considerations to more general kidney-exchange mechanisms.

翻译：Roth、Sönmez和Ünver的开创性工作表明，非二分匹配中的Edmonds-Gallai结构定理可用于设计一种随机算法，以在肾脏交换中实现患者-供者对匹配，并具有异常强大的性质。这一突破催生了随机多项式时间算法，用于寻找最大化个体公平性目标（通过节点被匹配的概率衡量）的最大基数匹配，例如纳什社会福利。然而，实际允许的交换超越了基数匹配，推广至加权变体并允许诸如路径和三循环等结构。我们证明，在加权设置下，针对匹配和双路径，能够保证相同公平性质的强多项式算法是可得的。尽管涉及长度至少为三的循环和路径的最大基数覆盖问题是NP难的，但我们提供了一个通用结果：任何优化子例程（针对所允许的结构）均可通过多项式次调用进行引导，从而产生具有与匹配所获公平性质类似的机制。我们通过计算实验（包括在经典合成数据集和实际数据样本上的实验）对这些理论结果进行了补充，展示了将公平性考虑纳入更一般的肾脏交换机制所产生的显著优势。