Optimizing for Fairness in Generalized Kidney Exchange: Theory and Computations

2026-05-19

AI Summary

This work addresses the joint optimization of matching efficiency and individual fairness in weighted generalized kidney exchange, particularly in settings involving complex structures such as chains and 3-cycles. By extending the Edmonds–Gallai structure theorem to weighted matchings with 2-paths, the authors devise the first strongly polynomial-time algorithm that maximizes total weight while ensuring fair allocation of matching probabilities among patients. They further introduce a general boosting framework capable of transforming any structural optimization subroutine into a mechanism that satisfies comparable fairness guarantees. Experimental results on both synthetic and real-world data demonstrate that the proposed approach significantly improves fairness metrics—such as Nash social welfare—while maintaining high matching cardinality, outperforming existing non-fair mechanisms.

Abstract

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.

Problem

• Kidney Exchange
• Fairness
• Matching
• Weighted Graphs
• NP-hard

Innovation

• fairness
• kidney exchange
• strongly polynomial algorithm
• Nash social welfare
• generalized matching