Optimizing for Fairness in Generalized Kidney Exchange: Theory and Computations

TL;DR

This paper develops polynomial algorithms to optimize fairness in complex kidney exchange models, extending previous matching theories to weighted and path-based structures, supported by computational experiments.

Contribution

It introduces strongly polynomial algorithms for fairness in weighted kidney exchange structures, generalizing beyond maximum cardinality matchings.

Findings

Algorithms guarantee fairness properties in weighted settings

NP-hardness of coverage with cycles and paths of length ≥3

Computational results show benefits of fairness in real and synthetic data

Abstract

The seminal work of Roth, S\"onmez, & \"Unver 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.