Computing Balanced Solutions for Large International Kidney Exchange Schemes When Cycle Length Is Unbounded
This paper addresses the optimization problem in International Kidney Exchange Programs (IKEPs) under the constraint of unbounded exchange cycle length, jointly maximizing social welfare and ensuring fairness in initial country-level allocations. We establish, for the first time, that this problem admits a polynomial-time exact algorithm; however, computing the lexicographically minimal deviation solution is shown to be feasible only under additional assumptions. To achieve fairness, we propose a cooperative-game-theoretic framework incorporating credit adjustment, integrating solution concepts such as the Shapley value and the nucleolus. Our scalable solver combines integer linear programming, graph matching, and cycle-covering algorithms. Extensive experiments on million-node instances demonstrate that unbounded-cycle exchanges significantly improve long-term stability and aggregate social welfare over 2-cycle–only schemes, while maintaining computational efficiency and scalability.