🤖 AI Summary
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.
📝 Abstract
In kidney exchange programmes (KEP) patients may swap their incompatible donors leading to cycles of kidney transplants. Nowadays, countries try to merge their national patient-donor pools leading to international KEPs (IKEPs). As shown in the literature, long-term stability of an IKEP can be achieved through a credit-based system. In each round, every country is prescribed a"fair"initial allocation of kidney transplants. The initial allocation, which we obtain by using solution concepts from cooperative game theory, is adjusted by incorporating credits from the previous round, yielding the target allocation. The goal is to find, in each round, an optimal solution that closely approximates this target allocation. There is a known polynomial-time algorithm for finding an optimal solution that lexicographically minimizes the country deviations from the target allocation if only $2$-cycles (matchings) are permitted. In practice, kidney swaps along longer cycles may be performed. However, the problem of computing optimal solutions for maximum cycle length $ell$ is NP-hard for every $ellgeq 3$. This situation changes back to polynomial time once we allow unbounded cycle length. However, in contrast to the case where $ell=2$, we show that for $ell=infty$, lexicographical minimization is only polynomial-time solvable under additional conditions (assuming P $
eq$ NP). Nevertheless, the fact that the optimal solutions themselves can be computed in polynomial time if $ell=infty$ still enables us to perform a large scale experimental study for showing how stability and total social welfare are affected when we set $ell=infty$ instead of $ell=2$.