New algorithms for hierarchical optimization in kidney exchange programs

Many kidney exchange programs (KEPs) use integer linear programming (ILP) based on a hierarchical set of objectives to determine optimal sets of transplants. We propose innovative techniques to remove barriers in existing mathematical models, vastly reducing solution times and allowing significant increases in potential KEP pool sizes. Our techniques include two methods to avoid unnecessary variables, and a diving algorithm that reduces the need to solve multiple complex ILP models while still guaranteeing optimality of a final solution. We also show how to transition between two existing formulations (namely, the cycle formulation and the position-indexed chain-edge formulation) when optimizing successive objective functions. We use this technique to devise a new algorithm, which, among other features, intelligently exploits the different advantages of the prior two models. We demonstrate the performance of our new algorithms with extensive computational experiments modeling the UK KEP, where we show that our improvements reduce running times by three orders of magnitude compared with the cycle formulation. We also provide substantial empirical evidence that the new methodology offers equally spectacular improvements when applied to the Spanish and Dutch KEP objectives, suggesting that our approach is not just viable, but a significant performance improvement, for many KEPs worldwide.