Iterative branch-and-price for large multi-criteria kidney exchange

Living donor kidney transplantation is the preferred treatment for patients with end stage renal disease. Unfortunately, living donors are often incompatible with their specified recipient due to physiological reasons, such as incompatible blood types. Kidney exchange is an increasing modality that allows the exchange of kidneys between such incompatible donor-patient pairs. Typically, the aim is to find an allocation of donors to patients that is optimal with respect to multiple hierarchically ordered criteria. In this paper we show why existing approaches to the optimization of kidney exhange cannot deal effectively with multiple hierarchical criteria or with large, sparse, multi-center pools, which now begin to arise in practice. We then present a generic iterative branch-and-price algorithm which can deal with such multi-criteria exchanges and we show how the pricing problem can be solved in polynomial time for a general class of criteria. Our algorithm is effective even for large realistic donor-patient pools. Moreover, the algorithm accomodates inclusion of altruistic donors (who have no specified recipient) and individual rationality constraints for hospitals, as these are of increasing importance in clinical practice. Our approach and its effects are demonstrated using data from the Dutch national kidney exchange program, which is the oldest nationally coordinated program.