Paired Kidney Donation and Listed Exchange

Deceased-donor and live-donor kidneys are the two sources for transplantation, and these sources are utilized via two different programs. One of these programs, a paired kidney donation (PKD), involves two donor-patient pairs, for each of whom transplantation from donor to intended recipient is not possible due to medical incompatibilities, but such that the patient in each couple could receive a transplant from the donor in the other pair. These two pairs can then exchange donated kidneys. Another possibility is a paired listed exchange (PLE): given two donor-patient pairs, the first donor provides a kidney to a candidate on the deceased-donor waiting list, the first patient receives the kidney of the second donor, and the second patient receives a priority on the waiting list. By allowing only PKD’s and PLE’s, we characterize the set of kidney exchanges, which include the maximum number of kidney transplants from the set of incompatible patient-donor pairs. This characterization generalizes the Gallai-Edmonds Decomposition Theorem, a well-known result from combinatorial optimization.