Maximum matchings in graphs for allocating kidney paired donation

Living donors are often incompatible with their intended recipients. Kidney paired donation matches one patient and his or her incompatible donor with another pair in the same situation for an exchange. Let patient-donor pairs be the vertices of an undirected graph $G$, with edges connecting reciprocally compatible vertices. A matching in $G$ is a feasible set of paired donations. Because the lifespan of a transplant depends on the immunologic concordance of donor and recipient, we weight the edges of $G$ and seek a maximum edge-weight matching. Unfortunately, such matchings might not have the maximum cardinality; there is a risk of an unpredictable trade-off between quality and quantity of paired donations. We prove that the number of paired donations is within a multiplicative factor of the maximum possible donations, where the factor depends on the edge weighting. We propose an edge weighting of $G$ which guarantees that every matching with maximum weight also has maximum cardinality, and also maximizes the number of transplants for an exceptional subset of recipients, while favoring immunologic concordance. We partially generalize this result to k-way exchange and chains, and we implement our weightings using a real patient dataset from Brazil.