Algorithms for Social Good: Kidney Exchange

Kidney exchange is a type of barter market where patients exchange willing but incompatible donors. These exchanges are conducted via cycles—where each incompatible patient-donor pair in the cycle both gives and receives a kidney—and chains, which are started by an altruist donor who does not need a kidney in return. Finding the best combination of cycles and chains is hard. The leading algorithms for this optimization problem use either branch and price—a combination of branch and bound and column generation—or constraint generation. Although these approaches handle cycles efficiently, they are limited by how they handle chains. Branch-and-price based solvers must perform a process called pricing. We begin with a proof of correctness for a polynomial time pricing algorithm for cycles, and a complementary hardness result for pricing chains. Next, we introduce a new integer programming formulation which bypasses this hardness result. On real data from the UNOS nationwide exchange in the United States and the NLDKSS nationwide exchange in the United Kingdom, as well as on generated realistic large-scale data, we show that our new formulation scales significantly better than other solvers—in many cases by orders of magnitude. Finally, we show how our new formulation can be modified in a straightforward way to take post-match edge failure into account, under the restriction that edges have equal probabilities of failure. Post-match edge failure is a primary source of inefficiency in presently-fielded kidney exchanges. We end with a polynomial time algorithm for cycle pricing in the failure-aware context.