Combinatorial Hodge theory for equitable kidney paired donation

Kidney Paired Donation (KPD) is a system whereby incompatible patient-donor pairs (PD pairs) are entered into a pool to find compatible cyclic kidney exchanges where each pair gives and receives a kidney. The donation allocation decision problem for a KPD pool has traditionally been viewed within an economic theory and integer-programming framework. While previous allocation schema work well to donate the maximum number of kidneys at a specific time, certain subgroups of patients are rarely matched in such an exchange. Consequently, these methods lead to systematic inequity in the exchange, where many patients are rejected a kidney repeatedly. Our goal is to investigate inequity within the distribution of kidney allocation among patients, and to present an algorithm which minimizes allocation disparities. The method presented is inspired by cohomology and describes the cyclic structure in a kidney exchange efficiently; this structure is then used to search for an equitable kidney allocation. Another key result of our approach is a score function defined on PD pairs which measures cycle disparity within a KPD pool; i.e., this function measures the relative chance for each PD pair to take part in the kidney exchange if cycles are chosen uniformly. Specifically, we show that PD pairs with underdemanded donors or highly sensitized patients have lower scores than typical PD pairs. Furthermore, our results demonstrate that PD pair score and the chance to obtain a kidney are positively correlated when allocation is done by utility-optimal integer programming methods. In contrast, the chance to obtain a kidney through our method is independent of score, and thus unbiased in this regard.