Generative Flow Networks, known as GFlowNets, have been introduced in recent times, presenting an exciting possibility for neural networks to model distributions across various data structures. In this paper, we broaden their applicability to encompass scenarios where the data structures are optimal solutions of a combinatorial problem. Concretely, we propose the use of GFlowNets to learn the distribution of optimal solutions for kidney exchange problems (KEPs), a generalized form of matching problems involving cycles.

Kidney transplant is the preferred method of treatment for patients suffering from kidney failure. However, not all patients can find a donor which matches their physiological characteristics. Kidney exchange programs (KEPs) seek to match such incompatible patient-donor pairs together, usually with the main objective of maximizing the total number of transplants. Since selecting one optimal solution translates to a decision on who receives a transplant, it has a major effect on the lives of patients. The current practice in selecting an optimal solution does not necessarily ensure fairness in the selection process. In this paper, the existence of multiple optimal plans for a KEP is explored as a mean to achieve individual fairness. We propose the use of randomized policies for selecting an optimal solution in which patients' equal opportunity to receive a transplant is promoted. Our approach gives rise to the problem of enumerating all optimal solutions, which we tackle using a hybrid of constraint programming and linear programming. The advantages of our proposed method over the common practice of using the optimal solution obtained by a solver are stressed through computational experiments. Our methodology enables decision makers to fully control KEP outcomes, overcoming any potential bias or vulnerability intrinsic to a deterministic solver.