Novel Integer Programming models for the stable kidney exchange problem

Kidney exchange programs (KEP's) represent an additional possibility of transplant for patients suffering from end stage kidney disease. If a patient has a willing living donor with whom the patient is not compatible, the pair patient--donor can join a pool of incompatible pairs and, if compatibility between patient and donor in two our more pairs exists, organs can be exchanged between them. The problem can be modeled as an integer program that, in general, aims at finding the pairs that should be selected for transplant such that maximum number of transplants is performed. In this paper we consider that for each patient there may exist a preference order over the organs that he/she can receive, since a patient may be compatible with several donors but may have a better fit over some than over others. Under this setting, the aim is to find the maximum cardinality stable exchange, a solution where no blocking cycle exists. For this purpose we propose three novel integer programming models based on the well-known edge and cycle formulations. These formulations are adjusted for both finding stable and strongly stable exchanges under strict preferences and for the case when ties in preferences may exist. Furthermore, we study a situation when the stability requirement can be relaxed by addressing the trade-off between maximum cardinality versus number of blocking cycles allowed in a solution. The effectiveness of the proposed models is assessed through extensive computational experiments on a wide set of instances.