# Operation Frames and Clubs in Kidney Exchange

## TL;DR

This paper introduces generalized models for kidney exchanges, including organ clubs and operation frames, with theoretical complexity results and practical improvements demonstrated through experiments.

## Contribution

It proposes novel generalizations of kidney exchange models, including multiple donors and organ clubs, along with integer programming formulations and the concept of operation frames.

## Key findings

- Operation frames improve planning efficiency by 34-51%.
- Allowing multiple donors increases social welfare.
- The uncapped clearing problem is NP-complete.

## Abstract

A kidney exchange is a centrally-administered barter market where patients swap their willing yet incompatible donors. Modern kidney exchanges use 2-cycles, 3-cycles, and chains initiated by non-directed donors (altruists who are willing to give a kidney to anyone) as the means for swapping. We propose significant generalizations to kidney exchange. We allow more than one donor to donate in exchange for their desired patient receiving a kidney. We also allow for the possibility of a donor willing to donate if any of a number of patients receive kidneys. Furthermore, we combine these notions and generalize them. The generalization is to exchange among organ clubs, where a club is willing to donate organs outside the club if and only if the club receives organs from outside the club according to given specifications. We prove that unlike in the standard model, the uncapped clearing problem is NP-complete. We also present the notion of operation frames that can be used to sequence the operations across batches, and present integer programming formulations for the market clearing problems for these new types of organ exchanges. Experiments show that in the single-donation setting, operation frames improve planning by 34%--51%. Allowing up to two donors to donate in exchange for one kidney donated to their designated patient yields a further increase in social welfare.