We study the computational problem of identifying optimal
sets of kidney exchanges in the UK. We show how to expand an integer
programming-based formulation [1, 19] in order to model the criteria that
constitute the UK definition of optimality. The software arising from this
work has been used by the National Health Service Blood and Transplant
to find optimal sets of kidney exchanges for their National Living Donor
Kidney Sharing Schemes since July 2008.We report on the characteristics
of the solutions that have been obtained in matching runs of the scheme
since this time. We then present empirical results arising from the real
datasets that stem from these matching runs, with the aim of establishing
the extent to which the particular optimality criteria that are present
in the UK influence the structure of the solutions that are ultimately
computed. A key observation is that allowing 4-way exchanges would be
likely to lead to a significant number of additional transplants
