Branch-and-cut-and-price for the cardinality-constrained multi-cycle problem in kidney exchange

Abstract The establishment of kidney exchange programs has dramatically improved rates for kidney transplants by matching donors to compatible patients who would otherwise fail to receive a kidney for transplant. Rather than simply swapping kidneys between two patient-donor pairs, having multiple patient-donor pairs simultaneously donate kidneys in a cyclic manner enables more patients to receive a kidney for transplant. For practicality reasons, the cycles must be limited to short lengths. Finding these cycles can be accomplished by solving the Cardinality-constrained Multi-cycle Problem, which generalizes the Prize-collecting Assignment Problem with constraints that bound the length of the subtours. This paper presents a series of additions to existing works—new constraints, some polyhedral results, new separation algorithms and a new pricing algorithm—and integrates them in the first branch-and-cut-and-price model of the problem. The model is shown to empirically outperform the state-of-the-art by solving 149 of 160 standard benchmarks, compared to 115 by the position-indexed chain-edge formulation and 114 by the position-indexed edge formulation.