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

Edward Lam, Vicky Mak-Hau

Research output: Contribution to journalArticleResearchpeer-review


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.

Original languageEnglish
Article number104852
Number of pages11
JournalComputers and Operations Research
Publication statusPublished - 2019


  • Branch-and-cut-and-price
  • Column generation
  • Cycle
  • Facet
  • Kidney exchange
  • Pricing
  • Separation
  • Subtour

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