Computing Balanced Solutions for Large International Kidney Exchange Schemes When Cycle Length Is Unbounded
CoRR(2023)
摘要
In kidney exchange programmes (KEP) patients may swap their incompatible
donors leading to cycles of kidney transplants. Nowadays, countries try to
merge their national patient-donor pools leading to international KEPs (IKEPs).
As shown in the literature, long-term stability of an IKEP can be achieved
through a credit-based system. In each round, every country is prescribed a
"fair" initial allocation of kidney transplants. The initial allocation, which
we obtain by using solution concepts from cooperative game theory, is adjusted
by incorporating credits from the previous round, yielding the target
allocation. The goal is to find, in each round, an optimal solution that
closely approximates this target allocation. There is a known polynomial-time
algorithm for finding an optimal solution that lexicographically minimizes the
country deviations from the target allocation if only 2-cycles (matchings)
are permitted. In practice, kidney swaps along longer cycles may be performed.
However, the problem of computing optimal solutions for maximum cycle length
ℓ is NP-hard for every ℓ≥ 3. This situation changes back to
polynomial time once we allow unbounded cycle length. However, in contrast to
the case where ℓ=2, we show that for ℓ=∞, lexicographical
minimization is only polynomial-time solvable under additional conditions
(assuming P ≠ NP). Nevertheless, the fact that the optimal solutions
themselves can be computed in polynomial time if ℓ=∞ still enables us
to perform a large scale experimental study for showing how stability and total
social welfare are affected when we set ℓ=∞ instead of ℓ=2.
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