To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient-donor pools. We consider a recent credit system where in each round, countries are given an initial kidney transplant allocation which is adjusted by a credit function yielding a target allocation. The goal is to find a solution in the patient-donor compatibility graph that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We first give a polynomial-time algorithm for computing such matchings. We then perform, for the first time, a computational study for a large number of countries. For the initial allocations we use, besides two easy-to-compute solution concepts, two classical concepts: the Shapley value and nucleolus. These are hard to compute, but by using state-of-the-art software we show that they are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 52% more balanced.

中文翻译：

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大型国际肾脏交换计划的计算平衡解决方案

为了克服不相容性问题，肾病患者可以交换他们的供体。在国际肾脏交换计划 (IKEP) 中，各国合并其国家患者-捐赠者库。我们考虑最近的信用体系，其中在每一轮中，各国都会获得初始肾移植分配，该分配通过产生目标分配的信用函数进行调整。目标是在患者-供体兼容性图中找到一个尽可能接近目标分配的解决方案，以确保国际资源库的长期稳定性。作为解决方案，我们使用最大匹配按字典顺序最小化国家与目标分配的偏差。我们首先给出了计算这种匹配的多项式时间算法。然后，我们首次对大量国家进行了计算研究。对于我们使用的初始分配，除了两个易于计算的解决方案概念之外，还有两个经典概念：Shapley 值和核仁。这些很难计算，但通过使用最先进的软件，我们表明现在多达 15 个国家/地区的 IKEP 可以使用它们。我们的实验表明，使用按字典顺序排列的最小最大匹配而不是仅最小化与目标分配的最大偏差（如之前所做的那样）可以使 IKEP 的平衡性提高 52%。