The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchange -- comprising cycles or chains starting from altruistic donors -- such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. Due to constraints in medical resources, some limits are often imposed on the lengths of these cycles and chains. These exchanges create a network of transplants aimed at maximizing the total number, $t$, of successful transplants. Recently, this problem was deterministically solved in $O^*(14.34^t)$ time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in $O^*(6.855^t)$ time.

翻译：肾脏交换问题是医疗与经济领域的一项突出挑战，源于器官移植场景。它在人工智能与优化领域已被广泛研究。在肾脏交换中，考虑一组供体-受体配对及利他捐赠者，目标在于识别出一系列交换序列——包括循环或始于利他捐赠者的链——使得每位供体将肾脏提供给下一配对中的兼容受体。受医疗资源约束，这些循环与链的长度通常存在限制。此类交换构建了一个旨在最大化成功移植总数$t$的移植网络。近期，该问题在$O^*(14.34^t)$时间内被确定性求解（IJCAI 2024）。本文针对肾脏交换问题引入代表集技术，证明该问题可在$O^*(6.855^t)$时间内被确定性求解。