The kidney exchange mechanism allows many patient-donor pairs who are otherwise incompatible with each other to come together and exchange kidneys along a cycle. However, due to infrastructure and legal constraints, kidney exchange can only be performed in small cycles in practice. In reality, there are also some altruistic donors who do not have any paired patients. This allows us to also perform kidney exchange along paths that start from some altruistic donor. Unfortunately, the computational task is NP-complete. To overcome this computational barrier, an important line of research focuses on designing faster algorithms, both exact and using the framework of parameterized complexity. The standard parameter for the kidney exchange problem is the number $t$ of patients that receive a healthy kidney. The current fastest known deterministic FPT algorithm for this problem, parameterized by $t$, is $O^\star\left(14^t\right)$. In this work, we improve this by presenting a deterministic FPT algorithm that runs in time $O^\star\left((4e)^t\right)\approx O^\star\left(10.88^t\right)$. This problem is also known to be W[1]-hard parameterized by the treewidth of the underlying undirected graph. A natural question here is whether the kidney exchange problem admits an FPT algorithm parameterized by the pathwidth of the underlying undirected graph. We answer this negatively in this paper by proving that this problem is W[1]-hard parameterized by the pathwidth of the underlying undirected graph. We also present some parameterized intractability results improving the current understanding of the problem under the framework of parameterized complexity.

翻译：肾脏交换机制使得许多原本互不兼容的患者-供体配对能够通过循环交换肾脏。然而，由于基础设施和法律限制，实际中肾脏交换只能在小型循环中进行。现实中还存在一些没有配对患者的利他性供体，这使得我们能够从利他性供体出发沿路径进行肾脏交换。遗憾的是，该计算任务属于NP完全问题。为克服这一计算障碍，一个重要研究方向聚焦于设计更快的算法，包括精确算法及基于参数化复杂度框架的算法。肾脏交换问题的标准参数是获得健康肾脏的患者数量$t$。目前已知最快的确定性FPT算法（以$t$为参数）时间复杂度为$O^\star\left(14^t\right)$。本研究中，我们提出一种确定性FPT算法，其运行时间为$O^\star\left((4e)^t\right)\approx O^\star\left(10.88^t\right)$，从而改进了现有结果。已知该问题在底层无向图的树宽参数化下属于W[1]-难问题。一个自然的问题是：肾脏交换问题是否允许基于底层无向图路径宽度的FPT算法？本文通过证明该问题在底层无向图路径宽度参数化下属于W[1]-难问题，给出了否定答案。我们同时提出若干参数化难解性结果，深化了在参数化复杂度框架下对该问题的理解。