Parameterized Algorithms for Kidney Exchange
Arnab Maiti, Palash Dey
TL;DR
The paper studies Kidney Exchange through the lens of parameterized complexity, modeling the problem on a directed compatibility graph with altruistic donors and patient–donor pairs. It develops FPT algorithms parameterized by the number of kidney recipients $t$, by the graph's treewidth together with the maximum of path and cycle lengths $\ell=\max\{l_p,l_c\}$, and by the number of vertex types $\theta$ when $l_p\le l_c$, supported by an MSO$_2$ formulation whose size is bounded by $\ell$. It also proves para-NP-hardness for parameterization by $(\Delta + l_p + l_c)$, provides a polynomial kernel for $t+\Delta$ under constant $\ell$, and establishes a kernel lower bound for $t+\Delta+\max\{l_p,l_c\}$, along with a $(\tfrac{16}{9}+\varepsilon)$-approximation for the special case of cycles only (length $\le 3$). Collectively, these results substantially advance the theoretical understanding of Kidney Exchange, offering scalable algorithms for practical instances and revealing fundamental limits on kernelization and approximation in constrained settings.
Abstract
In kidney exchange programs, multiple patient-donor pairs each of whom are otherwise incompatible, exchange their donors to receive compatible kidneys. The Kidney Exchange problem is typically modelled as a directed graph where every vertex is either an altruistic donor or a pair of patient and donor; directed edges are added from a donor to its compatible patients. The computational task is to find if there exists a collection of disjoint cycles and paths starting from altruistic donor vertices of length at most l_c and l_p respectively that covers at least some specific number t of non-altruistic vertices (patients). We study parameterized algorithms for the kidney exchange problem in this paper. Specifically, we design FPT algorithms parameterized by each of the following parameters: (1) the number of patients who receive kidney, (2) treewidth of the input graph + max{l_p, l_c}, and (3) the number of vertex types in the input graph when l_p <= l_c. We also present interesting algorithmic and hardness results on the kernelization complexity of the problem. Finally, we present an approximation algorithm for an important special case of Kidney Exchange.