Kidney Exchange: Faster Parameterized Algorithms and Tighter Lower Bounds

TL;DR

This paper advances the computational efficiency of kidney exchange algorithms by providing a faster fixed-parameter tractable algorithm based on the number of patients receiving kidneys, and establishes complexity bounds related to graph parameters.

Contribution

It introduces a faster deterministic FPT algorithm for kidney exchange parameterized by the number of patients, and proves W[1]-hardness results for parameterization by pathwidth.

Findings

New FPT algorithm runs in time O*((4e)^t)

Proves W[1]-hardness for parameterization by pathwidth

Improves understanding of parameterized complexity in kidney exchange

Abstract

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.