Parameterized Algorithms for Kidney Exchange

TL;DR

This paper develops fixed-parameter tractable algorithms and explores complexity results for the kidney exchange problem, improving computational methods for matching incompatible patient-donor pairs.

Contribution

It introduces new FPT algorithms based on various parameters and analyzes kernelization complexity, also providing an approximation algorithm for a key case.

Findings

FPT algorithms parameterized by number of patients, treewidth, and vertex types.

Hardness results on kernelization complexity.

An approximation algorithm for a special case.

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.