A Branch-and-Price Algorithm Enhanced by Decision Diagrams for the Kidney Exchange Problem

TL;DR

This paper introduces a novel branch-and-price algorithm enhanced with decision diagrams to solve the kidney exchange problem efficiently, handling long cycles and chains in large instances, outperforming existing methods.

Contribution

It presents the first decomposition approach capable of managing long cycles and chains in large kidney exchange instances using multi-valued decision diagrams.

Findings

Successfully solves almost all instances in the PrefLib library

Outperforms state-of-the-art methods in computational experiments

Provides a new, stronger upper bound on the optimal value of KEP

Abstract

Kidney paired donation programs allow patients registered with an incompatible donor to receive a suitable kidney from another donor, as long as the latter's co-registered patient, if any, also receives a kidney from a different donor. The kidney exchange problem (KEP) aims to find an optimal collection of kidney exchanges taking the form of cycles and chains. Existing exact solution methods for KEP either are designed for the case where only cyclic exchanges are considered, or can handle long chains but are scalable as long as cycles are short. We develop the first decomposition method that is able to deal with long cycles and long chains for large realistic instances. More specifically, we propose a branch-and-price framework, in which the pricing problems are solved (for the first time in packing problems in a digraph) through multi-valued decision diagrams. Also, we present a new upper bound on the optimal value of KEP, stronger than the one proposed in the literature, which is obtained via our master problem. Computational experiments show superior performance of our method over the state of the art by optimally solving almost all instances in the PrefLib library for multiple cycle and chain lengths.