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

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

An MDD-Based Lagrangian Approach to the Multicommodity Pickup-and-Delivery TSP

We address the one-to-one multicommodity pickup-and-delivery traveling salesman problem, a challenging variant of the traveling salesman problem that includes the transportation of commodities between locations. The goal is to find a minimum cost tour such that each commodity is delivered to its destination and the maximum capacity of the vehicle is never exceeded. We propose an exact approach that uses a discrete relaxation based on multivalued decision diagrams (MDDs) to better represent the combinatorial structure of the problem. We enhance our relaxation by using the MDDs as a subproblem to a Lagrangian relaxation technique, leading to significant improvements in both bound quality and run-time performance. Our work extends the use of MDDs for solving routing problems by presenting new construction methods and filtering rules based on capacity restrictions. Experimental results show that our approach outperforms state-of-the-art methodologies, closing 33 open instances from the literature, with 27 of those closed by our best variant.The authors gratefully acknowledge funding from the NaturalSciences and Engineering Research Council of Canada(NSERC) and CONICYT (Becas Chile)