Solving the Maximum Weighted Length-Constrained Simple Path

The Kidney Exchange Problem (KEP) aims at finding the best ≤ k-way exchanges in a barter market where agents are patients with a willing but incompatible donor. In 2007, Abraham et al. [1] introduced a natural (exponential) integer programming formulation called the cycle formulation, which maximizes the sum of chosen disjoint cycles over all cycles of length at most k in a compatibility graph. They could solve it efficiently by a branch-and-price approach, where the pricing problem is to find a cycle of length at most k of positive reduced cost. Recently, several countries allowed for the participation of altruistic donors in the exchanges. This implies the possibility to add simple chains, with an altruistic donor as starting point and a length limited to l (usually l ≥ k), in the solution. The “cycle” formulation still holds in this case, but must include these chains, becoming the path formulation. This variant of the KEP is harder to solve as the pricing problem becomes NP-complete because of the new question: does a simple path of length at most l and starting from an altruistic donor has a positive reduced cost? The associated optimization problem is related to the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). We seek a maximum (instead of minimum) weighted elementary path of limited length (the resource constraint).