Parameterized Complexity of Kidney Exchange Revisited

As of January 2023, there are more than 90,000 people on the national transplant waiting list in need of a kidney in the United States. These patients often have a friend or family member who is willing to donate, but whose kidney type might not be compatible. To help match these patients to suitable donors, patient-donor compatibility can be modeled as a directed graph. Specifically, in the Kidney Exchange problem, the input is a directed graph G, a subset B of vertices (altruistic donors), and two integers l_p and l_c. An altruistic donor is a donor who is not paired with a patient, and the remaining vertices are patient-donor pairs. Whenever a donor is compatible with a patient from a patient-donor pair, we place a directed edge from the donor vertex to the patient-donor pair. Here the donor vertex can be either altruistic or non-altruistic. The goal is to find a collection of vertex-disjoint cycles and paths covering the maximum number of patients such that each cycle has length at most l_c, and such that each path has length at most l_p and begins at a vertex in B. The path and cycle lengths are bounded so that the surgeries for a given path or cycle can be performed simultaneously. Kidney Exchange has received a great deal of attention in recent years. We contribute to this line of work by closing two open problems from IJCAI '18 and IJCAI '22: "Is Kidney Exchange FPT when parameterized by (i) the treewidth (omega) of G and (ii) the number of vertex types in G?'' Two vertices have the same vertex type if they have the same in- and out-neighborhoods. We show that Kidney Exchange is FPT parameterized by the number of vertex types. On the other hand, we show W[1]-hardness with respect to omega. We also design a randomized 4^t * n^O(1)-time algorithm parameterized by t, the number of patients helped, significantly improving upon the previous state of the art, which was 161^t * n^O(1).