{"type":"ScholarlyArticle","title":"Parameterized Complexity of Kidney Exchange Revisited","authors":[{"name":"Úrsula Hébert-Johnson","slug":"ursula-hebert-johnson","given":"Úrsula","family":"Hébert-Johnson","affiliation":"UC Santa Barbara"},{"name":"Daniel Lokshtanov","slug":"daniel-lokshtanov","given":"Daniel","family":"Lokshtanov","affiliation":"UC Santa Barbara"},{"name":"Chinmay Sonar","slug":"chinmay-sonar","given":"Chinmay","family":"Sonar","affiliation":"UC Santa Barbara"},{"name":"Vaishali Surianarayanan","slug":"vaishali-surianarayanan","given":"Vaishali","family":"Surianarayanan","affiliation":"UC Santa Barbara"}],"abstract":"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.\r\n\r\nKidney 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).","doi":"10.24963/ijcai.2024/9","url":"https://ijcai.org/proceedings/2024/9","pdf":"https://ijcai.org/proceedings/2024/0009.pdf","markdown":"https://ijcai.org/proceedings/2024/9.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/9","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Agent-based and Multi-agent Systems","track":"Main Track","number":9,"pages":{"first":76,"last":84},"keywords":["Constraint Satisfaction and Optimization: CSO: Constraint optimization problems","Agent-based and Multi-agent Systems: MAS: Resource allocation","Game Theory and Economic Paradigms: GTEP: Auctions and market-based systems","Game Theory and Economic Paradigms: GTEP: Computational social choice"],"videos":[],"references":[{"position":1,"raw":"[Abraham et al., 2007] David J Abraham, Avrim Blum, and Tuomas Sandholm. 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