Perfect Matchings and Popularity in the Many-To-Many Setting

Authors Telikepalli Kavitha , Kazuhisa Makino

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Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India
Kazuhisa Makino
  • Research Institute for Mathematical Sciences, Kyoto University, Japan


Thanks to Naoyuki Kamiyama for asking us about the computational complexity of the min-cost popular maximum matching problem when vertices have capacities.

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Telikepalli Kavitha and Kazuhisa Makino. Perfect Matchings and Popularity in the Many-To-Many Setting. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We consider a matching problem in a bipartite graph G where every vertex has a capacity and a strict preference list ranking its neighbors. We assume that G admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible here and we seek one that is popular within the set of perfect matchings - it is known that such a matching exists in G and can be efficiently computed. Now we are in the weighted setting, i.e., there is a cost function on the edge set, and we seek a min-cost popular perfect matching in G. We show that such a matching can be computed in polynomial time. Our main technical result shows that every popular perfect matching in a hospitals/residents instance G can be realized as a popular perfect matching in the marriage instance obtained by cloning vertices. Interestingly, it is known that such a mapping does not hold for popular matchings in a hospitals/residents instance.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Bipartite graphs
  • Matchings under preferences
  • Capacities
  • Dual certificates


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