Rank-Maximal Matchings – Structure and Algorithms

  • Pratik Ghosal
  • Meghana Nasre
  • Prajakta Nimbhorkar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)


Let \(G = (\mathcal {A}\cup \mathcal {P}, E)\) be a bipartite graph where \(\mathcal {A}\) denotes a set of agents, \(\mathcal {P}\) denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching \(M\) in \(G\) is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on.

In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is \(\#P\)-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Pratik Ghosal
    • 1
  • Meghana Nasre
    • 2
  • Prajakta Nimbhorkar
    • 3
  1. 1.University of WrocławWrocławPoland
  2. 2.Indian Institute of TechnologyMadrasIndia
  3. 3.Chennai Mathematical InstituteChennaiIndia

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