Diverse Pairs of Matchings

Authors Fedor V. Fomin , Petr A. Golovach , Lars Jaffke , Geevarghese Philip , Danil Sagunov

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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Lars Jaffke
  • University of Bergen, Norway
Geevarghese Philip
  • Chennai Mathematical Institute, India
  • UMI ReLaX, Chennai, India
Danil Sagunov
  • St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russia
  • JetBrains Research, St. Petersburg, Russia


We thank Günter Rote for pointing out to us that, given a maximum matching M, we can find a maximum matching M' such that |M △ M'| is maximum in polynomial time by the reduction to the Minimum Cost Maximum Matching problem.

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Fedor V. Fomin, Petr A. Golovach, Lars Jaffke, Geevarghese Philip, and Danil Sagunov. Diverse Pairs of Matchings. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on 𝒪(k²) vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Matching
  • Solution Diversity
  • Fixed-Parameter Tractability


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