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Fairly Popular Matchings and Optimality

Author Telikepalli Kavitha

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Bipartite graphs
  • Stable matchings
  • Mixed matchings
  • Polytopes

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Abstract

We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. 
Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. 
We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K.

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Telikepalli Kavitha. Fairly Popular Matchings and Optimality. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.41

Author Details

Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

  • Kavitha, Telikepalli: Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001.

Acknowledgements

Thanks to the reviewers for their helpful comments and suggestions. Thanks to Yuri Faenza and Jaikumar Radhakrishnan for useful discussions.

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