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Popular matchings with two-sided preferences and one-sided ties

Author

Listed:
  • Agnes Cseh

    (Institute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, and Corvinus University of Budapest)

  • Chien-Chung Huang

    (Ecole Normale Superieure, Paris, France)

  • Telikepalli Kavitha

    (Tata Institute of Fundamental Research, Mumbai, India)

Abstract

We are given a bipartite graph G = (A[B;E) where each vertex has a preference list ranking its neighbors: in particular, every a 2 A ranks its neighbors in a strict order of preference, whereas the preference list of any b 2 B may contain ties. A matching M is popular if there is no matching M0 such that the number of vertices that prefer M0 to M exceeds the number of vertices that prefer M to M0. We show that the problem of deciding whether G admits a popular matching or not is NP-hard. This is the case even when every b 2 B either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each b 2 B puts all its neighbors into a single tie. That is, all neighbors of b are tied in b's list and b desires to be matched to any of them. Our main result is an O(n2) algorithm (where n = jA [ Bj) for the popular matching problem in this model. Note that this model is quite di erent from the model where vertices in B have no preferences and do not care whether they are matched or not.

Suggested Citation

  • Agnes Cseh & Chien-Chung Huang & Telikepalli Kavitha, 2017. "Popular matchings with two-sided preferences and one-sided ties," CERS-IE WORKING PAPERS 1723, Institute of Economics, Centre for Economic and Regional Studies.
    • Handle: RePEc:has:discpr:1723
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    References listed on IDEAS

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    1. Eric McDermid & Robert W. Irving, 2011. "Popular matchings: structure and algorithms," Journal of Combinatorial Optimization, Springer, vol. 22(3), pages 339-358, October.
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    Cited by:

    1. Aleksei Yu. Kondratev & Alexander S. Nesterov, 2018. "Random Paths to Popularity in Two-Sided Matching," HSE Working papers WP BRP 195/EC/2018, National Research University Higher School of Economics.
    2. Kondratev, Aleksei Y. & Nesterov, Alexander S., 2022. "Minimal envy and popular matchings," European Journal of Operational Research, Elsevier, vol. 296(3), pages 776-787.
    3. Agnes Cseh & Yuri Faenza & Telikepalli Kavitha & Vladlena Powers, 2020. "Understanding popular matchings via stable matchings," CERS-IE WORKING PAPERS 2003, Institute of Economics, Centre for Economic and Regional Studies.

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    More about this item

    Keywords

    popular matching; NP-complete; polynomial algorithm; ties;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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