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Stable Marriage with General Preferences

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Abstract

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.

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Acknowledgments

We thank the anonymous SAGT and TOCS reviewers for their thorough and careful review. We highly appreciate their insightful suggestions that led to a substantial improvement of the paper.

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Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Linda Farczadi

  2. Ryerson University, Toronto, Canada

    Konstantinos Georgiou

  3. University of Waterloo, Waterloo, Canada

    Jochen Könemann

Authors
  1. Linda Farczadi
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  2. Konstantinos Georgiou
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  3. Jochen Könemann
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Corresponding author

Correspondence to Linda Farczadi.

Additional information

An extended abstract of this work appeared in the 7th International Symposium on Algorithmic Game Theory (SAGT) 2014 [6]. The work of all three authors was supported by NSERC’s Discovery Grant program.

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Farczadi, L., Georgiou, K. & Könemann, J. Stable Marriage with General Preferences. Theory Comput Syst 59, 683–699 (2016). https://doi.org/10.1007/s00224-016-9687-z

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