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Stable Marriage with Covering Constraints–A Complete Computational Trichotomy

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Algorithmic Game Theory (SAGT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 10504))

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Abstract

We consider Stable Marriage with Covering Constraints (SMC): in this variant of Stable Marriage, we distinguish a subset of women as well as a subset of men, and we seek a matching with fewest number of blocking pairs that matches all of the distinguished people. We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem.

Our main result is a complete complexity trichotomy that, for each choice of the studied parameters, classifies SMC as polynomial-time solvable, \(\mathsf {NP}\)-hard and fixed-parameter tractable, or \(\mathsf {NP}\)-hard and \(\mathsf {W}[1]\)-hard. We also classify all cases of one-sided constraints where only women may be distinguished.

M. Mnich—Supported by ERC Starting Grant 306465 (BeyondWorstCase).

I. Schlotter—Supported by the Hungarian National Research Fund (OTKA grants no. K-108383 and no. K-108947).

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Notes

  1. 1.

    For background on parameterized complexity, we refer to the recent monograph [9].

  2. 2.

    Restrictions without parameters are classified as polynomial-time solvable or \(\mathsf {NP}\)-hard.

  3. 3.

    Hamada et al. claim only a run time \(O((|\mathcal W||\mathcal M|)^{b+1})\), but their algorithm can easily be implemented to run in time \(O(|I|^{b+1})\).

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Authors and Affiliations

  1. Universität Bonn, Bonn, Germany

    Matthias Mnich

  2. Maastricht University, Maastricht, The Netherlands

    Matthias Mnich

  3. Budapest University of Technology and Economics, Budapest, Hungary

    Ildikó Schlotter

Authors
  1. Matthias Mnich
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  2. Ildikó Schlotter
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Correspondence to Ildikó Schlotter .

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Editors and Affiliations

  1. University of Salento, Lecce, Italy

    Vittorio Bilò

  2. University of L'Aquila, L'Aquila, Italy

    Michele Flammini

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Mnich, M., Schlotter, I. (2017). Stable Marriage with Covering Constraints–A Complete Computational Trichotomy. In: Bilò, V., Flammini, M. (eds) Algorithmic Game Theory. SAGT 2017. Lecture Notes in Computer Science(), vol 10504. Springer, Cham. https://doi.org/10.1007/978-3-319-66700-3_25

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  • DOI: https://doi.org/10.1007/978-3-319-66700-3_25

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