IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v23y1998i4p874-891.html
   My bibliography  Save this article

The Geometry of Fractional Stable Matchings and Its Applications

Author

Listed:
  • Chung-Piaw Teo

    (Department of Decision Sciences, Faculty of Business Administration, National University of Singapore)

  • Jay Sethuraman

    (Operations Research Center, MIT, Cambridge, Massachusetts 02139)

Abstract

We study the classical stable marriage and stable roommates problems using a polyhedral approach. We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching. Furthermore, for certain special weight functions on the edges, we construct a 2-approximation algorithm for the optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions. This also leads to a genuinely simple proof of the integrality of the stable marriage polytope.

Suggested Citation

  • Chung-Piaw Teo & Jay Sethuraman, 1998. "The Geometry of Fractional Stable Matchings and Its Applications," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 874-891, November.
  • Handle: RePEc:inm:ormoor:v:23:y:1998:i:4:p:874-891
    DOI: 10.1287/moor.23.4.874
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/moor.23.4.874
    Download Restriction: no

    References listed on IDEAS

    as
    1. Roth, Alvin E, 1984. "The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory," Journal of Political Economy, University of Chicago Press, vol. 92(6), pages 991-1016, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kesten, Onur & Ünver, M. Utku, 2015. "A theory of school choice lotteries," Theoretical Economics, Econometric Society, vol. 10(2), May.
    2. Pavlos Eirinakis & Dimitrios Magos & Ioannis Mourtos & Panayiotis Miliotis, 2014. "Polyhedral Aspects of Stable Marriage," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 656-671, August.
    3. Jay Sethuraman & Teo Chung Piaw & Rakesh V. Vohra, 2003. "Integer Programming and Arrovian Social Welfare Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 309-326, May.
    4. Haris Aziz & Bettina Klaus, 2019. "Random matching under priorities: stability and no envy concepts," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(2), pages 213-259, August.
    5. Agnes Cseh & Robert W. Irving & David F. Manlove, 2017. "The Stable Roommates problem with short lists," IEHAS Discussion Papers 1726, Institute of Economics, Centre for Economic and Regional Studies.
    6. Afacan, Mustafa Oǧuz, 2018. "The object allocation problem with random priorities," Games and Economic Behavior, Elsevier, vol. 110(C), pages 71-89.
    7. Haris Aziz & Bettina Klaus, 2017. "Random Matching under Priorities: Stability and No Envy Concepts," Cahiers de Recherches Economiques du Département d'Econométrie et d'Economie politique (DEEP) 17.09, Université de Lausanne, Faculté des HEC, DEEP.
    8. Manjunath, Vikram & Turhan, Bertan, 2016. "Two school systems, one district: What to do when a unified admissions process is impossible," Games and Economic Behavior, Elsevier, vol. 95(C), pages 25-40.
    9. Shuji Kijima & Toshio Nemoto, 2012. "On Randomized Approximation for Finding a Level Ideal of a Poset and the Generalized Median Stable Matchings," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 356-371, May.
    10. Jay Sethuraman & Chung-Piaw Teo & Liwen Qian, 2006. "Many-to-One Stable Matching: Geometry and Fairness," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 581-596, August.
    11. Martin Van der Linden, 2019. "Deferred acceptance is minimally manipulable," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 609-645, June.
    12. Yang Liu & Zhi-Ping Fan & Yan-Ping Jiang, 2018. "Satisfied surgeon–patient matching: a model-based method," Quality & Quantity: International Journal of Methodology, Springer, vol. 52(6), pages 2871-2891, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormoor:v:23:y:1998:i:4:p:874-891. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Matthew Walls). General contact details of provider: http://edirc.repec.org/data/inforea.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.