IDEAS home Printed from https://ideas.repec.org/a/eee/matsoc/v56y2008i2p254-268.html
   My bibliography  Save this article

Computing the minimal covering set

Author

Listed:
  • Brandt, Felix
  • Fischer, Felix

Abstract

We present the first polynomial-time algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set-the minimal upward covering set and the minimal downward covering set-unless P equals NP. Finally, we observe a strong relationship between von Neumann-Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.

Suggested Citation

  • Brandt, Felix & Fischer, Felix, 2008. "Computing the minimal covering set," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 254-268, September.
  • Handle: RePEc:eee:matsoc:v:56:y:2008:i:2:p:254-268
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0165-4896(08)00036-X
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. van Damme, Eric, 1998. "On the State of the Art in Game Theory: An Interview with Robert Aumann," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 181-210, July.
    2. Laffond G. & Laslier J. F. & Le Breton M., 1993. "The Bipartisan Set of a Tournament Game," Games and Economic Behavior, Elsevier, vol. 5(1), pages 182-201, January.
    3. Bordes, Georges, 1983. "On the possibility of reasonable consistent majoritarian choice: Some positive results," Journal of Economic Theory, Elsevier, vol. 31(1), pages 122-132, October.
    4. Josep E. Peris & BegoÓa Subiza, 1999. "Condorcet choice correspondences for weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(2), pages 217-231.
    5. Laffond G. & Laslier, J. F. & Le Breton, M., 1996. "Condorcet choice correspondences: A set-theoretical comparison," Mathematical Social Sciences, Elsevier, vol. 31(1), pages 59-59, February.
    6. Gerhard J. Woeginger, 2003. "Banks winners in tournaments are difficult to recognize," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 20(3), pages 523-528, June.
    7. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    8. Bhaskar Dutta & Jean-Francois Laslier, 1999. "Comparison functions and choice correspondences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(4), pages 513-532.
    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. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2013. "Computing tournament solutions using relation algebra and RelView," European Journal of Operational Research, Elsevier, vol. 226(3), pages 636-645.
    2. repec:hal:cesptp:hal-00756696 is not listed on IDEAS
    3. Berghammer, Rudolf & Schnoor, Henning, 2015. "Control of Condorcet voting: Complexity and a Relation-Algebraic approach," European Journal of Operational Research, Elsevier, vol. 246(2), pages 505-516.
    4. Felix Brandt & Felix Fischer & Paul Harrenstein & Maximilian Mair, 2010. "A computational analysis of the tournament equilibrium set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(4), pages 597-609, April.
    5. Demuynck, Thomas, 2011. "The computational complexity of rationalizing boundedly rational choice behavior," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 425-433.
    6. repec:hal:wpaper:hal-00756696 is not listed on IDEAS
    7. Felix Brandt & Christian Geist & Paul Harrenstein, 2016. "A note on the McKelvey uncovered set and Pareto optimality," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(1), pages 81-91, January.
    8. Carroll, Daniel R. & Dolmas, James & Young, Eric R., 2015. "Majority Voting: A Quantitative Investigation," Working Paper 1442, Federal Reserve Bank of Cleveland.
    9. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    10. John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    11. Felix Brandt & Markus Brill & Felix Fischer & Paul Harrenstein, 2014. "Minimal retentive sets in tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(3), pages 551-574, March.
    12. repec:hal:journl:halshs-00639942 is not listed on IDEAS

    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:eee:matsoc:v:56:y:2008:i:2:p:254-268. 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: (Dana Niculescu). General contact details of provider: http://www.elsevier.com/locate/inca/505565 .

    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.