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Computing the minimal covering set

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  • 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
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    References listed on IDEAS

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    1. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    2. 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.
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    7. 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.
    8. 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.
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    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. Thomas Demuynck, 2014. "The computational complexity of rationalizing Pareto optimal choice behavior," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(3), pages 529-549, March.
    3. Fabrice Talla Nobibon & Laurens Cherchye & Yves Crama & Thomas Demuynck & Bram De Rock & Frits C. R. Spieksma, 2016. "Revealed Preference Tests of Collectively Rational Consumption Behavior: Formulations and Algorithms," Operations Research, INFORMS, vol. 64(6), pages 1197-1216, December.
    4. 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.
    5. Daniel Carroll & Jim Dolmas & Eric Young, 2021. "The Politics of Flat Taxes," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 39, pages 174-201, January.
    6. 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.
    7. Demuynck, Thomas, 2011. "The computational complexity of rationalizing boundedly rational choice behavior," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 425-433.
    8. Brandl, Florian & Brandt, Felix, 0. "A natural adaptive process for collective decision-making," Theoretical Economics, Econometric Society.
    9. repec:hal:wpaper:hal-00756696 is not listed on IDEAS
    10. Daniel R. Carroll & Jim Dolmas & Eric Young, 2015. "Majority Voting: A Quantitative Investigation," Working Papers (Old Series) 1442, Federal Reserve Bank of Cleveland.
    11. Felix Brandt & Florian Grundbacher, 2023. "The Banks Set and the Bipartisan Set May be Disjoint," Papers 2308.01881, arXiv.org.
    12. 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.
    13. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    14. Felix Brandt & Chris Dong, 2022. "On Locally Rationalizable Social Choice Functions," Papers 2204.05062, arXiv.org, revised Mar 2024.
    15. John Duggan, 2011. "Uncovered Sets," Wallis Working Papers WP63, University of Rochester - Wallis Institute of Political Economy.
    16. John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    17. 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.

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