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Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners

  • Christopher M. Homan
  • Lane A. Hemaspaandra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

Dodgson’s election system elegantly satisfies the Condorcet criterion. However, determining the winner of a Dodgson election is known to be \({\mathrm{\Theta}^{\mathit{p}}_2}\)-complete ([1], see also [2]), which implies that unless P = NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates (although the number of voters may still be polynomial in the number of candidates), a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.

Keywords

Greedy Algorithm Vote System Condorcet Winner Success Frequency Polynomial Hierarchy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM 44(6), 806–825 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bartholdi III, J., Tovey, C., Trick, M.: Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare 6, 157–165 (1989)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Condorcet, M.: Essai sur l’Application de L’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix (1785); Facsimile reprint of original published in Paris, The Imprimerie Royale (1972)Google Scholar
  4. 4.
    McLean, I., Urken, A.: Classics of Social Choice. University of Michigan Press, Ann Arbor (1995)Google Scholar
  5. 5.
    Dodgson, C.: A method of taking votes on more than two issues. Clarendon Press, Oxford, pamphet (1876)Google Scholar
  6. 6.
    Black, D.: The Theory of Committees and Elections. Cambridge University Press, Cambridge (1958)MATHGoogle Scholar
  7. 7.
    Nanson, E.: Methods of election. Transactions and Proceedings of the Royal Society of Victoria 19, 197–240 (1882)Google Scholar
  8. 8.
    Borda, J.C.d.: Mémoire sur les élections au scrutin. Histoire de L’Académie Royale des Sciences Année 1781 (1784)Google Scholar
  9. 9.
    Papadimitriou, C., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) GI-TCS 1983. LNCS, vol. 145, pp. 269–276. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  10. 10.
    Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Theoretical Computer Science 349(3), 382–391 (2005)MATHMathSciNetGoogle Scholar
  11. 11.
    Rothe, J., Spakowski, H., Vogel, J.: Exact complexity of the winner problem for Young elections. Theory of Computing Systems 36(4), 375–386 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press/McGraw Hill (2001)Google Scholar
  13. 13.
    Ausiello, G., Crescenzi, P., Protasi, M.: Approximate solution of NP optimization problems. Theoretical Computer Science 150(1), 1–55 (1995)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaporis, A.C., Kirousis, L.M., Lalas, E.G.: The probabilistic analysis of a greedy satisfiability algorithm. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 574–585. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Chang, L., Korsh, J.: Canonical coin changing and greedy solutions. Journal of the ACM 23(3), 418–422 (1976)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Protasi, M., Talamo, M.: A new probabilistic model for the study of algorithmic properties of random graph problems. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 360–367. Springer, Heidelberg (1983)Google Scholar
  17. 17.
    Slavik, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25(2), 237–254 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Brown, D.: A probabilistic analysis of a greedy algorithm arising from computational biology. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 206–207. ACM Press, New York (2001)Google Scholar
  19. 19.
    Goldberg, A.V., Marchetti-Spaccamela, A.: On finding the exact solution of a zero-one knapsack problem. In: Proceedings of the 16th ACM Symposium on Theory of Computing, pp. 359–368 (1984)Google Scholar
  20. 20.
    Downey, R., Fellows, M.: Parameterized complexity. Springer, Heidelberg (1999)Google Scholar
  21. 21.
    Levin, L.: Average case complete problems. SIAM Journal on Computing (1986)Google Scholar
  22. 22.
    Spielman, D., Teng, S.: Smoothed analysis: Why the simplex algorithm usually takes polynomial time. Journal of the ACM 51(3), 385–463 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hemaspaandra, E., Hemaspaandra, L.A.: Computational politics: Electoral systems. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 64–83. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  24. 24.
    Spakowski, H., Vogel, J.: Θ2 p-completeness: A classical approach for new results. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 348–360. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  25. 25.
    Spakowski, H., Vogel, J.: The complexity of Kemeny’s voting system. In: Proceedings of the Workshop Argentino de Informática Teórica. Anales Jornadas Argentinas de Informática e Investigación Operativa, vol. 30, pp. 157–168. SADIO (2001)Google Scholar
  26. 26.
    Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Technical Report Math/Inf/14/03, Institut für Informatik, Friedrich-Schiller-Universität, Jena, Germany (2003)Google Scholar
  27. 27.
    Wagner, K.: More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science 51(1–2), 53–80 (1987)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The boolean hierarchy I: Structural properties. SIAM Journal on Computing 17(6), 1232–1252 (1988)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kadin, J.: The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing 17(6), 1263–1282 (1988); Erratum appears in the same journal, 20(2), 404Google Scholar
  30. 30.
    Raffaelli, G., Marsili, M.: Statistical mechanics model for the emergence of consensus. Physical Review E 72(1), 016114 (2005)Google Scholar
  31. 31.
    Chernoff, H.: A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics 23, 493–509 (1952)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley–Interscience, Chichester (2000)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christopher M. Homan
    • 1
  • Lane A. Hemaspaandra
    • 2
  1. 1.Rochester Institute of TechnologyRochesterUSA
  2. 2.University of RochesterRochesterUSA

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