Fixed-Parameter Algorithms for Kemeny Scores

  • Nadja Betzler
  • Michael R. Fellows
  • Jiong Guo
  • Rolf Niedermeier
  • Frances A. Rosamond
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5034)

Abstract

The Kemeny Score problem is central to many applications in the context of rank aggregation. Given a set of permutations (votes) over a set of candidates, one searches for a “consensus permutation” that is “closest” to the given set of permutations. Computing an optimal consensus permutation is NP-hard. We provide first, encouraging fixed-parameter tractability results for computing optimal scores (that is, the overall distance of an optimal consensus permutation). Our fixed-parameter algorithms employ the parameters “score of the consensus”, “maximum distance between two input permutations”, and “number of candidates”. We extend our results to votes with ties and incomplete votes, thus, in both cases having no longer permutations as input.

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References

  1. 1.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. In: Proc. 37th STOC, pp. 684–693. ACM, New York (2005)Google Scholar
  2. 2.
    Bartholdi III, J., Tovey, C.A., Trick, M.A.: 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.
    Chen, J.: Personal communication (December 2007)Google Scholar
  4. 4.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: Proc. 40th STOC, ACM, New York (2008)Google Scholar
  5. 5.
    Conitzer, V., Davenport, A., Kalagnanam, J.: Improved bounds for computing Kemeny rankings. In: Proc. 21st AAAI, pp. 620–626 (2006)Google Scholar
  6. 6.
    Davenport, A., Kalagnanam, J.: A computational study of the Kemeny rule for preference aggregation. In: Proc. 19th AAAI, pp. 697–702 (2004)Google Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the Web. In: Proc. WWW, pp. 613–622 (2001)Google Scholar
  9. 9.
    Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation revisited (manuscript, 2001)Google Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  11. 11.
    Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Theoretical Computer Science 349, 382–391 (2005)MATHMathSciNetGoogle Scholar
  12. 12.
    Kenyon-Mathieu, C., Schudy, W.: How to rank with few errors. In: Proc. 39th STOC, pp. 95–103. ACM, New York (2007)Google Scholar
  13. 13.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  14. 14.
    Raman, V., Saurabh, S.: Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG. Information Processing Letters 104(2), 65–72 (2007)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Truchon, M.: An extension of the Condorcet criterion and Kemeny orders. Technical report, cahier 98-15 du Centre de Recherche en Économie et Finance Appliquées (1998)Google Scholar
  16. 16.
    van Zuylen, A., Williamson, D.P.: Deterministic algorithms for rank aggregation and other ranking and clustering problems. In: Proc. 5th WAOA. LNCS, vol. 4927, pp. 260–273. Springer, Heidelberg (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nadja Betzler
    • 1
  • Michael R. Fellows
    • 2
  • Jiong Guo
    • 1
  • Rolf Niedermeier
    • 1
  • Frances A. Rosamond
    • 2
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.PC Research Unit, Office of DVC (Research)University of NewcastleCallaghanAustralia

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