Parameterized Enumeration of (Locally-) Optimal Aggregations

  • Naomi Nishimura
  • Narges Simjour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

We present a parameterized enumeration algorithm for Kemeny Rank Aggregation, the problem of determining an optimal aggregation, a total order that is at minimum total τ-distance (kt) from the input multi-set of m total orders (votes) over a set of alternatives (candidates), where the τ-distance between two total orders is the number of pairs of candidates ordered differently. Our \(O^*(4^{k_t\over m})\)-time algorithm constitutes a significant improvement over the previous \(O^*(36^{k_t\over m})\) upper bound.

The analysis of our algorithm relies on the notion of locally-optimal aggregations, total orders whose total τ-distances from the votes do not decrease by any single swap of two candidates adjacent in the ordering. As a consequence of our approach, we provide not only an upper bound of \(4^{k_t\over m}\) on the number of optimal aggregations, but also the first parameterized bound, \(4^{k_t\over m}\), on the number of locally-optimal aggregations, and demonstrate that it is tight. Furthermore, since our results rely on a known relation to Weighted Directed Feedback Arc Set, we obtain new results for this problem along the way.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. J. ACM 55(5), 1–27 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alon, N., Lokshtanov, D., Saurabh, S.: Fast fast. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Bartholdi, J.J., Tovey, C.A., Trick, M.A.: The computational difficulty of manipulating an election. Social Choice and Welfare 6(3), 227–241 (1989)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Betzler, N., Bredereck, R., Chen, J., Niedermeier, R.: Studies in computational aspects of voting - a parameterized complexity perspective. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift 2012. LNCS, vol. 7370, pp. 318–363. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Betzler, N., Bredereck, R., Niedermeier, R.: Partial kernelization for rank aggregation: Theory and experiments. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 26–37. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Betzler, N., Fellows, M.R., Guo, J., Niedermeier, R., Rosamond, F.A.: Fixed-parameter algorithms for kemeny scores. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 60–71. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Betzler, N., Fellows, M.R., Guo, J., Niedermeier, R., Rosamond, F.A.: Fixed-parameter algorithms for Kemeny rankings. Theor. Comput. Sci. 410(45), 4554–4570 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Biedl, T.C., Brandenburg, F., Deng, X.: On the complexity of crossings in permutations. Discrete Mathematics 309(7), 1813–1823 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Borda, J.: Mémoire sur les élections au scrutin. Histoire de l’Académie Royale des Sciences (1781)Google Scholar
  10. 10.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5) (2008)Google Scholar
  11. 11.
    Condorcet, M.: Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. L’imprimerie royale (1785)Google Scholar
  12. 12.
    Conitzer, V., Davenport, A., Kalagnanam, J.: Improved bounds for computing Kemeny rankings. In: AAAI 2006: Proc. of the 21st Nat. Conf. on Artificial Intelligence, vol. 1, pp. 620–626 (2006)Google Scholar
  13. 13.
    Coppersmith, D., Fleischer, L., Rudra, A.: Ordering by weighted number of wins gives a good ranking for weighted tournaments. In: SODA 2006: Proc. of the 17th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 776–782 (2006)Google Scholar
  14. 14.
    Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the web. In: WWW 2001: Proc. of the 10th Int. Conf. on World Wide Web, pp. 613–622 (2001)Google Scholar
  15. 15.
    Ephrati, E., Rosenschein, J.S.: The Clarke tax as a consensus mechanism among automated agents. In: AAAI 1991: Proc. of the 9th Nat. Conf. on Artificial Intelligence, vol. 1, pp. 173–178 (1991)Google Scholar
  16. 16.
    Fernau, H., Fomin, F.V., Lokshtanov, D., Mnich, M., Philip, G., Saurabh, S.: Ranking and drawing in subexponential time. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 337–348. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Theor. Comput. Sci. 349(3), 382–391 (2005)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jackson, B.N., Schnable, P.S., Aluru, S.: Consensus genetic maps as median orders from inconsistent sources. IEEE/ACM Trans. Comput. Biol. Bioinformatics 5(2), 161–171 (2008)CrossRefGoogle Scholar
  19. 19.
    Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 3–14. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Kemeny, J.G.: Mathematics without numbers. Daedalus 88, 575–591 (1959)Google Scholar
  21. 21.
    Kenyon-Mathieu, C., Schudy, W.: How to rank with few errors. In: STOC 2007: Proc. of the 39th Annual ACM Symp. on Theory of Computing, pp. 95–103 (2007)Google Scholar
  22. 22.
    Simjour, N.: Improved parameterized algorithms for the Kemeny aggregation problem. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 312–323. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  23. 23.
    Van Zuylen, A., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. Mathematics of Operations Research 34(3), 594–620 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Naomi Nishimura
    • 1
  • Narges Simjour
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations