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Schulze and ranked-pairs voting are fixed-parameter tractable to bribe, manipulate, and control

  • Lane A. Hemaspaandra
  • Rahman Lavaee
  • Curtis Menton
Article

Abstract

Schulze and ranked-pairs elections have received much attention recently, and the former has quickly become a quite widely used election system. For many cases these systems have been proven resistant to bribery, control, or manipulation, with ranked pairs being particularly praised for being NP-hard for all three of those. Nonetheless, the present paper shows that with respect to the number of candidates, Schulze and ranked-pairs elections are fixed-parameter tractable to bribe, control, and manipulate: we obtain uniform, polynomial-time algorithms whose running times’ degrees do not depend on the number of candidates. We also provide such algorithms for some weighted variants of these problems.

Keywords

Elections Computational social choice 

Mathematics Subject Classification (2010)

91B12 91B14 91B10 68Q17 

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References

  1. 1.
    Bartholdi III, J., Tovey, C., Trick, M.: The computational difficulty of manipulating an election. Soc. Choice Welf. 6(3), 227–241 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartholdi III, J., Tovey, C., Trick, M.: How hard is it to control an election? Math. Comput. Model. 16(8/9), 27–40 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Betzler, N.: A multivariate complexity analysis of voting problems. Ph.D. thesis, Friedrich-Schiller-Universität Jena, Jena, Germany (2010)Google Scholar
  4. 4.
    Betzler, N., Bredereck, R., Chen, J., Niedermeier, R.: Studies in computational aspects of voting—A parameterized complexity perspective. In: The Multivariate Algorithmic Revolution and Beyond, 318–363. Springer-Verlag Lecture Notes in Computer Science #7370 (2012)Google Scholar
  5. 5.
    Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 43–53 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brill, M., Fischer, F.: The price of neutrality for the ranked pairs method. In: Proceedings of the 26th AAAI Conference on Artificial Intelligence, pp. 1299–1305. AAAI Press (2012)Google Scholar
  7. 7.
    Conitzer, V., Sandholm, T., Lang, J.: When are elections with few candidates hard to manipulate? J. ACM 54(3) (2007). Article 14Google Scholar
  8. 8.
    Dorn, B., Schlotter, I.: Multivariate complexity analysis of swap bribery. Algorithmica 64(1), 126–151 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer-Verlag (1999)Google Scholar
  10. 10.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: How hard is bribery in elections? J. Artif. Intell. Res. 35, 485–532 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: Multimode control attacks on elections. J. Artif. Intell. Res. 40, 305–351 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: Weighted electoral control. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, 367–374. International Foundation for Autonomous Agents and Multiagent Systems (2013)Google Scholar
  13. 13.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: The complexity of manipulative attacks in nearly single-peaked electorates. Artif. Intell. 207, 69–99 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Llull and Copeland voting computationally resist bribery and constructive control. J. Artif. Intell. Res. 35, 275–341 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer-Verlag (2006)Google Scholar
  16. 16.
    Gaspers, S., Kalinowski, T., Narodytska, N., Walsh, T.: Coalitional manipulation for Schulze’s rule. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, 431–438. International Foundation for Autonomous Agents and Multiagent Systems (2013)Google Scholar
  17. 17.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  18. 18.
    Hemaspaandra, E., Hemaspaandra, L., Menton, C.: Search versus decision for election manipulation problems. In: Proceedings of the 30th Annual Symposium on Theoretical Aspects of Computer Science, 377–388. Leibniz International Proceedings in Informatics (LIPIcs) (2013)Google Scholar
  19. 19.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Anyone but him: The complexity of precluding an alternative. Artif. Intell. 171(5–6), 255–285 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hemaspaandra, L., Lavaee, R., Menton, C.: Schulze and ranked-pairs voting are fixed-parameter tractable to bribe, manipulate, and control. Tech. Rep. arXiv: http://arxiv.org/abs/1301.6118 [cs.GT], Computing Res. Rep., http://arxiv.org/corr/ Revised June 2014 (2012)
  21. 21.
    Hemaspaandra, L., Lavaee, R., Menton, C.: Schulze and ranked-pairs voting are fixed-parameter tractable to bribe, manipulate, and control. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, 1345–1346. International Foundation for Autonomous Agents and Multiagent Systems (2013)Google Scholar
  22. 22.
    Cai, L., Chen, J., Downey, R., Fellows, M.: Advice classes of parameterized tractability. Annals of Pure and Applied Logic 84(1), 119–138 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lenstra Jr., H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lin, A.: Solving hard problems in election systems. Ph.D. thesis, Rochester Institute of Technology, Rochester, NY (2012)Google Scholar
  25. 25.
    Liu, H., Zhu, D.: Parameterized complexity of control problems in maximin election. Inf. Process. Lett. 110(10), 383–388 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    McGarvey, D.: A theorem on the construction of voting paradoxes. Econometrica 21(4), 608–610 (1953)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Menton, C., Singh, P.: Manipulation and control complexity of Schulze voting. Tech. Rep. arXiv: http://arxiv.org/abs/1206.2111v1 (version 1) [cs.GT], Computing Research Repository, http://arxiv.org/corr/ (2012)
  28. 28.
    Menton, C., Singh, P.: Control complexity of Schulze voting. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, 286–292. AAAI Press (2013)Google Scholar
  29. 29.
    Menton, C., Singh, P.: Manipulation and control complexity of Schulze voting. Tech. Rep. arXiv: http://arxiv.org/abs/1206.2111 (version 4) [cs.GT], Computing Research Repository, http://arxiv.org/corr/ (2013)
  30. 30.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Parkes, D., Xia, L.: A complexity-of-strategic-behavior comparison between Schulze’s rule and ranked pairs. In: Proceedings of the 26th AAAI Conference on Artificial Intelligence, 1429–1435. AAAI Press (2012)Google Scholar
  32. 32.
    Rothe, J., Schend, L.: Challenges to complexity shields that are supposed to protect elections against manipulation and control: A survey. Ann. Math. Artif. Intell. 68(1–3), 161–193 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Russell, N.: Complexity of control of Borda count elections. Master’s thesis, Rochester Institute of Technology (2007)Google Scholar
  34. 34.
    Schulze, M.: A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method. Soc. Choice Welf. 36, 267–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tideman, T.: Collective Decisions and Voting: The Potential for Public Choice. Ashgate Publishing (2006)Google Scholar
  36. 36.
    Wikipedia: Schulze method en.wikipedia.org/wiki/Schulze_method (2013)
  37. 37.
    Xia, L.: Computing the margin of victory for various voting rules. In: Proceedings of the 13th ACM Conference on Electronic Commerce, 982–999 (2012)Google Scholar
  38. 38.
    Xia, L., Zuckerman, M., Procaccia, A., Conitzer, V., Rosenschein, J.: Complexity of unweighted manipulation under some common voting rules. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence, 348–353 (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Rahman Lavaee
    • 1
  • Curtis Menton
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Google Inc.Mountain ViewUSA

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