Elections with Few Candidates: Prices, Weights, and Covering Problems

  • Robert Bredereck
  • Piotr Faliszewski
  • Rolf Niedermeier
  • Piotr Skowron
  • Nimrod Talmon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9346)


We show that a number of election-related problems with prices (such as, for example, bribery) are fixed-parameter tractable (in \({\mathsf {FPT}}\)) when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems. Our results follow by a general technique that formulates voting problems as covering problems and extends the classic approach of using integer linear programming and the algorithm of Lenstra [19]. In this context, our central result is that Weighted Set Multicover parameterized by the universe size is fixed-parameter tractable. Our approach is also applicable to weighted electoral control for Approval voting. We improve previously known \({\mathsf {XP}}\)-memberships to \({\mathsf {FPT}}\)-memberships. Our preliminary experiments on real-world-based data show the practical usefulness of our approach for instances with few candidates.


  1. 1.
    Bartholdi III, J.J., Tovey, C.A., Trick, M.A.: The computational difficulty of manipulating an election. Soc. Choice Welf. 6(3), 227–241 (1989)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bartholdi III, J.J., Tovey, C.A., Trick, M.A.: How hard is it to control an election. Math. Comput. Model. 16(8–9), 27–40 (1992)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Baumeister, D., Faliszewski, P., Lang, J., Rothe, J.: Campaigns for lazy voters: truncated ballots. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2012), pp. 577–584, June 2012Google Scholar
  4. 4.
    Bredereck, R., Chen, J., Faliszewski, P., Guo, J., Niedermeier, R., Woeginger, G.J.: Parameterized algorithmics for computational social choice: nine research challenges. Tsinghua Sci. Technol. 19(4), 358–373 (2014a)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bredereck, R., Chen, J., Faliszewski, P., Nichterlein, A., Niedermeier, R.: Prices matter for the parameterized complexity of shift bribery. In: Proceedings of The Twenty-Eighth AAAI Conference on Artificial Intelligence (AAAI 2014), pp. 1398–1404 (2014b)Google Scholar
  6. 6.
    Conitzer, V., Sandholm, T., Lang, J.: When are elections with few candidates hard to manipulate? J. ACM 54(3), 1–33 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization lower bounds through colors and IDs. ACM Trans. Algorithm. 11(2), 13:1–13:20 (2014)MATHMathSciNetGoogle Scholar
  8. 8.
    Dorn, B., Schlotter, I.: Multivariate complexity analysis of swap bribery. Algorithmica 64(1), 126–151 (2012)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013)MATHCrossRefGoogle Scholar
  10. 10.
    Elkind, E., Faliszewski, P., Slinko, A.: Swap bribery. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 299–310. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  11. 11.
    Erdélyi, G., Fellows, M., Rothe, J., Schend, L.: Control complexity in Bucklin and fallback voting: an experimental analysis. J. Comput. Syst. Sci. 81(4), 661–670 (2015)CrossRefGoogle Scholar
  12. 12.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: How hard is bribery in elections? J. Artif. Intell. Res. 35, 485–532 (2009)MathSciNetGoogle Scholar
  13. 13.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: Multimode control attacks on elections. J. Artif. Intell. Res. 40, 305–351 (2011)MATHMathSciNetGoogle Scholar
  14. 14.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: Weighted electoral control. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2013), pp. 367–374 (2013)Google Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  16. 16.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Anyone but him: the complexity of precluding an alternative. Artif. Intell. 171(5–6), 255–285 (2007)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Mattei, N., Walsh, T.: PrefLib: a library for preferences In: Perny, P., Pirlot, M., Tsoukiàs, A. (eds.) ADT 2013. LNCS, vol. 8176, pp. 259–270. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  21. 21.
    Miasko, T., Faliszewski, P.: The complexity of priced control in elections. Manuscript (2014).
  22. 22.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)MATHCrossRefGoogle Scholar
  23. 23.
    Schlotter, I., Faliszewski, P., Elkind, E.: Campaign management under approval-driven voting rules. In: Proceedings of the Twenty-Fifth Conference on Artificial Intelligence (AAAI 2011), pp. 726–731, August 2011Google Scholar
  24. 24.
    Xia, L.: Computing the margin of victory for various voting rules. In: Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), pp. 982–999, June 2012Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Robert Bredereck
    • 1
  • Piotr Faliszewski
    • 2
  • Rolf Niedermeier
    • 1
  • Piotr Skowron
    • 3
  • Nimrod Talmon
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.AGH University of Science and TechnologyKrakowPoland
  3. 3.University of WarsawWarsawPoland

Personalised recommendations