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Algorithmica

, Volume 64, Issue 1, pp 126–151 | Cite as

Multivariate Complexity Analysis of Swap Bribery

  • Britta Dorn
  • Ildikó SchlotterEmail author
Article

Abstract

We consider the computational complexity of a problem modeling bribery in the context of voting systems. In the scenario of Swap Bribery, each voter assigns a certain price for swapping the positions of two consecutive candidates in his preference ranking. The question is whether it is possible, without exceeding a given budget, to bribe the voters in a way that the preferred candidate wins in the election.

We initiate a parameterized and multivariate complexity analysis of Swap Bribery, focusing on the case of k-approval. We investigate how different cost functions affect the computational complexity of the problem. We identify a special case of k-approval for which the problem can be solved in polynomial time, whereas we prove NP-hardness for a slightly more general scenario. We obtain fixed-parameter tractability as well as W[1]-hardness results for certain natural parameters.

Keywords

Computational social choice Parameterized complexity Swap Bribery 

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References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Betzler, N.: On problem kernels for possible winner determination under the k-approval protocol. In: MFCS 2010: Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 6281, pp. 114–125. Springer, Berlin (2010) CrossRefGoogle Scholar
  3. 3.
    Betzler, N., Dorn, B.: Towards a dichotomy for the Possible Winner problem in elections based on scoring rules. J. Comput. Syst. Sci. 76, 812–836 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 5425–5442 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Betzler, N., Hemmann, S., Niedermeier, R.: A multivariate complexity analysis of determining possible winners given incomplete votes. In: IJCAI’09: Proceedings of the 21st International Joint Conference on Artificial Intelligence, pp. 53–58 (2009) Google Scholar
  6. 6.
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: IWPEC 2009: Proceedings of the 4th International Workshop on Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 5917, pp. 17–37. Springer, Berlin (2009) CrossRefGoogle Scholar
  7. 7.
    Brams, S.J., Fishburn, P.C.: Voting procedures. In: Arrow, K.J., Sen, A.K., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, vol. 1, pp. 173–236. Elsevier, Amsterdam (2002) CrossRefGoogle Scholar
  8. 8.
    Christian, R., Fellows, M., Rosamond, F., Slinko, A.: On complexity of lobbying in multiple referenda. Rev. Econ. Des. 11(3), 217–224 (2007) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Conitzer, V., Sandholm, T., Lang, J.: When are elections with few candidates hard to manipulate? J. ACM 54(3), 1–33 (2007) MathSciNetGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. II. On completeness for W[1]. Theor. Comput. Sci. 141(1–2), 109–131 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999) CrossRefGoogle Scholar
  12. 12.
    Elkind, E., Faliszewski, P.: Approximation algorithms for campaign management. In: WINE 2010: Proceedings of the 6th Workshop on Internet and Network Economics. Lecture Notes in Computer Science, vol. 6484, pp. 473–482. Springer, Berlin (2010) Google Scholar
  13. 13.
    Elkind, E., Faliszewski, P., Slinko, A.M.: Swap bribery. In: SAGT 2009: Proceedings of the Second International Symposium on Algorithmic Game Theory. Lecture Notes in Computer Science, vol. 5814, pp. 299–310. Springer, Berlin (2009) CrossRefGoogle Scholar
  14. 14.
    Faliszewski, P.: Nonuniform bribery. In: AAMAS 2008: 7th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 1569–1572 (2008) Google Scholar
  15. 15.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: The complexity of bribery in elections. In: AAAI’06: Proceedings of the 21st National Conference on Artificial Intelligence, pp. 641–646 (2006) Google Scholar
  16. 16.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Llull and Copeland voting computationally resist bribery and constructive control. J. Artif. Intell. Res. 35, 275–341 (2009) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: Multimode control attacks on elections. J. Artif. Intell. Res. 40, 305–351 (2011) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410, 53–61 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, New York (2006) Google Scholar
  20. 20.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962) zbMATHGoogle Scholar
  21. 21.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007) CrossRefGoogle Scholar
  22. 22.
    Hemaspaandra, E., Hemaspaandra, L.A.: Dichotomy for voting systems. J. Comput. Syst. Sci. 73(1), 73–83 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Konczak, K., Lang, J.: Voting procedures with incomplete preferences. In: Proceedings of the IJCAI-2005 Multidisciplinary Workshop on Advances in Preference Handling, pp. 124–129 (2005) Google Scholar
  24. 24.
    Lenstra, H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Liu, H., Feng, H., Zhu, D., Luan, J.: Parameterized computational complexity of control problems in voting systems. Theor. Comput. Sci. 410(27–29), 2746–2753 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006) zbMATHCrossRefGoogle Scholar
  27. 27.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: STACS 2010: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, pp. 17–32 (2010) Google Scholar
  28. 28.
    Xia, L., Conitzer, V.: Determining possible and necessary winners under common voting rules given partial orders. J. Artif. Intell. Res. 41, 25–67 (2011) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Fakultät für Mathematik und WirtschaftswissenschaftenUniversität UlmUlmGermany
  2. 2.Department of Computer Science and Information TheoryBudapest University of Technology and EconomicsBudapestHungary

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