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Multivariate Complexity Analysis of Swap Bribery

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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.

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References

  1. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 5425–5442 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  8. Christian, R., Fellows, M., Rosamond, F., Slinko, A.: On complexity of lobbying in multiple referenda. Rev. Econ. Des. 11(3), 217–224 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Conitzer, V., Sandholm, T., Lang, J.: When are elections with few candidates hard to manipulate? J. ACM 54(3), 1–33 (2007)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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. 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)

    MathSciNet  MATH  Google Scholar 

  17. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: Multimode control attacks on elections. J. Artif. Intell. Res. 40, 305–351 (2011)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, New York (2006)

    Google Scholar 

  20. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  21. Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)

    Article  Google Scholar 

  22. Hemaspaandra, E., Hemaspaandra, L.A.: Dichotomy for voting systems. J. Comput. Syst. Sci. 73(1), 73–83 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. Lenstra, H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  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. Xia, L., Conitzer, V.: Determining possible and necessary winners under common voting rules given partial orders. J. Artif. Intell. Res. 41, 25–67 (2011)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Ildikó Schlotter.

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I. Schlotter is supported by the Hungarian National Research Fund (OTKA 67651), and by the European Union and the European Social Fund (grant TÁMOP 4.2.1./B-09/1/KMR-2010-0003).

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Dorn, B., Schlotter, I. Multivariate Complexity Analysis of Swap Bribery. Algorithmica 64, 126–151 (2012). https://doi.org/10.1007/s00453-011-9568-4

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