Abstract
The Possible Winner problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes (partial orders) are extended into complete ones (linear orders) in a favorable way. Under the k-approval protocol, for every voter, the best k candidates of his or her preference order get one point. A candidate with maximum total number of points wins. The Possible Winner problem for k-approval is NP-complete even if there are only two votes (and k is part of the input). In addition, it is NP-complete for every fixed k ∈ {2, ..., m − 2} with m denoting the number of candidates if the number of votes is unbounded. We investigate the parameterized complexity with respect to the combined parameter k and “number of incomplete votes” t, and with respect to the combined parameter k′: = m − k and t. For both cases, we use kernelization to show fixed-parameter tractability. However, we show that whereas there is a polynomial-size problem kernel with respect to (t,k′), it is very unlikely that there is a polynomial-size kernel for (t,k). We provide additional fixed-parameter algorithms for some special cases.
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References
Bachrach, Y., Betzler, N., Faliszewski, P.: Probabilistic possible winner determination. In: Proc. of 24th AAAI (to appear 2010)
Baumeister, D., Rothe, J.: Taking the final step to a full dichotomy of the Possible Winner problem in pure scoring rules. In: Proc. of 19th ECAI (2010) (short paper)
Betzler, N., Dorn, B.: Towards a complexity dichotomy of finding possible winners in elections based on scoring rules. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 124–136. Springer, Heidelberg (2009)
Betzler, N., Guo, J., Niedermeier, R.: Parameterized computational complexity of Dodgson and Young elections. Inform. Comput. 208(2), 165–177 (2010)
Betzler, N., Hemmann, S., Niedermeier, R.: A multivariate complexity analysis of determining possible winners given incomplete votes. In: Proc. of 21st IJCAI, pp. 53–58 (2009)
Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)
Chor, B., Fellows, M., Juedes, D.W.: Linear kernels in linear time, or how to save k colors in o(n 2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Elkind, E., Faliszewski, P., Slinko, A.: Swap bribery. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) Algorithmic Game Theory. LNCS, vol. 5814, pp. 299–310. Springer, Heidelberg (2009)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proc. of 40th STOC, pp. 133–142. ACM, New York (2008)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)
Hemaspaandra, E., Hemaspaandra, L.A.: Dichotomy for voting systems. J. Comput. Syst. Sci. 73(1), 73–83 (2007)
Konczak, K., Lang, J.: Voting procedures with incomplete preferences. In: Proc. of IJCAI 2005 Multidisciplinary Workshop on Advances in Preference Handling (2005)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Pini, M.S., Rossi, F., Venable, K.B., Walsh, T.: Incompleteness and incomparability in preference aggregation. In: Proc. of 20th IJCAI, pp. 1464–1469 (2007)
Xia, L., Conitzer, V.: Determining possible and necessary winners under common voting rules given partial orders. In: Proc. of 23rd AAAI, pp. 196–201. AAAI Press, Menlo Park (2008)
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Betzler, N. (2010). On Problem Kernels for Possible Winner Determination under the k-Approval Protocol. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_12
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DOI: https://doi.org/10.1007/978-3-642-15155-2_12
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