High-multiplicity election problems
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The computational study of elections generally assumes that the preferences of the electorate come in as a list of votes. Depending on the context, it may be much more natural to represent the list succinctly, as the distinct votes of the electorate and their counts, i.e., high-multiplicity representation. We consider how this representation affects the complexity of election problems. High-multiplicity representation may be exponentially smaller than standard representation, and so many polynomial-time algorithms for election problems in standard representation become exponential-time. Surprisingly, for polynomial-time election problems, we are often able to either adapt the same approach or provide new algorithms to show that these problems remain polynomial-time in the high-multiplicity case; this is in sharp contrast to the case where each voter has a weight, where the complexity usually increases. In the process we explore the relationship between high-multiplicity scheduling and manipulation of high-multiplicity elections. And we show that for any fixed set of job lengths, high-multiplicity scheduling on uniform parallel machines is in P, which was previously known for only two job lengths. We did not find any natural case where a polynomial-time election problem does not remain in P when moving to high-multiplicity representation. However, we found one natural NP-hard election problem where the complexity does increase, namely winner determination for Kemeny elections.
KeywordsComputational social choice Elections Manipulative actions High-multiplicity representation Scheduling
We thank the referees for their many helpful comments and suggestions. And we thank the AAAI-17 Student Abstract referees for their helpful comments and suggestions on our preliminary work on this topic . This work was supported in part by a National Science Foundation Graduate Research Fellowship under NSF Grant No. DGE-1102937.
- 1.Bachrach, Y., Lev, O., Lewenberg, Y., & Zick, Y. (July 2016). Misrepresentation in district voting. In Proceedings of the 25th international joint conference on artificial intelligence (pp. 81–87). IJCAI/AAAI Press.Google Scholar
- 2.Bartholdi, J., III., Tovey, C., & Trick, M. (1989). The computational difficulty of manipulating an election. Social Choice and Welfare, 6(3), 227–241.Google Scholar
- 3.Bartholdi, J., III., Tovey, C., & Trick, M. (1992). How hard is it to control an election? Mathematical and Computer Modeling, 16(8/9), 27–40.Google Scholar
- 5.Betzler, N., Niedermeier, R., & Woeginger, G. (July 2011). Unweighted coalitional manipulation under the Borda rule is NP-hard. In Proceedings of the 22nd international joint conference on artificial intelligence (pp. 55–60). IJCAI/AAAI Press.Google Scholar
- 9.Conitzer, V., Sandholm, T., & Lang, J. (2007). When are elections with few candidates hard to manipulate? Journal of the ACM, 54(3), Article 14.Google Scholar
- 10.Davies, J., Katsirelos, G., Narodytska, N., & Walsh, T. (August 2011). Complexity and algorithms for Borda manipulation. In Proceedings of the 25th AAAI conference on artificial intelligence (pp. 657–662). AAAI Press.Google Scholar
- 11.Dwork, C., Kumar, R., Naor, M., & Sivakumar, D. (March 2001). Rank aggregation methods for the web. In Proceedings of the 10th international world wide web conference (pp. 613–622). ACM Press.Google Scholar
- 16.Fitzsimmons, Z., & Hemaspaandra, E. (February 2017). The complexity of succinct elections. In Proceedings of the 31st AAAI conference on artificial intelligence (Student Abstract) (pp. 4921–4922). AAAI Press.Google Scholar
- 17.Fitzsimmons, Z., & Hemaspaandra, E. (July 2018). High-multiplicity election problems. In Proceedings of the 17th international conference on autonomous agents and multiagent systems (pp. 1558–1566). IFAAMAS.Google Scholar
- 19.Goemans, M., & Rothvoß, T. (January 2014). Polynomiality for bin packing with a constant number of item types. In Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms (pp. 830–839). SIAM.Google Scholar
- 24.Hemaspaandra, E., Hemaspaandra, L., & Schnoor, H. (July 2014). A control dichotomy for pure scoring rules. In Proceedings of the 28th AAAI conference on artificial intelligence (pp. 712–720). AAAI Press.Google Scholar
- 25.Hemaspaandra, E., Hemaspaandra, L., & Schnoor, H. (April 2014). A control dichotomy for pure scoring rules. Technical Report. arXiv:1404.4560 [cs.GT].
- 26.Hemaspaandra, E., & Schnoor, H. (April 2016). Complexity dichotomies for unweighted scoring rules. Technical Report. arXiv:1604.05264 [cs.CC].
- 27.Hemaspaandra, E., & Schnoor, H. (August/September 2016). Dichotomy for pure scoring rules under manipulative electoral actions. In Proceedings of the 22nd European conference on artificial intelligence (pp. 1071–1079). IOS Press.Google Scholar
- 30.Karp, R. (March 1972). Reducibilities among combinatorial problems. In Proceedings of a symposium on the complexity of computer computations (pp. 85–103). Plenum Press.Google Scholar
- 31.Kemeny, J. (1959). Mathematics without numbers. Daedalus, 88, 577–591.Google Scholar
- 32.Lenstra, H., Jr. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 538–548.Google Scholar
- 34.Lin, A. (2012). Solving hard problems in election systems. PhD thesis, Rochester Institute of Technology, Rochester, NY.Google Scholar
- 35.Mattei, N., & Walsh, T. (November 2013). PrefLib: A library for preferences. In Proceedings of the 3rd international conference on algorithmic decision theory (pp. 259–270). Springer.Google Scholar
- 38.Russell, N. (2007). Complexity of control of Borda count elections. Master’s thesis, Rochester Institute of Technology.Google Scholar
- 40.Xia, L., Conitzer, V., & Procaccia, A. (June 2010). A scheduling approach to coalitional manipulation. In Proceedings of the 11th ACM conference on electronic commerce (pp. 275–284). ACM Press.Google Scholar