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High-multiplicity election problems

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Abstract

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.

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Notes

  1. Xia, Conitzer, and Procaccia [40] have a notion of “divisible” weights, but they allow noninteger divisions and so this is very different from high-multiplicity. It is interesting that their paper reduces manipulation for this notion to a scheduling problem, though a different scheduling problem than what we use.

  2. This can also be thought of as specifying the number of manipulators, k, in unary.

  3. Notice that the ILP above checks that if we add the maximum number of voters such that the score of each of the non-p candidates is at most \(s_p + j\alpha \), then we have added at least j voters. This implies that we can add j voters in such a way that p is still a winner.

  4. Note that the weight of a vertex-weighted graph is the sum of the weights of each of its vertices.

  5. For two disjoint graphs \(G_1\) and \(G_2\), the join \(G_1 + G_2\) is defined as follows: \(V(G_1 + G_2) = V(G_1) \cup V(G_2)\) and \(E(G_1 + G_2) = E(G_1) \cup E(G_2) \cup \{\{v,w\} \ | \ v \in V(G_1), w \in V(G_2)\}\).

References

  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.

  2. Bartholdi, J., III., Tovey, C., & Trick, M. (1989). The computational difficulty of manipulating an election. Social Choice and Welfare, 6(3), 227–241.

  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.

  4. Betzler, N., & Dorn, B. (2010). Towards a dichotomy of finding possible winners in elections based on scoring rules. Journal of Computer and System Sciences, 76(8), 812–836.

    Article  MathSciNet  MATH  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.

  6. Borosh, I., & Treybig, L. (1976). Bounds on positive integral solutions of linear Diophantine equations. Proceedings of the American Mathematical Society, 55(2), 299–304.

    Article  MathSciNet  MATH  Google Scholar 

  7. Brandt, F., Conitzer, V., Endriss, U., Lang, J., & Procaccia, A. (2016). Handbook of Computational Social Choice. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  8. Clifford, J., & Posner, M. (2001). Parallel machine scheduling with high multiplicity. Mathematical Programming, 89(3), 359–383.

    Article  MathSciNet  MATH  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.

  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.

  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.

  12. Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. (2009). How hard is bribery in elections? Journal of Artificial Intelligence Research, 35, 485–532.

    Article  MathSciNet  MATH  Google Scholar 

  13. Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. (2015). Weighted electoral control. Journal of Artificial Intelligence Research, 52, 507–542.

    Article  MathSciNet  MATH  Google Scholar 

  14. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., & Rothe, J. (2009). Llull and Copeland voting computationally resist bribery and constructive control. Journal of Artificial Intelligence Research, 35, 275–341.

    Article  MathSciNet  MATH  Google Scholar 

  15. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., & Rothe, J. (2011). The shield that never was: Societies with single-peaked preferences are more open to manipulation and control. Information and Computation, 209(2), 89–107.

    Article  MathSciNet  MATH  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.

  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.

  18. Garey, M., & Johnson, D. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco: W. H. Freeman and Company.

    MATH  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.

  20. Hemachandra, L. (1989). The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3), 299–322.

    Article  MathSciNet  MATH  Google Scholar 

  21. Hemaspaandra, E., & Hemaspaandra, L. (2007). Dichotomy for voting systems. Journal of Computer and System Sciences, 73(1), 73–83.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hemaspaandra, E., Hemaspaandra, L., & Rothe, J. (2009). Hybrid elections broaden complexity-theoretic resistance to control. Mathematical Logic Quarterly, 55(4), 397–424.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hemaspaandra, E., Hemaspaandra, L., & Rothe, J. (2014). The complexity of online manipulation of sequential elections. Journal of Computer and System Sciences, 80(4), 697–710.

    Article  MathSciNet  MATH  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.

  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.

  28. Hemaspaandra, E., Spakowski, H., & Vogel, J. (2005). The complexity of Kemeny elections. Theoretical Computer Science, 349(3), 382–391.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hochbaum, D., & Shamir, R. (1991). Strongly polynomial algorithms for the high multiplicity scheduling problem. Operations Research, 39(4), 648–653.

    Article  MATH  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.

  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.

  33. Leung, J. (1982). On scheduling independent tasks with restricted execution times. Operations Research, 30(1), 163–171.

    Article  MATH  Google Scholar 

  34. Lin, A. (2012). Solving hard problems in election systems. PhD thesis, Rochester Institute of Technology, Rochester, NY.

  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.

  36. McCormick, T., Smallwood, S., & Spieksma, F. (2001). A polynomial algorithm for multiprocessor scheduling with two job lengths. Mathematics of Operations Research, 26(1), 31–49.

    Article  MathSciNet  MATH  Google Scholar 

  37. McGarvey, D. (1953). A theorem on the construction of voting paradoxes. Econometrica, 21(4), 608–610.

    Article  MathSciNet  Google Scholar 

  38. Russell, N. (2007). Complexity of control of Borda count elections. Master’s thesis, Rochester Institute of Technology.

  39. Schrijver, A. (2003). Combinatorial optimization: Polyhedra and efficiency. Berlin: Springer.

    MATH  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.

  41. Young, H., & Levenglick, A. (1978). A consistent extension of Condorcet’s election principle. SIAM Journal on Applied Mathematics, 35(2), 285–300.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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 [16]. This work was supported in part by a National Science Foundation Graduate Research Fellowship under NSF Grant No. DGE-1102937.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA, 01610, USA

    Zack Fitzsimmons

  2. Department of Computer Science, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Edith Hemaspaandra

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  1. Zack Fitzsimmons
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  2. Edith Hemaspaandra
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Correspondence to Zack Fitzsimmons.

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The conference version of this paper [17] appears in the Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems.

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Fitzsimmons, Z., Hemaspaandra, E. High-multiplicity election problems. Auton Agent Multi-Agent Syst 33, 383–402 (2019). https://doi.org/10.1007/s10458-019-09410-4

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