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High-multiplicity election problems

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 \(E(G_1 + G_2) = E(G_1) \cup E(G_2) \cup \{\{v,w\} \ | \ v \in V(G_1), w \in V(G_2)\}\).

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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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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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Keywords

  • Computational social choice
  • Elections
  • Manipulative actions
  • High-multiplicity representation
  • Scheduling