## 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

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.

This can also be thought of as specifying the number of manipulators,

*k*, in unary.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.Note that the weight of a vertex-weighted graph is the sum of the weights of each of its vertices.

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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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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DOI: https://doi.org/10.1007/s10458-019-09410-4

### Keywords

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