## Abstract

We study the complexity of Destructive Shift Bribery. In this problem, we are given an election with a set of candidates and a set of voters (each ranking the candidates from the best to the worst), a despised candidate *d*, a budget *B*, and prices for shifting *d* back in the voters’ rankings. The goal is to ensure that *d* is not a winner of the election. We show that this problem is polynomial-time solvable for scoring protocols (encoded in unary), the Bucklin and Simplified Bucklin rules, and the Maximin rule, but is \(\mathrm {NP}\)-hard for the Copeland rule. This stands in contrast to the results for the constructive setting (known from the literature), for which the problem is polynomial-time solvable for *k*-Approval family of rules, but is \(\mathrm {NP}\)-hard for the Borda, Copeland, and Maximin rules. We complement the analysis of the Copeland rule showing \(\mathrm {W}\)-hardness for the parameterization by the budget value, and by the number of affected voters. We prove that the problem is \(\mathrm {W}\)-hard when parameterized by the number of voters even for unit prices. From the positive perspective we provide an efficient algorithm for solving the problem parameterized by the combined parameter the number of candidates and the maximum bribery price (alternatively the number of different bribery prices).

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## Notes

The authors’ definition of the Condorcet rule slightly differs from the standard one usually seen in the literature. They assume that if there is no unique Condorcet winner the rule returns the whole set of candidates instead of an empty set (well established as a return value for this case in the literature.)

For function \(f_i^k\) it is never necessary to shift

*d*to position lower than \(k+1\), which is why we consider \(j \in \{0\} \cup [k]\).If the bribery price functions allow for shifting

*d*back at zero cost in some votes, we shift*d*as much as possible at zero cost as a preprocessing step.For binary encoding of the prices and the scores we give an \(\mathrm {NP}\)-completeness result.

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## Acknowledgements

We are grateful to the AAMAS and JAAMAS reviewers for their helpful comments. Piotr Faliszewski was supported by the AGH University Grant 11.11.230.124 (statutory research). Andrzej Kaczmarczyk was partially supported by the AGH University Grant 11.11.230.124 (statutory research) and partially by the DFG Project AFFA (BR 5207/1 and NI 369/15).

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Kaczmarczyk, A., Faliszewski, P. Algorithms for destructive shift bribery.
*Auton Agent Multi-Agent Syst* **33**, 275–297 (2019). https://doi.org/10.1007/s10458-019-09403-3

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DOI: https://doi.org/10.1007/s10458-019-09403-3