## 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).

This is a preview of subscription content, access via your institution.

## Notes

- 1.
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.)

- 2.
For function \(f_i^k\) it is never necessary to shift

*d*to position lower than \(k+1\), which is why we consider \(j \in \{0\} \cup [k]\). - 3.
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. - 4.
For binary encoding of the prices and the scores we give an \(\mathrm {NP}\)-completeness result.

## References

- 1.
Baumeister, D., Erdélyi, G., & Rothe, J. (2011). How hard is it to bribe judges? A study of the complexity of bribery in judgement aggregation. In

*Proceedings of the 2nd international conference on algorithmic decision theory*(pp. 1–15). - 2.
Baumeister, D., Faliszewski, P., Lang, J., & Rothe, J. (2012). Campaigns for lazy voters: Truncated ballots. In

*Proceedings of the 11th international conference on autonomous agents and multiagent systems*(pp. 577–584). - 3.
Binkele-Raible, D., Erdélyi, G., Fernau, H., Goldsmith, J., Mattei, N., & Rothe, J. (2014). The complexity of probabilistic lobbying.

*Discrete Optimization*,*11*, 1–21. - 4.
Bredereck, R., Chen, J., Faliszewski, P., Nichterlein, A., & Niedermeier, R. (2016). Prices matter for the parameterized complexity of shift bribery.

*Information and Computation*,*251*, 140–164. - 5.
Bredereck, R., Faliszewski, P., Kaczmarczyk, A., Niedermeier, R., Skowron, P., & Talmon, N. (2017). Robustness among multiwinner voting rules. In

*Proceedings of the 10th international symposium on algorithmic game theory*(pp. 80–92). - 6.
Bredereck, R., Faliszewski, P., Niedermeier, R., & Talmon, N. (2016). Large-scale election campaigns: Combinatorial shift bribery.

*Journal of Artificial Intelligence Research*,*55*, 603–652. - 7.
Cary, D. (2011). Estimating the margin of victory for instant-runoff voting. Presented at the 2011 Electronic Voting Technology Workshop/Workshop on Trustworthy Elections

- 8.
Christian, R., Fellows, M., Rosamond, F., & Slinko, A. (2007). On complexity of lobbying in multiple referenda.

*Review of Economic Design*,*11*(3), 217–224. - 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.
Cygan, M., Fomin, F., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., et al. (2015).

*Parameterized algorithms*. Berlin: Springer. - 11.
Dey, P., & Narahari, Y. (2015). Estimating the margin of victory of an election using sampling. In

*Proceedings of the 24th international joint conference on artificial intelligence*(pp. 1120–1126). - 12.
Dorn, B., & Krüger, D. (2016). On the hardness of bribery variants in voting with CP-nets.

*Annals of Mathematics and Artificial Intelligence*,*77*(3–4), 251–279. - 13.
Dorn, B., & Schlotter, I. (2012). Multivariate complexity analysis of swap bribery.

*Algorithmica*,*64*(1), 126–151. - 14.
Downey, R. G., & Fellows, M. R. (1995). Fixed-parameter tractability and completeness II: On completeness for \(W\)[1].

*Theoretical Computer Science*,*141*, 109–131. - 15.
Elkind, E., & Faliszewski, P. (2010). Approximation algorithms for campaign management. In

*Proceedings of the 6th international workshop on internet and network economics*(pp. 473–482). - 16.
Elkind, E., Faliszewski, P., & Slinko, A. (2009). Swap bribery. In

*Proceedings of the 2nd international symposium on algorithmic game theory*(pp. 299–310). - 17.
Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. (2009). How hard is bribery in elections?

*Journal of Artificial Intelligence Research*,*35*, 485–532. - 18.
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. - 19.
Faliszewski, P., Manurangsi, P., & Sornat, K. (2019). Approximation and hardness of shift-bribery. In

*Proceedings of the 33rd AAAI conference on artificial intelligence*(To appear). - 20.
Faliszewski, P., Reisch, Y., Rothe, J., & Schend, L. (2015). Complexity of manipulation, bribery, and campaign management in Bucklin and fallback voting.

*Autonomous Agents and Multiagent Systems*,*29*(6), 1091–1124. - 21.
Faliszewski, P., & Rothe, J. (2016). Control and bribery in voting. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.),

*Handbook of computational social choice*(pp. 146–168). Cambridge: Cambridge University Press. - 22.
Fellows, M. R., Hermelin, D., Rosamond, F., & Vialette, S. (2009). On the parameterized complexity of multiple-interval graph problems.

*Theoretical Computer Science*,*410*, 53–61. - 23.
Hazon, N., Lin, R., & Kraus, S. (2013). How to change a group’s collective decision? In

*Proceedings of the 23rd international joint conference on artificial intelligence*(pp. 198–205). - 24.
Hemaspaandra, E., Hemaspaandra, L., & Rothe, J. (2007). Anyone but him: The complexity of precluding an alternative.

*Artificial Intelligence*,*171*(5–6), 255–285. - 25.
Karp, R. (1972). Reducibilities among combinatorial problems. In

*Proceedings of a Symposium on the Complexity of Computer Computations*(pp. 85–103). - 26.
Knop, D., Koutecký, M., & Mnich, M. (2017). Voting and bribing in single-exponential time. In

*Proceedings of the 34th symposium on theoretical aspects of computer science*(pp. 46:1–46:14). - 27.
Knop, D., Koutecký, M., & Mnich, M. (2018). A unifying framework for manipulation problems. In

*Proceedings of the 17th international conference on autonomous agents and multiagent systems*(pp. 256–264). - 28.
Magrino, T., Rivest, R., Shen, E., & Wagner, D. (2011). Computing the margin of victory in IRV elections. Presented at 2011 Electronic Voting Technology Workshop/Workshop on Trushworthy Elections

- 29.
Maran, A., Maudet, N., Pini, M., Rossi, F., & Venable, K. (2013). A framework for aggregating influenced CP-nets and its resistance to bribery. In

*Proceedings of the 27th AAAI conference on artificial intelligence*(pp. 668–674). - 30.
Mattei, N., Pini, M., Rossi, F., & Venable, K. (2013). Bribery in voting with CP-nets.

*Annals of Mathematics and Artificial Intelligence*,*68*(1–3), 135–160. - 31.
Mattei, N., & Walsh, T. (2013). Preflib: A library for preferences. In

*Proceedings of the 3rd international conference on algorithmic decision theory*(pp. 259–270). - 32.
Maushagen, C., Nevelling, M., Rothe, J., & Selker, A. (2018). Complexity of shift bribery in iterative elections. In

*Proceedings of the 17th international conference on autonomous agents and multiagent systems*(pp. 1567–1575). - 33.
Nehama, I. (2015). Complexity of optimal lobbying in threshold aggregation. In

*Proceedings of the 4th international conference on algorithmic decision theory*(pp. 379–395). - 34.
Niedermeier, R. (2006).

*Invitation to fixed-parameter algorithms*. Oxford: Oxford University Press. - 35.
Obraztsova, S., & Elkind, E. (2011). On the complexity of voting manipulation under randomized tie-breaking. In

*Proceedings of the 22nd international joint conference on artificial intelligence*(pp. 319–324). - 36.
Obraztsova, S., Elkind, E., & Hazon, N. (2011). Ties matter: Complexity of voting manipulation revisited. In

*Proceedings of the 10th international conference on autonomous agents and multiagent systems*(pp. 71–78). - 37.
Papadimitriou, C. H. (1994).

*Computational complexity*. Boston: Addison-Wesley. - 38.
Pietrzak, K. (2003). On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems.

*Journal of Computer and System Sciences*,*4*, 757–771. - 39.
Schlotter, I., Faliszewski, P., & Elkind, E. (2017). Campaign management under approval-driven voting rules.

*Algorithmica*,*77*, 84–115. - 40.
Shiryaev, D., Yu, L., & Elkind, E. (2013). On elections with robust winners. In

*Proceedings of the 12th international conference on autonomous agents and multiagent systems*(pp. 415–422). - 41.
Xia, L. (2012). Computing the margin of victory for various voting rules. In

*Proceedings of the 13th ACM conference on electronic commerce*(pp. 982–999).

## 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).

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

An early version of this paper was presented at the AAMAS 2016 conference.

## Rights and permissions

## About this article

### Cite this article

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

Published:

Issue Date:

### Keywords

- Elections
- Bribery
- Computational complexity
- Copeland
- Scoring protocols
- Bucklin
- Maximin