Abstract
The notion of robustness in the context of committee elections was introduced by Bredereck et al. [SAGT 2018] [2] to capture the impact of small changes in the input preference orders, depending on the voting rules used. They show that for certain voting rules, such as Chamberlin-Courant, checking if an election instance is robust, even to the extent of a small constant, is computationally hard. More specifically, it is NP-hard to determine if one swap in any of the votes can change the set of winning committees with respect to the Chamberlin-Courant voting rule. Further, the problem is also \(\mathsf {W[1]}\)-hard when parameterized by the size of the committee, k. We complement this result by suggesting an algorithm that is in \(\mathsf {XP}\) with respect to k. We also show that on nearly-structured profiles, the problem of robustness remains NP-hard. We also address the case of approval ballots, where we show a hardness result analogous to the one established in [2] about rankings and again demonstrate an \(\mathsf {XP}\) algorithm.
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- 1.
We refer the reader to the section on Preliminaries for the definition of \(\ell \)-single-crossing domains. Some definitions and results are deferred to the full version due to lack of space and are marked with a \((\star )\).
- 2.
Note the slight abuse of terminology here: when referring to \(C_W\) as a hitting set, we are referring to the elements of U corresponding to the candidates in \(C_W\). As long as this is clear from the context, we will continue to use this convention.
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Misra, N., Sonar, C. (2019). Robustness Radius for Chamberlin-Courant on Restricted Domains. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_27
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