Given an election, a preferred candidate p, and a budget, the SHIFT BRIBERY problem asks whether p can win the election after shifting p higher in some voters' preference orders. Of course, shifting comes at a price (depending on the voter and on the extent of the shift) and one must not exceed the given budget. We study the (parameterized) computational complexity of S HIFT BRIBERY for multiwinner voting rules where winning the election means to be part of some winning committee. We focus on the well-established SNTV, Bloc, k-Borda, and Chamberlin-Courant rules, as well as on approximate variants of the Chamberlin-Courant rule, since the original rule is NP-hard to compute. We show that SHIFT BRIBERY tends to be harder in the multiwinner setting than in the single-winner one by showing settings where SHIFT BRIBERY is easy in the single-winner cases, but is hard (and hard to approximate) in the multiwinner ones. Moreover, we show that the non-monotonicity of those rules which are based on approximation algorithms for the Chamberlin-Courant rule sometimes affects the complexity of SHIFT BRIBERY.

We study the (parameterized) complexity of Shift Bribery for multiwinner voting rules. We focus on the SNTV, Bloc, k-Borda, and Chamberlin-Courant rules, as well as on approximate variants of the Chamberlin-Courant rule, since the original rule is NP-hard to compute. We show that Shift Bribery tends to be significantly harder in the multiwinner setting than in the single-winner one by showing settings where Shift Bribery is easy in the single-winner cases, but is hard (and hard to approximate) in the multiwinner ones. We show that the non-monotonicity of those rules which are based on approximation algorithms for the Chamberlin--Courant rule sometimes affects the complexity of Shift Bribery.

In the Shift Bribery problem, we are given an election (based on preference orders), a preferred candidate p, and a budget. The goal is to ensure that p wins by shifting p higher in some voters' preference orders. However, each such shift request comes at a price (depending on the voter and on the extent of the shift) and we must not exceed the given budget. We study the parameterized computational complexity of Shift Bribery with respect to a number of parameters (pertaining to the nature of the solution sought and the size of the election) and several classes of price functions. When we parameterize Shift Bribery by the number of affected voters, then for each of our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If, instead, we parameterize by the number of positions by which p is shifted in total, then the problem is fixed-parameter tractable for Borda and Maximin, and is W[1]-hard for Copeland. If we parameterize by the budget for the cost of shifting, then the results depend on the price function class. We also show that Shift Bribery tends to be tractable when parameterized by the number of voters, but that the results for the number of candidates are more enigmatic.