Prices matter for the parameterized complexity of shift bribery

In the SHIFT BRIBERY problem, we are given an election, a preferred candidate p, and a budget. The goal is to ensure p's victory by shifting p higher in some voters’ preference orders. However, each such shift request comes at a price and we must not exceed the given budget. We study the parameterized computational complexity of SHIFT BRIBERY for a number of parameters and several classes of price functions: For the number of affected voters, SHIFT BRIBERY is W[2]-hard for Borda, Maximin, and Copeland. For the number of positions by which p is shifted in total, the problem is fixed-parameter tractable for Borda and Maximin, and is W[1]-hard for Copeland. For the budget, the results depend on the price function class. Finally, SHIFT BRIBERY tends to be tractable when parameterized by the number of voters, but the results for the number of candidates are more enigmatic.