Resolute control: Forbidding candidates from winning an election is hard

We study a set of voting problems where given an election E=(C,Π_V) (where C is the set of candidates and Π_V is a set of votes), and a non-empty subset of candidates J, the question under consideration is: Can we modify the election in a way so that none of the candidates in J wins the election? The modification operations allowed are that of either adding or deleting some candidates. Yang and Wang (2017) [44] introduced these problems as the RESOLUTE CONTROL problem, a generalization of the destructive control problem where J is a singleton. They studied parameterized complexity of RESOLUTE CONTROL for voting rules Borda (both addition and deletion), Maximin (addition), and Copeland (both addition and deletion). They primarily consider |J| as parameter. In this paper we study RESOLUTE CONTROL parameterized by the other natural parameters viz., the number of candidates added or deleted. We show that the RESOLUTE CONTROL for Borda (both addition and deletion), Maximin (addition) and Copeland (deletion) are W[2]-hard. We complement this by showing that when the number of voters is odd, Copeland (deletion) is FPT parameterized by the sum of the number of deleted candidates and the size of the feedback arc set of the majority graph of the election.