Controlling Borda Elections by Adding or Deleting either Votes or Candidates: Complete and Top-Truncated Votes

An election is defined as a pair of a set of candidates C=\{c_1,\cdots,c_m\} and a multiset of votes V=\{v_1,\cdots,v_n\}, where each vote is a linear order of the candidates. The Borda election rule is characterized by a vector \langle m-1,m-2,\cdots,0\rangle, which means that the candidate ranked at the i-th position of a vote v receives a score m-i from v, and the candidate receiving the most score from all votes wins the election. Here, we consider the control problems of a Borda election, where the chair of the election attempts to influence the election outcome by adding or deleting either votes or candidates with the intention to make a special candidate win (constructive control) or lose (destructive control) the election. Control problems have been extensively studied for Borda elections from both classical and parameterized complexity viewpoints. We complete the parameterized complexity picture for Borda control problems by showing W[2]-hardness with the number of additions/deletions as parameter for constructive control by deleting votes, adding candidates, or deleting candidates. The hardness result for deleting votes settles an open problem posed by Liu and Zhu. Following the suggestion by Menon and Larson, we also investigate the impact of introducing top-truncated votes, where each voter ranks only t out of the given m candidates, on the classical and parameterized complexity of Borda control problems. Constructive Borda control problems remain NP-hard even with t being a small constant. Moreover, we prove that in the top-truncated case, constructive control by adding/deleting votes problems are FPT with the number \ell of additions/deletions and t as parameters, while for every constant t\geq 2, constructive control by adding/deleting candidates problems are W[2]-hard with respect to \ell.