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.

翻译：选举定义为由候选人集合C={c_1,\cdots,c_m}和选票多重集V={v_1,\cdots,v_n}构成的二元组，其中每张选票是候选人的线性排序。博尔达选举规则由向量\langle m-1,m-2,\cdots,0\rangle刻画，即一张选票v中排名第i位的候选人从该选票获得m-i分，获得所有选票总分最高的候选人赢得选举。本文研究博尔达选举的控制问题，即选举主持人试图通过增加或删除选票或候选人影响选举结果，其意图是使特定候选人获胜（建设性控制）或落选（破坏性控制）。此前，博尔达选举的控制问题已从经典复杂性和参数化复杂性两个视角得到广泛研究。我们通过证明以下问题的W[2]-困难性（以增加/删除数量为参数）完善了博尔达控制问题的参数化复杂性图景：通过删除选票、增加候选人或删除候选人实施建设性控制。其中删除选票的建设性控制困难结果解决了Liu与Zhu提出的开放问题。遵循Menon与Larson的建议，我们还探究了引入截断排名选票（每位选民仅对给定m个候选人中的t个进行排名）对博尔达控制问题经典复杂性和参数化复杂性的影响。即便t为小常数，建设性博尔达控制问题仍保持NP困难性。此外，我们证明在截断排名情形下，通过增加/删除选票的建设性控制问题关于参数（增加/删除数量\ell与t）是固定参数可解的（FPT），而t\geq 2为任意常数时，通过增加/删除候选人的建设性控制问题关于参数\ell是W[2]-困难的。