On the Complexity of Borda Control in Single-Peaked Elections

Recent research reveals that many NP-hard voting problems in general become polynomial-time solvable in single-peaked elections. In contrast to these results, we prove for the first time that constructive control by adding/deleting votes for Borda are NP-hard even in single-peaked elections. On the other side, we prove that constructive control by adding/deleting votes/candidates for Borda are polynomial-time solvable in single-dived elections, which are elections obtained from single-peaked elections by reversing voters' preferences. Finally, we study constructive control by adding/deleting votes/candidates for Borda in single-peaked elections with k-truncated votes, i.e., each voter ranks only her top-k candidates, aiming at investigating how the values of $k$ affect the complexity of these problems. For this purpose, we adopt the voting correspondences Borda↑, Borda↓ and Bordaav. We obtain many polynomial-time solvable results for k being a constant.

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