The Complexity of Determining the Necessary and Possible Top-k Winners in Partial Voting Profiles

Abstract: When voter preferences are known in an incomplete (partial) manner, winner determination is commonly treated as the identification of the necessary and possible winners; these are the candidates who win in all completions or at least one completion, respectively, of the partial voting profile. In the case of a positional scoring rule, the winners are the candidates who receive the maximal total score from the voters. Yet, the outcome of an election might go beyond the absolute winners to the top-$k$ winners, as in the case of committee selection, primaries of political parties, and ranking in recruiting. We investigate the computational complexity of determining the necessary and possible top-$k$ winners over partial voting profiles. Our results apply to general classes of positional scoring rules and focus on the cases where $k$ is given as part of the input and where $k$ is fixed.