  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.
