Consider elections where the set of candidates is partitioned into parties, and each
party must nominate exactly one candidate. The P ossible P resident problem asks whether
some candidate of a given party can become the unique winner of the election for some
nominations from other parties. We perform a multivariate computational complexity
analysis of P ossible P resident for several classes of elections based on positional
scoring rules. We consider the following parameters: the size of the largest party,
the number of parties, the number of voters and the number of voter types. We provide
a complete computational map of P ossible P resident in the sense that for each choice
of the four possible parameters as (i) constant, (ii) parameter, or (iii) unbounded,
we classify the computational complexity of the resulting problem as either polynomial-time
solvable or -complete, and for parameterized versions as either fixed-parameter tractable
or [1]-hard with respect to the parameters considered.