Abstract
Consider elections where the set of candidates is partitioned into parties, and each party must nominate exactly one candidate. The Possible President 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 Possible President 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 Possible President 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 NP-complete, and for parameterized versions as either fixed-parameter tractable or W[1]-hard with respect to the parameters considered.
