Candidate nomination for Condorcet-consistent voting rules

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 winner of the election for some nominations from other parties. We perform a multivariate computational complexity analysis of Possible President for a range of Condorcet-consistent voting rules, namely for Copeland$^\alpha$ for $\alpha \in [0,1]$ and Maximin. The parameters we study are the number of voters, the number of parties, and the maximum size of a party. For all voting rules under consideration, we obtain dichotomies based on the number of voters, classifying $\mathsf{NP}$-complete and polynomial-time solvable cases. Moreover, for each $\mathsf{NP}$-complete variant, we determine the parameterized complexity of every possible parameterization with the studied parameters as either (a) fixed-parameter tractable, (b) $\mathsf{W}[1]$-hard but in $\mathsf{XP}$, or (c) $\mathsf{paraNP}$-hard, outlining the limits of tractability for these problems.