Candidate Nomination for Condorcet-consistent Voting Rules

TL;DR: This study analyzes the computational complexity of the Possible President problem for Condorcet-consistent voting rules, including Copelandα and Maximin, under various parameters, revealing dichotomies and parameterized complexities for NP-complete and polynomial-time solvable cases.

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 Copelandα for α ∈ [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 NP-complete and polynomial-time solvable cases. Moreover, for each NP-complete variant, we determine the parameterized complexity of every possible parameterization with the studied parameters as either (a) fixed-parameter tractable, (b) W[1]-hard but in XP, or (c) para-NP-hard, outlining the limits of tractability for these problems.