Candidate nomination for Condorcet-consistent voting rules

TL;DR

This paper analyzes the computational complexity of the Possible President problem in elections with party nominations under Condorcet-consistent rules, providing classifications based on voters, parties, and party size.

Contribution

It offers a comprehensive complexity analysis of the Possible President problem for various Condorcet-consistent voting rules, including dichotomies and parameterized complexity results.

Findings

Classifies NP-complete and polynomial cases based on voter count.

Determines fixed-parameter tractability and hardness for different parameters.

Provides limits of computational tractability for the problem.

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.