Navigating the Complexity Landscape of Nominee Selection in Schulze Voting

2026-04-13

AI Summary

This study investigates the computational complexity of determining whether a party’s nominee can possibly or necessarily win under the Schulze voting rule, depending on the coalition of nominees from other parties. Employing parameterized complexity theory, it systematically analyzes how three natural parameters—the number of voters, the maximum party size, and the number of parties—affect the difficulty of these two decision problems. The work provides the first complete fine-grained complexity classification across all combinations of these parameters, revealing a sharp dichotomy with respect to the number of voters. By establishing NP- and coNP-completeness results where appropriate, it fully characterizes the complexity landscape of both problems, substantially extending and refining prior findings in the literature.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Security and Privacy: Large-scale security measurements

Abstract

We study the Possible President problem and the Necessary President problem for Schulze voting, a rule that, due to its many desirable axiomatic properties, is popular in practice. In both problems, we are given an election with the candidates partitioned into a set of parties, and we are interested in questions about a given distinguished party. In the Possible President problem, we ask whether it is possible for the parties to each nominate exactly one candidate such that the nominee of the distinguished party is a Schulze winner of the resulting election with only the nominees running. In the Necessary President problem, we ask whether the distinguished party's nominee is a Schulze winner of the resulting election, irrespective of the nomination from the other parties. Rothe and Woitaschik have shown that Possible President is NP-complete and Necessary President is coNP-complete for Schulze elections. We complement and improve their results by a more fine-grained analysis: we determine the parameterized complexity of both problems with respect to all possible parameterizations, where we consider each of three natural parameters -- the number of voters, the maximum party size, and the number of parties -- to be either a constant, a parameter, or unbounded. In particular, we obtain dichotomies regarding the number of voters for both problems.

Problem

• Schulze voting
• Possible President
• Necessary President
• parameterized complexity
• nominee selection

Innovation

• Schulze voting
• parameterized complexity
• Possible President
• Necessary President
• dichotomy