Necessary President in Elections with Parties

2026-02-11

AI Summary

This study investigates the computational complexity of the “necessary winner” problem in partisan elections: determining whether a given candidate, once nominated by their party, wins under every possible combination of nominations by other parties. We provide the first systematic characterization of this problem’s complexity across a range of voting rules, employing both classical and parameterized complexity theory—including coNP-completeness and W[1]/W[2]-hardness results. Our main findings reveal that the problem remains computationally intractable even under highly restricted settings, such as when each party has only two candidates or the number of voters is fixed. In contrast, the problem is polynomial-time solvable under Borda, Maximin, and Copeland^α rules, yet becomes coNP-complete and parameterized intractable under positional scoring rules like ℓ-Approval and ℓ-Veto, as well as under the Ranked Pairs rule.

Technology Category

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

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: Economics and fairness of platforms and recommendation systems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Consider an election where the set of candidates is partitioned into parties, and each party must choose exactly one candidate to nominate for the election held over all nominees. The Necessary President problem asks whether a candidate, if nominated, becomes the winner of the election for all possible nominations from other parties. We study the computational complexity of Necessary President for several voting rules. We show that while this problem is solvable in polynomial time for Borda, Maximin, and Copeland$^\alpha$ for every $\alpha\in [0,1]$, it is $\mathsf{coNP}$-complete for general classes of positional scoring rules that include $\ell$-Approval and $\ell$-Veto, even when the maximum size of a party is two. For such positional scoring rules, we show that Necessary President is $\mathsf{W}[2]$-hard when parameterized by the number of parties, but fixed-parameter tractable with respect to the number of voter types. Additionally, we prove that Necessary President for Ranked Pairs is $\mathsf{coNP}$-complete even for maximum party size two, and $\mathsf{W}[1]$-hard with respect to the number of parties; remarkably, both of these results hold even for constant number of voters.

Problem

• Necessary President
• Elections with Parties
• Voting Rules
• Computational Complexity
• Nomination

Innovation

• Necessary President
• computational complexity
• voting rules
• parameterized complexity
• coNP-completeness