Necessary President in Elections with Parties

TL;DR

This paper investigates the computational complexity of the Necessary President problem in elections with party-based candidate nominations, revealing polynomial-time solvability for some rules and hardness results for others, including parameterized complexities.

Contribution

It provides a comprehensive complexity analysis of Necessary President across various voting rules, identifying cases of polynomial solvability and establishing hardness results.

Findings

Polynomial-time solvability for Borda, Maximin, and Copeland$^eta$ rules.

coNP-completeness for positional scoring rules like $ ext{l}$-Approval and $ ext{l}$-Veto.

W[2]-hardness and fixed-parameter tractability results based on the number of parties and voter types.

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.