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Candidate nomination for Condorcet-consistent voting rules

Ildikó Schlotter, Katarína Cechlárová

TL;DR

This work provides a thorough multivariate complexity analysis of the Possible President problem under Condorcet-consistent rules, focusing on Copeland$^\alpha$ and Maximin in a party-nomination setting. It delivers a complete map of tractability: Copeland$^\alpha$ exhibits a sharp dichotomy by the number of voters (polynomial for $n=2$ when $\alpha=1$, NP-hard for $n=2$ with $\alpha<1$, and NP-hard for $n\ge3$ with various parameterized hardness results), while Maximin shows a dichotomy by the number of voters (polynomial for $n\le3$, NP-hard for $n\ge4$) but is fixed-parameter tractable when parameterized by the number of parties $t$. The authors introduce and exploit flat elections and build reductions from Maximum Matching with Couples and Multicolored Clique, plus an FPT algorithm via Partitioned Subdigraph Isomorphism, to delineate precise hardness and tractability frontiers. These results illuminate the computational limits of predicting or influencing outcomes in party-nomination elections under these rules, with implications for the design of nomination processes and primary strategies.

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$^α$ for $α\in [0,1]$ 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.

Candidate nomination for Condorcet-consistent voting rules

TL;DR

This work provides a thorough multivariate complexity analysis of the Possible President problem under Condorcet-consistent rules, focusing on Copeland and Maximin in a party-nomination setting. It delivers a complete map of tractability: Copeland exhibits a sharp dichotomy by the number of voters (polynomial for when , NP-hard for with , and NP-hard for with various parameterized hardness results), while Maximin shows a dichotomy by the number of voters (polynomial for , NP-hard for ) but is fixed-parameter tractable when parameterized by the number of parties . The authors introduce and exploit flat elections and build reductions from Maximum Matching with Couples and Multicolored Clique, plus an FPT algorithm via Partitioned Subdigraph Isomorphism, to delineate precise hardness and tractability frontiers. These results illuminate the computational limits of predicting or influencing outcomes in party-nomination elections under these rules, with implications for the design of nomination processes and primary strategies.

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

Paper Structure

This paper contains 32 sections, 31 theorems, 10 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copeland$^\alpha$ voting rule
  4. Two Voters
  5. Three Voters
  6. A special case of Maximum Matching with Couples
  7. Flat elections with three voters
  8. Reduction for Theorem \ref{['thm:3voters-NPh']}
  9. Four or More Voters
  10. Few Parties
  11. Maximin voting rule
  12. Few Voters
  13. Few Parties
  14. Conclusions and future research
  15. Additional related work
  16. ...and 17 more sections

Key Result

theorem 1

Let $n$ be a fixed integer and $\alpha \in [0,1]$. Then Possible President for Copeland$^\alpha$ is $\mathsf{NP}$-complete when restricted to instances with $n$ voters and maximum party size $\sigma=2$, if By contrast, Possible President for Copeland$^1$ (i.e., Llull) restricted to instances with $2$ voters is polynomial-time solvable.

Theorems & Definitions (38)

  • theorem 1
  • theorem 2
  • theorem 3: app:proof-of-Llulltwovoters
  • theorem 4: sec:proof-of-CopelandtwoNPc
  • theorem 5: sec:proof-of-3votersNPh
  • theorem 6: app:MC-variant
  • definition 1
  • theorem 7: sec:proof-of-4votersNPc
  • theorem 8
  • theorem 9: sec:param-t-even
  • ...and 28 more