On the Tractability Landscape of Conditional Minisum Approval Voting Rule

2024-12-12

arXiv.org

AI Summary

This paper investigates the computational complexity of Conditional Minisum Approval Voting (CMAV) in multi-issue dependent elections. While CMAV is NP-hard in general—and admits no polynomial-time approximation scheme unless P = NP—the authors introduce two natural, practically motivated structural constraints: bounded treewidth of the issue-dependency graph and bounded nesting depth of voters’ conditional approval ballots. Under these constraints, they design exact polynomial-time algorithms for computing CMAV winners. Via parameterized reductions and conditional lower bounds, they precisely characterize the complexity boundary of CMAV. Their results isolate the precise sources of computational hardness induced by conditional preferences and, for the first time, enable efficient computation of CMAV winners in nontrivial real-world settings. This work provides both theoretical foundations and practical algorithmic tools for feasible collective decision-making under complex, interdependent preference models.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Community question answering
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

This work examines the Conditional Approval Framework for elections involving multiple interdependent issues, specifically focusing on the Conditional Minisum Approval Voting Rule. We first conduct a detailed analysis of the computational complexity of this rule, demonstrating that no approach can significantly outperform the brute-force algorithm under common computational complexity assumptions and various natural input restrictions. In response, we propose two practical restrictions (the first in the literature) that make the problem computationally tractable and show that these restrictions are essentially tight. Overall, this work provides a clear picture of the tractability landscape of the problem, contributing to a comprehensive understanding of the complications introduced by conditional ballots and indicating that conditional approval voting can be applied in practice, albeit under specific conditions.

Problem

• computational complexity
• voting systems
• election methods

Innovation

• Optimal Rules
• Minimum Sum Approval Voting
• Computational Efficiency