On the Tractability Landscape of the Conditional Minisum Approval Voting Rule

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.