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.

翻译：本研究考察了涉及多个相互依赖议题的选举中的条件批准框架，特别聚焦于条件最小和批准投票规则。我们首先对该规则的计算复杂性进行了详细分析，证明在常见计算复杂性假设及多种自然输入限制下，任何方法都无法显著优于暴力算法。为此，我们提出了两种使该问题在计算上可处理的实际限制条件（此为文献中首次提出），并证明这些限制条件本质上是紧致的。总体而言，本研究清晰描绘了该问题的可处理性图景，有助于全面理解条件投票引入的复杂性，并表明条件批准投票可在特定条件下应用于实践。