On the Tractability Landscape of the Conditional Minisum Approval Voting Rule

TL;DR

The paper studies the computational tractability of the Conditional Minisum Approval Voting rule (cms) in multi-issue elections with interdependencies, showing that winner determination is hard in general and unlikely to admit algorithms substantially faster than brute-force under the Strong Exponential Time Hypothesis (SETH) and, under the Exponential Time Hypothesis (ETH), even when voter dependency graphs are simple. To restore practical applicability, it identifies two natural, tight tractability regimes: (i) group-dichotomous ballots on binary issues, which reduce cms to a minimum constraint satisfaction problem that can be solved in polynomial time via a min-cut reduction, and (ii) per-voter dependency graphs with bounded vertex cover number and maximum in-degree $\Delta=1$, which ensure the union of dependency graphs has bounded treewidth for a constant number of voters, enabling a polynomial-time algorithm by leveraging existing treewidth-based results. These results together map the tractability landscape, showing cms can be used in practice under concrete restrictions without sacrificing too much expressivity. The findings guide the design of practical conditional-approval systems and suggest directions for empirical evaluation and extensions to related voting settings such as committees or participatory budgeting.

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.

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Theorems & Definitions (15)

• Definition 1
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• Theorem 1
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• Definition 3
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