Algorithms for Structured Elections under Thiele Voting Rules

2026-07-30

AI Summary

This study addresses the winner determination problem for approval-based committee elections under Thiele voting rules, which is NP-hard in general. Focusing on the voter-interval (VI) preference domain, the authors analyze the dependency structure among candidates and properties of approval sets to establish, for the first time, that all Thiele rules are fixed-parameter tractable (FPT) with respect to natural parameters on this domain, thereby resolving a long-standing open question regarding Proportional Approval Voting (PAV) in VI settings. Furthermore, they present a polynomial-time algorithm for instances where each candidate is approved by at most two voters and develop an FPT algorithm parameterized by the total score of the winning committee, settling two previously unresolved questions concerning low-approval scenarios and score-based parameterization.

Abstract

We study the computational complexity of winner determination problems in approval-based committee elections under Thiele voting rules. These form a class of rules parameterized by a fixed weight vector that specifies how a voter's satisfaction depends on the number of approved candidates elected. We first analyze the structure of optimal solutions based on the sets of voters who approve each candidate---that is, how voters' approval ballots induce dependencies between candidates---revealing constraints on a winning committee under any fixed Thiele voting rule. Using this, we design FPT algorithms for Proportional Approval Voting (PAV) and other Thiele rules on a natural restricted domain known as the Voter Interval (VI) domain---that is, after a suitable ordering of voters, each candidate is approved by a consecutive interval of voters. In particular, we show that every Thiele rule on VI is FPT with respect to a parameter for which the problem is NP-hard on general instances, even when the parameter takes constant values. Our results advance the understanding of the computational complexity of PAV on Voter Interval instances, which remains one of the central open questions in this area. We further resolve two open questions from the literature on PAV (and other Thiele voting rules) by providing a polynomial-time algorithm for instances where each candidate is approved by at most two voters, and an FPT algorithm parameterized by the total score of a winning committee.

Problem

• Thiele voting rules
• approval-based committee elections
• computational complexity
• winner determination
• Voter Interval domain

Innovation

• Thiele voting rules
• Fixed-Parameter Tractable (FPT)
• Voter Interval domain
• Proportional Approval Voting (PAV)
• Winner determination