Algorithms for Structured Elections under Thiele Voting Rules
2026-07-30 • Computer Science and Game Theory
Computer Science and Game TheoryArtificial IntelligenceData Structures and AlgorithmsMultiagent Systems
AI summaryⓘ
The authors study how hard it is to pick winners in certain voting systems where people approve multiple candidates, focusing on rules called Thiele voting rules. They show that if the voters and candidates follow a special pattern (Voter Interval domain), picking winners becomes easier and can be done efficiently with specific algorithms. They also solve two open problems by creating a quick method to find winners when each candidate is approved by only a few voters, and another approach based on the total score of the winning committee. Their work helps better understand when these voting problems are computationally manageable.
Approval-based committee electionsThiele voting rulesProportional Approval VotingVoter Interval domainComputational complexityFixed-parameter tractability (FPT)NP-hardnessWinner determinationApproval ballotsCommittee selection
Authors
Alexandra Lassota, Krzysztof Sornat
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.