发表机构
TU Eindhoven; AGH University（埃因霍温理工大学; AGH大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文研究Thiele投票规则下结构化选举的获胜者确定问题，针对选民区间域等设计FPT算法，解决了PAV相关的两个开放问题，推进了此类选举的计算复杂性研究。
AI 中文摘要
我们研究了Thiele投票规则下基于认可的委员会选举中获胜者确定问题的计算复杂性。Thiele投票规则构成一类由固定权重向量参数化的规则，该向量指定选民的满意度如何依赖于当选的获认可候选人数量。我们首先基于每个候选人的获认可选民集合分析最优解的结构，即选民的认可选票如何在候选人之间引发依赖关系，揭示了任意固定Thiele投票规则下获胜委员会的约束条件。利用这一点，我们在称为选民区间（Voter Interval, VI）的自然受限域上，为比例认可投票（Proportional Approval Voting, PAV）及其他Thiele规则设计了固定参数可处理（Fixed-Parameter Tractable, FPT）算法——该域指在对选民进行适当排序后，每个候选人的获认可选民构成一个连续区间。特别地，我们证明VI域上的每一个Thiele规则，相对于一个参数而言都是FPT的，而该参数对应的问题在一般实例上是NP难的，即使该参数取常数值时亦是如此。我们的结果加深了对VI域上PAV计算复杂性的理解，而这仍是该领域的核心开放问题之一。我们进一步解决了文献中关于PAV（及其他Thiele投票规则）的两个开放问题：为每个候选人最多被两名选民认可的实例提供了多项式时间算法，并为以获胜委员会总得分作为参数的情况设计了FPT算法。
英文摘要
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.
Comments18 pages. A conference version of this work appeared in AAAI 2026
