强化完全正当代表性：高效验证与计算

中文摘要 AI 辅助

完全正当代表性（Full Justified Representation, FJR）是认可型委员会选举中已知可满足的最强比例公理之一。近期研究表明，多项式时间内可找到满足FJR的委员会，但验证给定委员会是否满足FJR仍为余NP完全问题。我们引入FJR+，这是对FJR和EJR+的严格强化，可在多项式时间内完成验证与满足。随后分析残差预算贪心（Residual-Budget Greedy, RBG）算法，证明其选择的部分委员会，其任意大小为k的补全均满足FJR+。该自由度使我们可使用序贯Phragmén方法获得可定价补全。所得规则始终满足FJR+和次核心条件，且只要至少k个候选人获得认可，即可实现可定价。我们还得到FJR+的德罗普配额（Droop-quota）版本，最后将FJR+扩展至项目成本任意的认可型参与式预算。针对项目的RBG版本可在多项式时间内计算该属性，且可延续至满足成本型次核心的可定价结果。

英文摘要

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

发表机构

• University of Oxford（牛津大学）

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