We study the proportional clustering problem of Chen et al. [ICML'19] and relate it to the area of multiwinner voting in computational social choice. We show that any clustering satisfying a weak proportionality notion of Brill and Peters [EC'23] simultaneously obtains the best known approximations to the proportional fairness notion of Chen et al. [ICML'19], but also to individual fairness [Jung et al., FORC'20] and the "core" [Li et al. ICML'21]. In fact, we show that any approximation to proportional fairness is also an approximation to individual fairness and vice versa. Finally, we also study stronger notions of proportional representation, in which deviations do not only happen to single, but multiple candidate centers, and show that stronger proportionality notions of Brill and Peters [EC'23] imply approximations to these stronger guarantees.

我们研究了陈等人的比例聚类问题，并将其与计算社会选择中的多胜者投票领域相关联。我们展示了任何满足Brill和Peters的弱比例概念的聚类同时获得陈等人比例公平概念、个体公平和“核心”的最佳已知拟合度。事实上，我们证明了任何对比例公平的近似也是对个体公平的近似，反之亦然。最后，我们还研究了比例代表性更强的概念，在这些概念中，偏差不仅发生在单个候选中心，还发生在多个候选中心，并且展示了Brill和Peters的更强比例概念对这些更严格的保证的拟合度。

聚类中的比例公平性：社会选择视角