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.

翻译：我们研究了Chen等人[ICML'19]提出的比例聚类问题，并将其与计算社会选择中的多赢家投票领域联系起来。我们证明了任何满足Brill和Peters[EC'23]弱比例性概念的聚类，同时能够获得Chen等人[ICML'19]比例公平性概念的最佳已知近似，而且也能近似个体公平性[Jung等人, FORC'20]和"核心"概念[Li等人, ICML'21]。事实上，我们证明了任何对比例公平性的近似也是对个体公平性的近似，反之亦然。最后，我们还研究了更强比例代表性的概念，其中偏离不仅发生在单个候选中心，而是多个候选中心，并证明了Brill和Peters[EC'23]的更强比例性概念意味着对这些更强保证的近似。