In this work, we study diversity-aware clustering problems where the data points are associated with multiple attributes resulting in intersecting groups. A clustering solution need to ensure that a minimum number of cluster centers are chosen from each group while simultaneously minimizing the clustering objective, which can be either $k$-median, $k$-means or $k$-supplier. We present parameterized approximation algorithms with approximation ratios $1+ \frac{2}{e}$, $1+\frac{8}{e}$ and $3$ for diversity-aware $k$-median, diversity-aware $k$-means and diversity-aware $k$-supplier, respectively. The approximation ratios are tight assuming Gap-ETH and FPT $\neq$ W[2]. For fair $k$-median and fair $k$-means with disjoint faicility groups, we present parameterized approximation algorithm with approximation ratios $1+\frac{2}{e}$ and $1+\frac{8}{e}$, respectively. For fair $k$-supplier with disjoint facility groups, we present a polynomial-time approximation algorithm with factor $3$, improving the previous best known approximation ratio of factor $5$.

翻译：本文研究了数据点具有多个属性并形成交叉群体的多样性感知聚类问题。聚类解决方案需要确保从每个群体中选择最少数量的聚类中心，同时最小化聚类目标函数（可以是$k$-中位数、$k$-均值或$k$-供应商）。我们针对多样性感知$k$-中位数、多样性感知$k$-均值和多样性感知$k$-供应商分别提出了近似比为$1+ \frac{2}{e}$、$1+\frac{8}{e}$和$3$的参数化近似算法。在Gap-ETH假设且FPT $\neq$ W[2]的条件下，这些近似比是紧的。对于具有不相交设施群体的公平$k$-中位数和公平$k$-均值问题，我们分别提出了近似比为$1+\frac{2}{e}$和$1+\frac{8}{e}$的参数化近似算法。对于具有不相交设施群体的公平$k$-供应商问题，我们提出了一个因数为$3$的多项式时间近似算法，改进了此前最优的因数为$5$的近似比。