Approximating Fair Clustering with Cascaded Norm Objectives

We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over $W$ of the $\ell_p$-norms of the weighted distances of points in $P$ from the centers. This generalizes various clustering problems, including Socially Fair $k$-Median and $k$-Means, and is closely connected to other problems such as Densest $k$-Subgraph and Min $k$-Union. We utilize convex programming techniques to approximate the $(p,q)$-Fair Clustering problem for different values of $p$ and $q$. When $p\geq q$, we get an $O(k^{(p-q)/(2pq)})$, which nearly matches a $k^{\Omega((p-q)/(pq))}$ lower bound based on conjectured hardness of Min $k$-Union and other problems. When $q\geq p$, we get an approximation which is independent of the size of the input for bounded $p,q$, and also matches the recent $O((\log n/(\log\log n))^{1/p})$-approximation for $(p, \infty)$-Fair Clustering by Makarychev and Vakilian (COLT 2021).