Approximation Algorithms for Fair Range Clustering

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick $k$ centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of $n$ points in a metric space $(P,d)$ where each point belongs to one of the $\ell$ different demographics (i.e., $P = P1 \uplus P2 \uplus \cdots \uplus P\ell$) and a set of $\ell$ intervals $[\alpha1, \beta1], \cdots, [\alpha\ell, \beta\ell]$ on desired number of centers from each group, the goal is to pick a set of $k$ centers $C$ with minimum $\ellp$-clustering cost (i.e., $(\sum{v\in P} d(v,C)^p){1/p}$) such that for each group $i\in \ell$, $|C\cap Pi| \in [\alphai, \betai]$. In particular, the fair range $\ellp$-clustering captures fair range $k$-center, $k$-median and $k$-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range $\ellp$-clustering for all values of $p\in [1,\infty)$.