Fair Representation Clustering with Several Protected Classes

We study the problem of fair $k$-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation $k$-median problem, we are given a set of points $X$ in a metric space. Each point $x\in X$ belongs to one of $\ell$ groups. Further, we are given fair representation parameters $\alphaj$ and $\betaj$ for each group $j\in [\ell]$. We say that a $k$-clustering $C1, \cdots, Ck$ fairly represents all groups if the number of points from group $j$ in cluster $Ci$ is between $\alphaj |Ci|$ and $\betaj |Ci|$ for every $j\in[\ell]$ and $i\in [k]$. The goal is to find a set $\mathcal{C}$ of $k$ centers and an assignment $\phi: X\rightarrow \mathcal{C}$ such that the clustering defined by $(\mathcal{C}, \phi)$ fairly represents all groups and minimizes the $\ell1$-objective $\sum{x\in X} d(x, \phi(x))$. We present an $O(\log k)$-approximation algorithm that runs in time $n^{O(\ell)}$. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both $k$ and $\ell$. We also consider an important special case of the problem where $\alphaj = \betaj = \frac{fj}{f}$ and $f_j, f \in \mathbb{N}$ for all $j\in [\ell]$. For this special case, we present an $O(\log k)$-approximation algorithm that runs in $(kf)^{{O(\ell)}\log} n + poly(n)$ time.