Constant-Factor Approximation Algorithms for Socially Fair $k$-Clustering

We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups, whose special cases include the socially fair $k$-median ($p=1$) and socially fair $k$-means ($p=2$) problems. We present (1) a polynomial-time $(5+2\sqrt{6})^{p$-approximation} with at most $k+m$ centers (2) a $(5+2\sqrt{6}+\epsilon)^{p$-approximation} with $k$ centers in time $n^{{2{O(p)}\cdot} m^2}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{{m}\cdot\text{poly}(n)$.} The first result is obtained via a refinement of the iterative rounding method using a sequence of linear programs. The latter two results are obtained by converting a solution with up to $k+m$ centers to one with $k$ centers using sparsification methods for (2) and via an exhaustive search for (3). We also compare the performance of our algorithms with existing bicriteria algorithms as well as exactly $k$ center approximation algorithms on benchmark datasets, and find that our algorithms also outperform existing methods in practice.