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

Abstract: We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups which include the socially fair $k$-median ($p=1$) and $k$-means ($p=2$). 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 $(nk)^{{2^{O(p)} m^2}/\epsilon}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{m}\cdot\text{poly}(n)$. The former is obtained by a refinement of the iterative rounding method via a sequence of linear programs. The latter two are obtained by converting a solution with up to $k+m$ centers to one with $k$ centers by sparsification methods for (2) and via an exhaustive search for (3). We also compare the performance of our algorithms with existing approximation algorithms on benchmark datasets, and find that our algorithms outperform existing methods.