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

TL;DR

This paper introduces new approximation algorithms for socially fair $k$-clustering problems, achieving constant-factor guarantees with improved efficiency and practical performance over existing methods.

Contribution

The paper presents novel approximation algorithms for socially fair $( ext{ell}_p, k)$-clustering, with improved approximation ratios and runtime, and demonstrates their effectiveness on benchmark datasets.

Findings

Achieved a polynomial-time $(5+2\sqrt{6})^p$-approximation with at most $k+m$ centers.

Developed a $(5+2\sqrt{6}+\epsilon)^p$-approximation with $k$ centers in specific runtime.

Outperformed existing bicriteria and exact algorithms on benchmark datasets.

Abstract

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.