We study variants of classical clustering formulations in the context of algorithmic fairness, known as diversity-aware clustering. This variant gives you a collection of a subset of facilities. The solution should contain at least the specified number of facilities from each subset while minimizing the clustering objective ($k$-median or $k$-means). We investigate the fixed-parameter tractability of these problems and show some negative difficulty and inapproximability results, even though we can tolerate exponential run times with respect to some parameters.
Motivated by these results, we identified the natural parameters of the problem and calculated the approximate ratios $\big(1 + \frac{2}{e} +\epsilon \big)$ and $\big(1 + \frac{ 8}{e}+ \epsilon \big)$ denote the diversity-adjusted $k-median and diversity-adjusted $k-mean respectively, and assuming the gap exponential time hypothesis, these ratios is inherently tight. Finally, we propose an efficient and practical heuristic. We evaluate the scalability and effectiveness of our method in a variety of rigorously conducted experiments, both on real and synthetic data.
