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In a recent work, [19] studied the following "fair" variants of classicalclustering problems such as $k$-means and $k$-median: given a set of $n$ datapoints in $\mathbb{R}^d$ and a binary type associated to each data point, thegoal is to cluster the points while ensuring that the proportion of each typein each cluster is roughly the same as its underlying proportion. Subsequentwork has focused on either extending this setting to when each data point hasmultiple, non-disjoint sensitive types such as race and gender [6], or toaddress the problem that the clustering algorithms in the above work do notscale well. The main contribution of this paper is an approach to clusteringwith fairness constraints that involve multiple, non-disjoint types, that isalso scalable. Our approach is based on novel constructions of coresets: forthe $k$-median objective, we construct an $\varepsilon$-coreset of size$O(\Gamma k^2 \varepsilon^{-d})$ where $\Gamma$ is the number of distinctcollections of groups that a point may belong to, and for the $k$-meansobjective, we show how to construct an $\varepsilon$-coreset of size $O(\Gammak^3\varepsilon^{-d-1})$. The former result is the first known coresetconstruction for the fair clustering problem with the $k$-median objective, andthe latter result removes the dependence on the size of the full dataset as in[39] and generalizes it to multiple, non-disjoint types. Plugging our coresetsinto existing algorithms for fair clustering such as [5] results in the fastestalgorithms for several cases. Empirically, we assess our approach over the\textbf{Adult}, \textbf{Bank}, \textbf{Diabetes} and \textbf{Athlete} dataset,and show that the coreset sizes are much smaller than the full dataset. We alsoachieve a speed-up to recent fair clustering algorithms [5,6] by incorporatingour coreset construction.

Coresets for Clustering with Fairness Constraints