We study the fair variant of the classic k-median problem introduced by (Chierichetti et al., NeurIPS 2017). In the standard k-median problem, given an input pointset P, the goal is to find k centers C and assign each input point to one of the centers in C such that the average distance of points to their cluster center is minimized. In the fair variant of k-median, the points are colored, and the goal is to minimize the same average distance objective while ensuring that all clusters have an “approximately equal” number of points of each color.

(Chierichetti et al., NeurIPS 2017) proposed a two-phase algorithm for fair k-median. In the first step, the pointset is partitioned into subsets called fairlets that satisfy the fairness requirement and approximately preserve the k-median objective. In the second step, fairlets are merged into k clusters by one of the existing k-median algorithms. The running time of this algorithm is dominated by the first step, which takes super-quadratic time.

In this paper, we present a practical approximate fairlet decomposition algorithm that runs in nearly linear time. We complement our theoretical bounds with empirical evaluation.