Approximating Fair Clustering with Cascaded Norm Objectives

Abstract

We introduce the (p,q)-Fair Clustering problem. In this problem, we are given a set of points P and a collection of different weight functions W. We would like to find a clustering which minimizes the _q-norm of the vector over W of the _p-norms of the weighted distances of points in P from the centers. This generalizes various clustering problems, including Socially Fair k-Median and k-Means, and is closely connected to other problems such as Densest k-Subgraph and Min k-Union. We utilize convex programming techniques to approximate the (p,q)-Fair Clustering problem for different values of p and q. When p q, we get an O(k^(p-q)/(2pq)), which nearly matches a k^((p-q)/(pq)) lower bound based on conjectured hardness of Min k-Union and other problems. When q p, we get an approximation which is independent of the size of the input for bounded p,q, and also matches the recent O(( n/( n))^1/p)-approximation for (p, )-Fair Clustering by Makarychev and Vakilian (COLT 2021).

Tasks

Reproductions