Diversity-Aware k-median: Clustering with Fair Center Representation

Abstract

We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost clustering of the data into k clusters so that a specified minimum number of cluster centers are chosen from each group. We thus require that all groups are represented in the clustering solution as cluster centers, according to specified requirements. More precisely, we are given a set of clients C, a set of facilities , a collection $\mathscr {F}=\{F_1,\dots ,F_t\}$ of facility groups , a budget k, and a set of lower-bound thresholds $R=\{r_1,\dots ,r_t\}$, one for each group in $\mathscr {F}$. The diversity-aware k-median problem asks to find a set S of k facilities in such that $|S \cap F_i| \ge r_i$, that is, at least $r_i$ centers in S are from group $F_i$, and the k-median cost $\sum _{c \in C} \min _{s \in S} d(c,s)$ is minimized. We show that in the general case where the facility groups may overlap, the diversity-aware k-median problem is $\mathbf {NP}$-hard, fixed-parameter intractable with respect to parameter k, and inapproximable to any multiplicative factor. On the other hand, when the facility groups are disjoint, approximation algorithms can be obtained by reduction to the matroid median and red-blue median problems. Experimentally, we evaluate our approximation methods for the tractable cases, and present a relaxation-based heuristic for the theoretically intractable case, which can provide high-quality and efficient solutions for real-world datasets.

This research is supported by the Academy of Finland projects AIDA (317085) and MLDB (325117), the ERC Advanced Grant REBOUND (834862), the EC H2020 RIA project SoBigData (871042), and the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.