A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering

Clustering is a fundamental task in theoretical computer science and machine learning aimed at dividing a set of data items into several groups or clusters, such that each group contains similar data items. Typically, the similarity between data items is measured using a metric distance function. Clustering is often modeled as an optimization problem where the objective is to minimize a global cost function that reflects the quality of the clusters; this function varies depending on the application. Among the many cost functions studied for clustering, the most popular are k-median, k-means, and k-center. These objectives generally aim to minimize the variance within the clusters, serving as a proxy for grouping similar data items In this work, we study clustering problems with fairness constraints, commonly known as fair clustering problems. Fair clustering emerged as one of the most active research areas in algorithms motivated by the recent trend of research on fairness in artificial intelligence. In a seminal work, Chierichetti et al. [18] introduced a fair clustering problem, where given a set R of red points, a set B of blue points, and an integer balance parameter t 1, a clustering is said to be balanced if, in every cluster, the number of red points is at least 1/t times the number of blue points and at most t times the number of blue points.