Fair Colorful k-Center Clustering

TL;DR

This paper introduces a fair colorful k-center clustering problem with color-based coverage constraints, providing an efficient approximation algorithm that nearly matches the classical k-center's approximation guarantee.

Contribution

It formulates a new fair clustering problem combining color constraints with k-center, and develops an approximation algorithm overcoming integrality gaps.

Findings

Achieves a 3-approximation guarantee for the problem.

Shows strong integrality gap lower bounds for natural LP relaxations.

Extends classical k-center problem to incorporate fairness constraints.

Abstract

An instance of colorful k-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius \r{ho} such that there exist balls of radius \r{ho} around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes.