Abstract
An instance of colorfulk-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 rho such that there exist balls of radius rho 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. algorithms either opened more than k centers or only worked in the special case when the input points are in the plane.
Highlights
In the colorful k-center problem introduced in [5], we are given a set of n points P in a metric space partitioned into a set R of red points and a set B of blue points, along with parameters k, r, and b.The goal is to find a set of k centers C ⊆ P that minimizes ρ so that balls of radius ρ around each point in C cover at least r red points and at least b blue points.More generally, the points can be partitioned into ω color classes C1, . . . , Cω, with coverage requirements p1, . . . , pω
Their notion of fairness requires each individual cluster to have a balanced number of points from each color class, which leads to very different algorithmic considerations and is motivated by other applications, such as “feature engineering”
One can observe that it generalizes the k-center problem with outliers, which is equivalent to only having red points and needing to cover at least r of them
Summary
In the colorful k-center problem introduced in [5], we are given a set of n points P in a metric space partitioned into a set R of red points and a set B of blue points, along with parameters k, r , and b. One can observe that it generalizes the k-center problem with outliers, which is equivalent to only having red points and needing to cover at least r of them This outlier version is already more challenging than the classic k-center problem: only recent results give tight 2-approximation algorithms [6,12], improving upon the 3-approximation guarantee of [7]. Our algorithm guesses certain centers of the optimal solution that it uses to partition the point set into a “dense” part Pd and a “sparse” part Ps. The dense part is clustered using a subset-sum instance while the sparse set is clustered using the techniques of Bandyapadhyay, Inamdar, Pai, and Varadarajan [5]
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