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 \(\rho \) 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.

彩色k -中心的实例由度量空间中红色或蓝色的点以及整数k和每种颜色的覆盖要求组成。目标是找到最小半径\(\rho \) ，使得满足覆盖要求的k 个点周围存在半径为\(\rho \)的球。这个问题背后的动机是双重的。首先，从公平性考虑：每个颜色/组都应该获得类似的服务保证；其次，从它带来的算法挑战：这个问题结合了聚类的困难以及子集和问题。特别是，我们表明这种组合会导致几个自然线性规划松弛的强完整性差距下界。我们的主要成果是一种有效的近似算法，它克服了这些困难，实现了 3 的近似保证，几乎与该问题推广的经典k中心问题的 2 的严格近似保证相匹配。算法要么打开超过k 个中心，要么仅在输入点位于平面内的特殊情况下起作用。