There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colors—settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan—and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that \(\mathtt {P}\ne \mathtt {NP}\). Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

最近，人们对将公平方面纳入经典聚类问题的兴趣激增。本着这种精神，最近引入的两个k中心问题的变体是由 Bandyapadhyay、Inamdar、Pai 和 Varadarajan 引入的彩色k中心问题和彩票模型，例如由 Harris、Pensyl、Srinivasan 引入的 Fair Robust k中心问题，和Trinh。为了解决公平方面的问题，这些模型与传统的k-Center，包括额外的覆盖约束。这些模型的先验近似结果需要放宽一些通常的硬约束，例如要打开的中心数量或涉及的覆盖约束，因此只能获得常数因子伪近似。在本文中，我们介绍了一种新方法来处理导致（真实）近似的此类覆盖约束，包括具有不断多种颜色的彩色k中心的 4 近似——解决了 Bandyapadhyay、Inamdar、Pai、和 Varadarajan — 以及 Fair Robust k的 4 近似值-Center，其中存在一个（真正的）常数因子近似也是开放的。我们通过表明如果允许无限数量的颜色来补充我们的结果，那么假设\(\mathtt {P}\ne \mathtt {NP}\) ，Colorful k -Center 不承认具有有限近似保证的近似算法。此外，在指数时间假设下，如果颜色数量的增长速度快于地面集大小的对数增长速度，则该问题是不可近似的。