In the non-uniform $k$-center problem, the objective is to cover points in a metric space with specified number of balls of different radii. Chakrabarty, Goyal, and Krishnaswamy [ICALP 2016, Trans. on Algs. 2020] (CGK, henceforth) give a constant factor approximation when there are two types of radii. In this paper, we give a constant factor approximation for the two radii case in the presence of outliers. To achieve this, we need to bypass the technical barrier of bad integrality gaps in the CGK approach. We do so using "the ellipsoid method inside the ellipsoid method": use an outer layer of the ellipsoid method to reduce to stylized instances and use an inner layer of the ellipsoid method to solve these specialized instances. This idea is of independent interest and could be applicable to other problems. Keywords: Approximation, Clustering, Outliers, and Round-or-Cut.

中文翻译：

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具有两种半径的强大的$ k $中心

在不均匀的$ k $中心问题中，目标是用指定数量的不同半径的球覆盖度量空间中的点。Chakrabarty，Goyal和Krishnaswamy [ICALP 2016，Trans。在阿尔格斯。[2020年]（以下简称CGK）给出两种半径类型时的常数因子近似值。在本文中，在存在离群值的情况下，我们给出了两个半径情况下的常数因子近似值。为了实现这一目标，我们需要绕过CGK方法中不良的完整性差距的技术障碍。我们使用“椭圆体方法内部的椭圆体方法”进行此操作：使用椭圆体方法的外层来简化为风格化的实例，并使用椭圆体方法的内层来解决这些特殊的实例。这个想法是独立利益的，可以适用于其他问题。关键字：近似值