We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clusetring problem, given a set P of points in R^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L_1-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r=0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given \epsilon>0, produces a hybrid k-clustering with balls of radius (1+\epsilon)r. This algorithm achieves a cost at most 1+\epsilon of the optimum, and it operates in time 2^{(kd/\epsilon)^{O(1)}} n^{O(1)}. Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clusetring.

翻译：我们提出了一种新颖的聚类模型，该模型涵盖了两种经典的聚类模型：k中心聚类与k中值聚类。在混合k聚类问题中，给定d维欧氏空间中的点集P、整数k以及非负实数r，我们的目标是在空间中放置k个半径为r的闭球，以最小化未被球覆盖的点到其最近球的距离之和。等价地，我们寻求在欧氏空间中用k个半径为r的球的并集对点集进行最优的L_1拟合。当r=0时，该问题退化为k中值聚类；当最小和为零（即所有点均被覆盖）时，则对应于k中心聚类。我们的主要成果是一个双准则近似算法：对于给定的ε>0，该算法能够生成使用半径为(1+ε)r的球的混合k聚类方案，其代价至多为最优解的1+ε倍，且运行时间为2^{(kd/ε)^{O(1)}} n^{O(1)}。值得注意的是，基于k中心与k中值问题已知的下界结果，我们的双准则近似可视为混合k聚类问题目前可能达到的最佳理论结果。