This paper explores three concepts: the k-center problem, some of its variants, and asymmetry. The k-center problem is a fundamental clustering problem, similar to the k-median problem. Variants of k-center may more accurately model real-life problems than the original formulation. Asymmetry is a significant impediment to approximation in many graph problems, such as k-center, facility location, k-median and the TSP. We demonstrate an O(log* n)-approximation algorithm for the asymmetric weighted k-center problem. Here, the vertices have weights and we are given a total budget for opening centers. In the p-neighbor variant each vertex must have p (unweighted) centers nearby: we give an O(log* k)-bicriteria algorithm using 2k centers, for small p. Finally, the following three versions of the asymmetric k-center problem we show to be inapproximable: priority k-center, k-supplier, and outliers with forbidden centers. © Springer-Verlag Berlin Heidelberg 2003.
Gørtz, I. L., & Wirth, A. (2003). Asymmetry in k-center variants. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2764, 59–70. https://doi.org/10.1007/978-3-540-45198-3_6