# Improved approximation algorithms for two Euclidean k-Center variants

The k-Center problem is one of the most popular clustering problems. After decades of work, the complexity of most of its variants on general metrics is now well understood. Surprisingly, this is not the case for a natural setting that often arises in practice, namely the Euclidean setting, in which the input points are points in ℝ^d, and the distance between them is the standard ℓ_2 Euclidean distance. In this work, we study two Euclidean k-Center variants, the Matroid Center problem on the real line and the Robust Euclidean k-Supplier problem, and provide algorithms that improve upon the best approximation guarantees known for these problems. In particular, we present a simple 2.5-approximation algorithm for the Matroid Center problem on the real line, thus improving upon the 3-approximation factor algorithm of Chen, Li, Liang, and Wang (2016) that works for general metrics. Moreover, we present a (1 + √(3))-approximation algorithm for the Robust Euclidean k-Supplier problem, thus improving upon the state-of-the-art 3-approximation algorithm for Robust k-Supplier on general metrics and matching the best approximation factor known for the non-robust setting by Nagarajan, Schieber and Shachnai (2020).

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