The $k$-Supplier problem is an important location problem that has been actively studied in both general and Euclidean metrics. Many of its variants have also been studied, primarily on general metrics. We study two variants of $k$-Supplier, namely Priority $k$-Supplier and $k$-Supplier with Outliers, in Euclidean metrics. We obtain $(1+\sqrt{3})$-approximation algorithms for both variants, which are the first improvements over the previously-known factor-$3$ approximation (that is known to be best-possible for general metrics). We also study the Matroid Supplier problem on Euclidean metrics, and show that it cannot be approximated to a factor better than $3$ (assuming $P\ne NP$); so the Euclidean metric offers no improvement in this case.

翻译：以美元计算的供应商问题是一个重要的地点问题,在一般指标和欧几里德指标中都对此进行了积极研究,还研究了许多变式,主要是一般指标。我们研究了两种变式,即欧几里德指标中的以美元计算的供应商优先价格和以美元计算的外部供应商优先价格。我们为这两种变式都获得了1美元(sqrt{3})以美元计算的配方算法,这是与先前已知的因数-3美元近似值相比的首次改进(据知,通用指标最有可能采用)。我们还研究了欧几里德指标中的以美元为单位的供应商问题,并表明不能将其近似于3美元(假定为美元);因此,欧几里德指标在本案中没有改进。