We develop two simple and efficient approximation algorithms for the continuous $k$-medians problems, where we seek to find the optimal location of $k$ facilities among a continuum of client points in a convex polygon $C$ with $n$ vertices in a way that the total (average) Euclidean distance between clients and their nearest facility is minimized. Both algorithms run in $\mathcal{O}(n + k + k \log n)$ time. Our algorithms produce solutions within a factor of 2.002 of optimality. In addition, our simulation results applied to the convex hulls of the State of Massachusetts and the Town of Brookline, MA show that our algorithms generally perform within a range of 5\% to 22\% of optimality in practice.

翻译：我们针对连续$k$-中位点问题提出了两种简单高效的近似算法，该问题旨在寻找凸多边形$C$（具有$n$个顶点）内连续客户点群中$k$个设施的最优位置，使得客户与其最近设施之间的总（平均）欧几里得距离最小化。两种算法的时间复杂度均为$\mathcal{O}(n + k + k \log n)$，且所得解的最优性近似比均达到2.002。此外，我们将算法应用于马萨诸塞州及布鲁克莱恩镇凸包的实际仿真结果表明，在实践中，算法性能通常处于最优解的5%至22%范围内。