We consider the approximability of center-based clustering problems where the points to be clustered lie in a metric space, and no candidate centers are specified. We call such problems "continuous", to distinguish from "discrete" clustering where candidate centers are specified. For many objectives, one can reduce the continuous case to the discrete case, and use an $\alpha$-approximation algorithm for the discrete case to get a $\beta\alpha$-approximation for the continuous case, where $\beta$ depends on the objective: e.g. for $k$-median, $\beta = 2$, and for $k$-means, $\beta = 4$. Our motivating question is whether this gap of $\beta$ is inherent, or are there better algorithms for continuous clustering than simply reducing to the discrete case? In a recent SODA 2021 paper, Cohen-Addad, Karthik, and Lee prove a factor-$2$ and a factor-$4$ hardness, respectively, for continuous $k$-median and $k$-means, even when the number of centers $k$ is a constant. The discrete case for a constant $k$ is exactly solvable in polytime, so the $\beta$ loss seems unavoidable in some regimes. In this paper, we approach continuous clustering via the round-or-cut framework. For four continuous clustering problems, we outperform the reduction to the discrete case. Notably, for the problem $\lambda$-UFL, where $\beta = 2$ and the discrete case has a hardness of $1.27$, we obtain an approximation ratio of $2.32 < 2 \times 1.27$ for the continuous case. Also, for continuous $k$-means, where the best known approximation ratio for the discrete case is $9$, we obtain an approximation ratio of $32 < 4 \times 9$. The key challenge is that most algorithms for discrete clustering, including the state of the art, depend on linear programs that become infinite-sized in the continuous case. To overcome this, we design new linear programs for the continuous case which are amenable to the round-or-cut framework.

翻译：我们认为以中心为主的群集问题的近似性, 其中要分组的点位于一个计量空间, 没有指定候选中心 。 我们称之为“ 持续” 问题, 以区别于指定候选中心 的“ 分解” 群集 。 对于许多目标, 人们可以减少离散案例的连续案例, 并且对离散案例使用 $alpha$- accolation 算法, 以获得 $\ bal- pal- acceration 的连续案例。 在最新的SOD 2021 纸上, Cone- adad, Karthik 和 Lee 证明, 美元取决于目标 : 例如, 美元- more, 美元, 美元= 2, 美元- betata = 美元, 美元 美元 = 4 美元 。 我们的不断变换的阵列程序, 美元 美元 直立地, 美元 直立地, 直立地, 直立地, 美元 直立地, 直立地。