Adaptive Sampling for Minimum-Norm $k$-Clustering

In $k$-clustering problems, we are given a metric space $(\mathcal{C}, d)$, and must choose a set $S$ of $k$ centers to open. Each client $j \in \mathcal{C}$ incurs an assignment cost, which is the distance between $j$ and center in $S$ that it has been assigned to. In this work, we study the \emph{minimum-norm $k$-clustering problem}, where we are given an arbitrary monotone symmetric norm $f$, and wish to open $k$ centers so as to minimize $f$(assignment-cost vector). This is a powerful generalization, encompassing many classical $k$-clustering problems including the $k$-median, $k$-means, and $k$-center problems. A simple and efficient algorithmic idea is that of \emph{adaptive sampling}, wherein we randomly choose the location of the next center to open with probability proportional to its ``cost" under the currently chosen set. While this has yielded fast algorithms for some $k$-clustering problem, little is known for settings \emph{without} ``min-sum" objectives. We devise the first adaptive-sampling-based bicriteria constant-factor approximation algorithm for general minimum-norm $k$-clustering, vastly expanding the scope of problems handled by adaptive sampling. For the special case of $\text{Top}_\ell$ norms, which form a building block of monotone symmetric norms, we show that adaptive sampling yields an $O(\log k)$-approximation algorithm.