AI 中文总结
研究最小范数k聚类问题，提出自适应采样算法，设计出首个基于此的双准则常数因子近似算法，对一般最小范数k聚类适用，对特殊的\(\text{Top}_\ell\)范数产生\(O(\log k)\)近似算法，扩展了自适应采样处理问题的范围。
AI 中文摘要
在k聚类问题中，给定一个度量空间(\(\mathcal{C}, d\))，必须选择k个中心的集合S来开放。每个客户端\(j \in \mathcal{C}\)会产生一个分配成本，即它被分配到的S中中心与它之间的距离。本文研究最小范数k聚类问题，给定任意单调对称范数f，希望开放k个中心以最小化f(分配成本向量)。这是一个强大的推广，涵盖了许多经典的k聚类问题。提出了自适应采样算法，随机选择下一个要开放的中心位置，概率与其在当前所选集合下的“成本”成正比。对于一般最小范数k聚类，设计了首个基于自适应采样的双准则常数因子近似算法，对于特殊的\(\text{Top}_\ell\)范数，自适应采样产生一个\(O(\log k)\)近似算法。
英文摘要
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.
