Coresets for $k$-means and $k$-median problems yield a small summary of the data, which preserve the clustering cost with respect to any set of $k$ centers. Recently coresets have also been constructed for constrained $k$-means and $k$-median problems. However, the notion of coresets has the drawback that (i) they can only be applied in settings where the input points are allowed to have weights, and (ii) in general metric spaces, the size of the coresets can depend logarithmically on the number of points. The notion of weak coresets, which have less stringent requirements than coresets, has been studied in the context of classical $k$-means and $k$-median problems. A weak coreset is a pair $(J,S)$ of subsets of points, where $S$ acts as a summary of the point set and $J$ as a set of potential centers. This pair satisfies the properties that (i) $S$ is a good summary of the data as long as the $k$ centers are chosen from $J$ only, and (ii) there is a good choice of $k$ centers in $J$ with cost close to the optimal cost. We develop this framework, which we call universal weak coresets, for constrained clustering settings. In conjunction with recent coreset constructions for constrained settings, our designs give greater data compression, are conceptually simpler, and apply to a wide range of constrained $k$-median and $k$-means problems.

翻译：针对$k$-均值和$k$-中位数问题的核心集可生成数据的小规模摘要，能保留关于任意$k$个中心点的聚类代价。近期，核心集也被构建用于带约束的$k$-均值和$k$-中位数问题。然而，核心集的概念存在以下缺陷：（i）仅适用于输入点允许带权重的场景；（ii）在一般度量空间中，核心集规模可能随点数量呈对数增长。弱核心集的概念对约束要求较低，已在经典$k$-均值和$k$-中位数问题背景下得到研究。弱核心集是一对点集$(J,S)$，其中$S$作为点集的摘要，$J$作为潜在中心集。该对满足以下性质：（i）只要$k$个中心点仅从$J$中选取，$S$即为数据良好的摘要；（ii）存在一组来自$J$、代价接近最优解的$k$个中心点。我们针对带约束聚类场景发展了该框架，称之为通用弱核心集。结合近期面向约束场景的核心集构建方法，我们的设计可实现更高数据压缩率，概念上更简洁，并适用于广泛的带约束$k$-中位数和$k$-均值问题。