分层$\mathcal{F}$-聚类：聚类成树和有界直径图的近似与难度
Hierarchical $\mathcal{F}$-Clustering: Approximation and Hardness of Clustering into Trees and Bounded Diameter Graphs
AI总结：
研究分层$\mathcal{F}$-聚类问题，针对$\mathcal{F}$为树和有界直径图的情况，给出多项式时间近似算法，主要贡献是基于线性规划的近似框架，还证明在小集扩展假设下这两种聚类不能在常数因子内近似。
AI中文摘要：
考虑分层聚类问题的如下变体：通常构建分层聚类时，会递归划分数据直到每个聚类成为单元素集。我们放宽递归过程的停止条件，当剩余聚类是属于类$\mathcal{F}$的图时停止。我们称此问题为分层$\mathcal{F}$-聚类，并用适配的达斯古普塔聚类目标来衡量任何解的质量。我们研究$\mathcal{F}$的两种自然选择：树和有界直径图。我们分别给出了用于聚类成树和有界直径图的首个多项式时间$\mathcal{O}(\log n\cdot\log\log n)$和$\mathcal{O}(\log n)$近似算法。我们的主要技术贡献是基于线性规划的近似此类问题的框架。事实上，我们刻画了可应用我们方法的图类$\mathcal{F}$，表明它包括树和有界直径图。然而，我们的想法不限于它们，可能对其他结构也有用。大致来说，只要相应的平坦聚类问题（我们称为$p_{\mathcal{F}}$-划分）允许自然的整数线性规划公式以及具有可证明近似保证的舍入过程，我们的框架就适用。直观地，给定一组称为终端的顶点，问题是找到一个边集，其移除会导致每个终端满足某些依赖顶点的结构谓词。然后我们使用这些要素构建具有上述近似保证的聚类树。为补充这些结果，我们表明在小集扩展假设下，分层聚类成树和有界直径图都不能在任何常数因子内近似。
英文摘要：
Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton. We relax the halting condition of the recursive process to stop whenever the remaining cluster is a graph belonging to a class $\mathcal{F}$. We call this problem Hierarchical $\mathcal{F}$-Clustering and we measure the quality of any solution using adapted Dasgupta's clustering objective. We study two natural choices of $\mathcal{F}$: trees and graphs of bounded diameter. We present the first polynomial time $\mathcal{O}(\log n\cdot\log\log n)$ and $\mathcal{O}(\log n)$-approximation algorithms for clustering into trees and bounded diameter graphs respectively. Our main technical contribution is a framework for approximating such problems based on linear programming. In fact, we characterize graphs classes $\mathcal{F}$ for which our approach can be applied and show that it includes both trees and bounded diameter graphs. However, our ideas are not limited to them and might be useful for other structures as well. Broadly speaking, our framework applies whenever the corresponding flat clustering problem, which we call $p_{\mathcal{F}}$-Partitioning, admits a natural ILP formulation together with a rounding procedure with provable approximation guarantees. Intuitively, given a set of vertices called terminals, the problem is to find an edge set whose removal results in satisfying certain vertex-dependent structural predicate for each terminal. We then use these ingredients to build clustering trees with the aforementioned approximation guarantees. To complement these results, we show that both Hierarchical Clustering into trees and into bounded diameter graphs cannot be approximated within any constant factor under the Small Set Expansion Hypothesis.
