We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k = 2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $\mathcal{X} \subseteq \mathbb{R}^d$; a problem which is equivalent to Min Sum 2-Clustering.

翻译：我们证明了针对两个自然欧几里得聚类问题的最简单局部搜索启发式算法是PLS完全的。首先，我们证明，即使当$k = 2$时，用于$k$-均值聚类的Hartigan--Wong方法也是PLS完全的。其次，我们证明，即使边权由某个集合$\mathcal{X} \subseteq \mathbb{R}^d$中点之间的（平方）欧几里得距离给出时，用于最大割的Flip启发式算法也具有相同结论；该问题等价于最小和双聚类。