We explore Cluster Editing and its generalization Correlation Clustering with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both problems are NP-hard, yet they exhibit significant differences in parameterized complexity and approximability. For Cluster Editing with Permissive Vertex Splitting, we show a polynomial kernel when parameterized by the solution size and develop a polynomial-time algorithm with approximation factor 7. In the case of Correlation Clustering, we establish para-NP-hardness when parameterized by solution size and demonstrate that computing an $n^{1-\epsilon}$-approximation is NP-hard for any constant $\epsilon > 0$. Additionally, we extend the established link between Correlation Clustering and Multicut to the setting with permissive vertex splitting.

翻译：我们探讨了聚类编辑及其推广形式——相关性聚类，引入了一种称为“许可性顶点分裂”的新操作，该操作旨在解决信息不确定条件下重叠聚类的发现问题。我们确定这两个问题均为NP难问题，但在参数化复杂度和可近似性方面存在显著差异。对于许可性顶点分裂的聚类编辑问题，我们证明了当以解大小作为参数时存在一个多项式核，并开发了一个具有近似因子7的多项式时间算法。在相关性聚类的情况下，我们建立了以解大小作为参数时的参数化NP难度，并证明对于任意常数$\epsilon > 0$，计算$n^{1-\epsilon}$-近似解是NP难的。此外，我们还将相关性聚类与多割之间已知的关联性推广到了具有许可性顶点分裂的设定中。