In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $ω: {V \choose 2}\to Z$ and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most $k$ is FPT with respect to $k$ but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by $k+d$, where $d$ is the degeneracy of the fuzzy edge graph, and when parameterized by $k+c$, where $c$ is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.

翻译：在关联聚类问题中，输入是一个图 $G=(V,E)$，带有权函数 $ω: {V \choose 2}\to Z$，任务是划分顶点集为若干簇，使得簇间边与簇内缺失边的总权重最小化。由于关联聚类与边多割问题之间的紧密联系，判定是否存在总代价至多为 $k$ 的划分相对于 $k$ 是固定参数可解的，但多项式核化器可能不存在。我们研究了模糊边图（即由权重为0的边诱导的图）结构对问题复杂度的影响。我们特别证明，当参数化为 $k+d$（其中 $d$ 为模糊边图的退化度）以及参数化为 $k+c$（其中 $c$ 为模糊边图的闭包）时，关联聚类承认多项式问题核。对于边和非边诱导的图具有非常受限结构的若干情形，我们通过证明困难性来补充这些正面结果。