In the Cluster Editing problem, sometimes known as (unweighted) Correlation Clustering, we must insert and delete a minimum number of edges to achieve a graph in which every connected component is a clique. Owing to its applications in computational biology, social network analysis, machine learning, and others, this problem has been widely studied for decades and is still undergoing active research. There exist several parameterized algorithms for general graphs, but little is known about the complexity of the problem on specific classes of graphs. Among the few important results in this direction, if only deletions are allowed, the problem can be solved in polynomial time on cographs, which are the $P_4$-free graphs. However, the complexity of the broader editing problem on cographs is still open. We show that even on a very restricted subclass of cographs, the problem is NP-hard, W[1]-hard when parameterized by the number $p$ of desired clusters, and that time $n^{o(p/\log p)}$ is forbidden under the ETH. This shows that the editing variant is substantially harder than the deletion-only case, and that hardness holds for the many superclasses of cographs (including graphs of clique-width at most $2$, perfect graphs, circle graphs, permutation graphs). On the other hand, we provide an almost tight upper bound of time $n^{O(p)}$, which is a consequence of a more general $n^{O(cw \cdot p)}$ time algorithm, where $cw$ is the clique-width. Given that forbidding $P_4$s maintains NP-hardness, we look at $\{P_4, C_4\}$-free graphs, also known as trivially perfect graphs, and provide a cubic-time algorithm for this class.

翻译：在聚类编辑问题（有时称为（非加权）相关聚类问题）中，我们必须插入和删除最少数量的边，使得所得图的每个连通分量都是一个团。由于该问题在计算生物学、社交网络分析、机器学习等领域的应用，几十年来已被广泛研究，并且目前仍处于活跃研究阶段。对于一般图，存在多种参数化算法，但该问题在特定图类上的复杂性却鲜为人知。在此方向的少数重要结果中，如果仅允许删除操作，该问题在cographs（即$P_4$-free图）上可在多项式时间内求解。然而，在cographs上更广泛的编辑问题的复杂性仍然悬而未决。我们证明，即使在cographs的一个非常受限的子类上，该问题也是NP难的，当以期望聚类数$p$为参数时是W[1]-难的，并且在ETH假设下不存在$n^{o(p/\log p)}$时间的算法。这表明编辑变体比仅允许删除的情况要困难得多，并且该困难性适用于cographs的许多超类（包括团宽度至多为$2$的图、完美图、圆图、置换图）。另一方面，我们给出了一个几乎紧的上界$n^{O(p)}$时间，这源于一个更一般的$n^{O(cw \cdot p)}$时间算法，其中$cw$是团宽度。鉴于禁止$P_4$子图仍保持NP难性，我们研究了$\{P_4, C_4\}$-free图（也称为平凡完美图），并为该类图提供了一个三次时间算法。