We introduce Cluster Edge Modification problems with constraints on the size of the clusters and study their complexity. A graph $G$ is a cluster graph if every connected component of $G$ is a clique. In a typical Cluster Edge Modification problem such as the widely studied Cluster Editing, we are given a graph $G$ and a non-negative integer $k$ as input, and we have to decide if we can turn $G$ into a cluster graph by way of at most $k$ edge modifications -- that is, by adding or deleting edges. In this paper, we study the parameterized complexity of such problems, but with an additional constraint: The size difference between any two connected components of the resulting cluster graph should not exceed a given threshold. Depending on which modifications are permissible -- only adding edges, only deleting edges, both adding and deleting edges -- we have three different computational problems. We show that all three problems, when parameterized by $k$, admit single-exponential time FPT algorithms and polynomial kernels. Our problems may be thought of as the size-constrained or balanced counterparts of the typical Cluster Edge Modification problems, similar to the well-studied size-constrained or balanced counterparts of other clustering problems such as $k$-Means Clustering.

翻译：我们引入带有簇大小约束的簇边修改问题并研究其复杂性。图$G$被称为簇图，当且仅当其每个连通分量都是团。在典型的簇边修改问题（如广泛研究的簇编辑问题）中，给定图$G$和非负整数$k$作为输入，需要判断能否通过最多$k$次边修改操作（即添加或删除边）将$G$转化为簇图。本文研究了此类问题的参数化复杂性，但附加了一个约束：所得簇图的任意两个连通分量之间的大小差不应超过给定阈值。根据允许的修改方式（仅添加边、仅删除边、同时添加和删除边），我们得到三个不同的计算问题。研究表明，当以$k$为参数时，这三个问题均存在单指数时间FPT算法和多项式核。我们的问题可视为典型簇边修改问题的尺寸约束或平衡版本，类似于已被广泛研究的其他聚类问题（如$k$-均值聚类）的尺寸约束或平衡版本。