In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a $2^{\mathcal O(k)} \cdot n^{\mathcal O(1)}$-time algorithm where $k$ is the number of edges to be selected and $n$ the number of vertices. We also prove the existence of a problem kernel of size $\mathcal O(k^{5/2} )$, resolving an open problem posed in the literature. We consider parameters that are smaller than $k$, the number of edges to be selected, and $r$, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between $k$ or $r$ and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is $W[1]$-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to slim tree-cut width.

翻译：[翻译摘要] 在带颜色聚类问题中，需要将带有颜色标记的边（超）图进行聚类，其中颜色代表交互类型。具体而言，目标是尽可能多地选择边，但不得选择共享同一端点且颜色不同的两条边。等价地，该目标也可描述为：为顶点分配颜色，使其尽可能匹配边着色方案。由于该问题属于NP难问题，我们基于前人工作，研究其参数化复杂度。我们给出一个时间复杂度为$2^{\mathcal O(k)} \cdot n^{\mathcal O(1)}$的算法，其中$k$为待选边数，$n$为顶点数。我们还证明存在大小为$\mathcal O(k^{5/2})$的问题核，解决了文献中提出的开放性问题。我们考虑小于$k$（待选边数）和$r$（可删除边数）的参数，此类较小参数通过考虑$k$或$r$与其下界之间的差值获得。我们针对此类参数化下的带颜色聚类问题，同时给出了算法与下界。最后，我们通过证明带颜色聚类关于顶点覆盖数与树割宽度均为$W[1]$-难问题，但关于瘦树割宽度具有固定参数可解性，确定了该问题关于结构图参数的参数化复杂度。