The Minimum Consistent Subset (MCS) problem arises naturally in the context of supervised clustering and instance selection. In supervised clustering, one aims to infer a meaningful partitioning of data using a small labeled subset. However, the sheer volume of training data in modern applications poses a significant computational challenge. The MCS problem formalizes this goal: given a labeled dataset $\mathcal{X}$ in a metric space, the task is to compute a smallest subset $S \subseteq \mathcal{X}$ such that every point in $\mathcal{X}$ shares its label with at least one of its nearest neighbors in $S$. Recently, the MCS problem has been extended to graph metrics, where distances are defined by shortest paths. Prior work has shown that MCS remains NP-hard even on simple graph classes like trees, though an algorithm with runtime $\mathcal{O}(2^{6c} \cdot n^6)$ is known for trees, where $c$ is the number of colors and $n$ the number of vertices. This raises the challenge of identifying graph classes that admit algorithms efficient in both $n$ and $c$. In this work, we study the Minimum Consistent Subset problem on graphs, focusing on two well-established measures: the vertex cover number ($vc$) and the neighborhood diversity ($nd$). We develop an algorithm with running time $vc^{\mathcal{O}(vc)}\cdot\text{Poly}(n,c)$, and another algorithm with runtime $nd^{\mathcal{O}(nd)}\cdot\text{Poly}(n,c)$. In the language of parameterized complexity, this implies that MCS is fixed-parameter tractable (FPT) parameterized by the vertex cover number and the neighborhood diversity. Notably, our algorithms remain efficient for arbitrarily many colors, as their complexity is polynomially dependent on the number of colors.

翻译：最小一致子集（MCS）问题在监督聚类和实例选择的背景下自然产生。在监督聚类中，目标是通过少量标记子集推断数据的有意义划分。然而，现代应用中训练数据的庞大规模带来了显著的计算挑战。MCS问题形式化了这一目标：给定度量空间中的标记数据集 $\mathcal{X}$，任务是计算最小子集 $S \subseteq \mathcal{X}$，使得 $\mathcal{X}$ 中的每个点与其在 $S$ 中的至少一个最近邻共享相同标签。最近，MCS问题已扩展至图度量空间，其中距离由最短路径定义。先前研究表明，即使在简单图类（如树结构）上，MCS问题仍保持NP难特性，尽管已知在树结构上存在运行时间为 $\mathcal{O}(2^{6c} \cdot n^6)$ 的算法，其中 $c$ 为颜色数量，$n$ 为顶点数量。这引出了识别在 $n$ 和 $c$ 上均具有高效算法的图类的挑战。在本研究中，我们探究图上的最小一致子集问题，重点关注两个经典度量指标：顶点覆盖数（$vc$）与邻域多样性（$nd$）。我们开发了运行时间为 $vc^{\mathcal{O}(vc)}\cdot\text{Poly}(n,c)$ 的算法，以及运行时间为 $nd^{\mathcal{O}(nd)}\cdot\text{Poly}(n,c)$ 的另一种算法。在参数化复杂度理论框架下，这表明MCS问题在顶点覆盖数和邻域多样性参数化下是固定参数可解的。值得注意的是，我们的算法在任意多颜色情况下仍保持高效性，因为其复杂度与颜色数量呈多项式相关。