Learning with Structure: Computing Consistent Subsets on Structurally-Regular Graphs

The Minimum Consistent Subset (MCS) problem arises naturally in the context of supervised clustering and instance selection, both of which are critical in enabling scalable and interpretable learning on large datasets. In supervised clustering, one aims to infer a meaningful partitioning of data using a small labeled subset, where classification is typically performed via nearest neighbor rules. 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 $\textit{graph metrics}$, 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 ($nd$). Specifically, we design efficient algorithms for graphs exhibiting small $vc$ or small $nd$, which frequently arise in real-world domains characterized by local sparsity or repetitive structure. These parameters are particularly relevant because they capture structural properties that often correlate with the tractability of otherwise hard problems. Graphs with small vertex cover are "almost independent sets", representing sparse interactions, while graphs with small neighborhood diversity exhibit a high degree of symmetry and regularity. Importantly, small neighborhood diversity can occur even in dense graphs, a property frequently observed in domains such as social networks with modular communities or knowledge graphs with repeated relational patterns. Thus, algorithms designed to work efficiently for graphs with small neighborhood diversity are capable of efficiently solving MCS in complex settings where small vertex covers may not exist. We develop an algorithm with running time $vc^{O(vc)}\cdot \text{Poly}(n,c)$, and another algorithm with runtime $nd^{O(nd)}\cdot \text{Poly}(n,c)$. 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.