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

2025-12-14

AI Summary

This paper studies the Minimum Consistent Subset (MCS) problem in graph metric spaces: given a labeled graph, find the smallest label subset such that every vertex has a nearest neighbor of the same label within that subset. MCS underpins supervised clustering and efficient instance selection but is computationally hard. We establish, for the first time, that MCS is fixed-parameter tractable (FPT) with respect to two structural graph parameters—vertex cover number (vc) and neighborhood diversity (nd)—surpassing prior results restricted to trees or other highly constrained graph classes. Our algorithms run in time $vc^{O(vc)} cdot ext{poly}(n,c)$ and $nd^{O(nd)} cdot ext{poly}(n,c)$, respectively, where $n$ is the number of vertices and $c$ the number of labels, maintaining polynomial dependence on $c$. The approach integrates structural graph analysis, dynamic programming, and enumeration pruning. These advances significantly enhance the feasibility and efficiency of computing MCS on large-scale, multi-labeled graph data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 $vc^{mathcal{O}(vc)}cdot ext{Poly}(n,c)$, and another algorithm with runtime $nd^{mathcal{O}(nd)}cdot ext{Poly}(n,c)$. 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.

Problem

• Computes smallest labeled subset for supervised clustering
• Extends Minimum Consistent Subset problem to graph metrics
• Develops efficient algorithms using vertex cover and neighborhood diversity

Innovation

• Parameterized algorithms using vertex cover number
• Fixed-parameter tractable approach with neighborhood diversity
• Polynomial dependence on colors for graph MCS