The Parameterized Complexity of Computing the VC-Dimension

2025-10-20

AI Summary

This paper investigates the exact and approximate computation of the Vapnik–Chervonenkis (VC) dimension within the framework of parameterized complexity. For hypergraphs, it establishes that the VC dimension admits both a 1-additive fixed-parameter tractable (FPT) approximation algorithm parameterized by maximum degree Δ and an FPT enumerative algorithm parameterized by dimension d—demonstrating that these are the only structural parameters yielding FPT solvability. Extending to graphs, the authors introduce a generalized VC dimension based on treewidth tw and devise a single-exponential FPT algorithm running in $2^{O(tw)} cdot ext{poly}(n)$ time. Through conditional lower bounds under the Exponential Time Hypothesis (ETH), treewidth-based decomposition, and a hypergraph-to-graph structural mapping, the work establishes fine-grained complexity boundaries for VC dimension computation. It provides the first systematic characterization of the completeness of admissible parameters and delineates the fundamental algorithmic limits for this problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

The VC-dimension is a fundamental and well-studied measure of the complexity of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $mathcal{H}=(mathcal{V},mathcal{E})$, we prove that the naive $2^{mathcal{O}(|mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a 1-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We show that it is fixed-parameter tractable parameterized by the treewidth of the graph (which, in the case of set systems, applies to the treewidth of its incidence graph). In contrast with closely related problems whose dependency on the treewidth is necessarily double-exponential (assuming the ETH), our algorithm has a relatively low dependency on the treewidth.

Problem

• Computing VC-dimension complexity under Exponential Time Hypothesis
• Developing approximation algorithms for VC-dimension with structural parameters
• Generalizing VC-dimension computation to graphs with treewidth parameterization

Innovation

• Proves tightness of naive VC-dimension algorithm under ETH
• Provides fixed-parameter approximations using structural parameters
• Achieves low treewidth dependency for generalized VC-dimension problem