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The Parameterized Complexity of Computing the VC-Dimension

Florent Foucaud, Harmender Gahlawat, Fionn Mc Inerney, Prafullkumar Tale

TL;DR

This work analyzes the computational complexity of VC-Dimension for set systems and graphs. It provides ETH-tight lower bounds for the naive brute-force approach and delivers a spectrum of parameterized results, including a $1$-additive approximation parameterized by the maximum degree $\Delta$ and a fixed-parameter algorithm by the dimension $D$, along with a fast $2^{\mathcal{O}({\rm tw}\cdot \log {\rm tw})}\cdot |V|$ algorithm for Gen-VC-Dimension on graphs of treewidth ${\rm tw}$. The results delineate tractable versus intractable parameter regimes and contrast the treewidth-based algorithm with existing double-exponential dependencies under ETH. They also establish strong lower bounds for related graph- and hypergraph-VC problems, guiding practical computation for graph-structured data and set systems. Overall, the paper significantly advances the understanding of VC-dimension computation from both algorithmic and complexity-theoretic viewpoints.

Abstract

The VC-dimension is a well-studied and fundamental complexity measure 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 design a $2^{\mathcal{O}(\rm{tw}\cdot \log \rm{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\rm{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

The Parameterized Complexity of Computing the VC-Dimension

TL;DR

This work analyzes the computational complexity of VC-Dimension for set systems and graphs. It provides ETH-tight lower bounds for the naive brute-force approach and delivers a spectrum of parameterized results, including a -additive approximation parameterized by the maximum degree and a fixed-parameter algorithm by the dimension , along with a fast algorithm for Gen-VC-Dimension on graphs of treewidth . The results delineate tractable versus intractable parameter regimes and contrast the treewidth-based algorithm with existing double-exponential dependencies under ETH. They also establish strong lower bounds for related graph- and hypergraph-VC problems, guiding practical computation for graph-structured data and set systems. Overall, the paper significantly advances the understanding of VC-dimension computation from both algorithmic and complexity-theoretic viewpoints.

Abstract

The VC-dimension is a well-studied and fundamental complexity measure 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 , we prove that the naive -time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a -additive fixed-parameter approximation algorithm when parameterized by the maximum degree of 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 design a -time algorithm for any graph of treewidth (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).
Paper Structure (6 sections, 11 theorems, 2 figures)

This paper contains 6 sections, 11 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithmic Lower Bounds
  4. Parameterized Complexity of VC-Dimension
  5. Tractability via Treewidth
  6. Conclusion

Key Result

Proposition 1

Assuming the $\mathsf{ETH}$, there exists a constant $\epsilon_c > 0$ such that 3-Coloring does not admit an algorithm running in $2^{\epsilon_c \cdot |V(G')|}\cdot |V(G')|^{\mathcal{O}\xspace(1)}$ time.

Figures (2)

  • Figure 1: The graph $G$ constructed from $G'$, where $X$ (vertex cover of $G$) and $Y= I_1\cup I_2 \cup I_{\geq 3}$ are independent sets. An edge between $w\in I_{\geq 3}$ and $U_i$ signifies that each vertex in $U_i$ is adjacent to $w$.
  • Figure 2: An illustration for the pattern graph $\mathcal{P}$. Here, $S\subseteq X$, $W\subseteq Y$, and $W$ witnesses $S$.

Theorems & Definitions (26)

  • Proposition 1: bookParameterized
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Claim 4
  • Claim 5
  • Claim 6
  • Claim 7
  • Lemma 8
  • ...and 16 more