The Parameterized Complexity of Computing the VC-Dimension
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909864551251968 |
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| author | Foucaud, Florent Gahlawat, Harmender Inerney, Fionn Mc Tale, Prafullkumar |
| author_facet | Foucaud, Florent Gahlawat, Harmender Inerney, Fionn Mc Tale, Prafullkumar |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Parameterized Complexity of Computing the VC-Dimension Foucaud, Florent Gahlawat, Harmender Inerney, Fionn Mc Tale, Prafullkumar Computational Complexity Artificial Intelligence Discrete Mathematics Machine Learning Combinatorics 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). |
| title | The Parameterized Complexity of Computing the VC-Dimension |
| topic | Computational Complexity Artificial Intelligence Discrete Mathematics Machine Learning Combinatorics |
| url | https://arxiv.org/abs/2510.17451 |