The Parameterized Complexity of Computing the VC-Dimension

• URL: http://arxiv.org/abs/2510.17451v2
• Date: Thu, 23 Oct 2025 10:56:40 GMT
• Title: The Parameterized Complexity of Computing the VC-Dimension
• Authors: Florent Foucaud, Harmender Gahlawat, Fionn Mc Inerney, Prafullkumar Tale,
• Abstract summary: 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.<n>We prove that naive $2mathcalO(mathcalV|)$-time algorithm is tight under the Exponential Time Hypothesis (ETH)
• Score: 5.53479503648814
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

Related papers

• Accelerated Evolving Set Processes for Local PageRank Computation [75.54334100808022]
  This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank computation.<n>We show that the time complexity of such localized methods is upper bounded by $mintildemathcalO(R2/epsilon2), tildemathcalO(m)$ to obtain an $epsilon$-approximation of the PPR vector.
  arXiv  Detail & Related papers  (2025-10-09T09:47:40Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Fast Maximum $k$-Plex Algorithms Parameterized by Small Degeneracy Gaps [30.06993761032835]
  The maximum $k$-plex problem is important but computationally challenging in applications such as graph mining and community detection. We present an exact algorithm parameterized by $g_k(G)$, which has the worst-case running time in the size of the input graph and exponential in $g_k(G)$. We further extend our discussion to an even smaller parameter $cg_k(G)$, the gap between the community-degeneracy bound and the size of the maximum $k$-plex.
  arXiv  Detail & Related papers  (2023-06-23T01:28:24Z)
• Detection-Recovery Gap for Planted Dense Cycles [72.4451045270967]
  We consider a model where a dense cycle with expected bandwidth $n tau$ and edge density $p$ is planted in an ErdHos-R'enyi graph $G(n,q)$. We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree algorithms.
  arXiv  Detail & Related papers  (2023-02-13T22:51:07Z)
• Private Isotonic Regression [54.32252900997422]
  We consider the problem of isotonic regression over a partially ordered set (poset) $mathcalX$ and for any Lipschitz loss function. We obtain a pure-DP algorithm that has an expected excess empirical risk of roughly $mathrmwidth(mathcalX) cdot log|mathcalX| / n$, where $mathrmwidth(mathcalX)$ is the width of the poset. We show that the bounds above are essentially the best that can be
  arXiv  Detail & Related papers  (2022-10-27T05:08:07Z)
• Inferring Hidden Structures in Random Graphs [13.031167737538881]
  We study the two inference problems of detecting and recovering an isolated community of emphgeneral structure planted in a random graph. We derive lower bounds for detecting/recovering the structure $Gamma_k$ in terms of the parameters $(n,k,q)$, as well as certain properties of $Gamma_k$, and exhibit computationally optimal algorithms that achieve these lower bounds.
  arXiv  Detail & Related papers  (2021-10-05T09:39:51Z)
• Hybrid Stochastic-Deterministic Minibatch Proximal Gradient: Less-Than-Single-Pass Optimization with Nearly Optimal Generalization [83.80460802169999]
  We show that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model. For loss factors, we prove that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model.
  arXiv  Detail & Related papers  (2020-09-18T02:18:44Z)
• On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems [86.92205445270427]
  We consider non-con minimax problems, $min_mathbfx max_mathhidoty f(mathbfdoty)$ efficiently.
  arXiv  Detail & Related papers  (2019-06-02T03:03:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.