The Parameterized Complexity of Computing the VC-Dimension

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

Keywords: VC-dimension, parameterized complexity, hypergraphs, graphs, treewidth

TL;DR: We establish new results on the complexity of computing the VC-dimension, including fixed-parameter algorithms and running time lower bounds under the Exponential Time Hypothesis.

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 $2^{\mathcal{O}(\texttt{tw}\cdot \log \texttt{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\texttt{tw}$ (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).

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 20810