The Parameterized Complexity of Computing the VC-Dimension

TL;DR

This paper investigates the computational complexity of determining the VC-dimension, establishing tight bounds, fixed-parameter algorithms, and a new graph-based approach that improves existing complexity results.

Contribution

It provides tight complexity bounds for VC-dimension computation, introduces fixed-parameter algorithms, and designs a novel graph-based algorithm with improved exponential dependency on treewidth.

Findings

Naive exponential algorithm is asymptotically tight under ETH.

Fixed-parameter approximation algorithm based on maximum degree.

Efficient algorithm for VC-dimension via graph treewidth with single exponential dependency.

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).