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).
Categories: cs.CC cs.AI cs.DM cs.LG math.CO