The Parameterized Complexity of Computing the VC-Dimension

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 H=(V,E), we prove that the naive 2O(|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 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^O(rmtw· log rmtw)· |V|-time algorithm for any graph G=(V,E) of treewidth rmtw (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).