Parameterized and approximation complexity of Partial VC Dimension

Cristina Bazgan, Florent Foucaud, Florian Sikora
2019 Theoretical Computer Science  
We introduce the problem Partial VC Dimension that asks, given a hypergraph H = (X, E) and integers k and ℓ, whether one can select a set C ⊆ X of k vertices of H such that the set {e ∩ C, e ∈ E} of distinct hyperedge-intersections with C has size at least ℓ. The sets e ∩ C define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case ℓ = 2 k , and of Distinguishing Transversal, which corresponds to the case ℓ = |E| (the latter is
more » ... known as Test Cover in the dual hypergraph). We also introduce the associated fixed-cardinality maximization problem Max Partial VC Dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices. We study the algorithmic complexity of Partial VC Dimension and Max Partial VC Dimension both on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs. ⋆ A preliminary version of this work containing the approximation complexity results appeared in [8] .
doi:10.1016/j.tcs.2018.09.013 fatcat:7f62vhskhbetbawvjngdjae4oi