On the Approximability of Partial VC Dimension [chapter]

Cristina Bazgan, Florent Foucaud, Florian Sikora
2016 Lecture Notes in 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 approximation complexity of Max Partial VC Dimension on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.
doi:10.1007/978-3-319-48749-6_7 fatcat:niczarrae5d2dnozkgp5qjr25e