Parameterized and approximation complexity of Partial VC Dimension

Cristina Bazgan (1) , Florent Foucaud (2) , Florian Sikora (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
2 LIMOS - Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes

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= 2k, and of Distinguishing Transversal, which corresponds to the case ℓ=|E|(the latter is also 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.

Mots clés: Hypergraphs; Parameterized complexity; Approximation complexity; VC dimension; Distinguishing Transversal; Partial problems

Domaines: Informatique [cs]

Cristina Bazgan, Florent Foucaud, Florian Sikora. Parameterized and approximation complexity of Partial VC Dimension. Theoretical Computer Science, 2019, 766, ⟨10.1016/j.tcs.2018.09.013⟩. ⟨hal-02091342⟩