Parameterized and approximation complexity of Partial VC Dimension

Cristina Bazgan, Florent Foucaud, Florian Sikora

Abstract: We introduce the problem Partial VC Dimension that asks, given a hypergraph H=(X,E)<math><mi is="true">H</mi><mo is="true">=</mo><mo stretchy="false" is="true">(</mo><mi is="true">X</mi><mo is="true">,</mo><mi is="true">E</mi><mo stretchy="false" is="true">)</mo></math> and integers k and ℓ, whether one can select a set C⊆X<math><mi is="true">C</mi><mo is="true">⊆</mo><mi is="true">X</mi></math> of k vertices of H such that the set {e∩C,e∈E}<math><mo stretchy="false" is="true">{</mo><mi is="true">e</mi><mo is="true">∩</mo><mi is="true">C</mi><mo is="true">,</mo><mi is="true">e</mi><mo is="true">∈</mo><mi is="true">E</mi><mo stretchy="false" is="true">}</mo></math> of distinct hyperedge-intersections with C has size at least ℓ. The sets e∩C<math><mi is="true">e</mi><mo is="true">∩</mo><mi is="true">C</mi></math> define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case ℓ=2k<math><mi is="true">ℓ</mi><mo is="true">=</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math>, and of Distinguishing Transversal, which corresponds to the case ℓ=|E|<math><mi is="true">ℓ</mi><mo is="true">=</mo><mo stretchy="false" is="true">|</mo><mi is="true">E</mi><mo stretchy="false" is="true">|</mo></math> (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.