On the Approximability of Partial VC Dimension

We introduce the problem Partial VC Dimension that asks, given a hypergraph \(H=(X,E)\) and integers k and \(\ell \), whether one can select a set \(C\subseteq X\) of k vertices of H such that the set \(\{e\cap C, e\in E\}\) of distinct hyperedge-intersections with C has size at least \(\ell \). The sets \(e\cap C\) define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case \(\ell =2^k\), and of Distinguishing Transversal, which corresponds to the case \(\ell =|E|\) (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 approximation complexity of Max Partial VC Dimension on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.

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