The ε -t-Net Problem

We study a natural generalization of the classical ε-net problem (Haussler and Welzl in Discrete Comput. Geom. 2(2), 127–151 (1987)), which we call the ε–t-net problem: Given a hypergraph on n vertices and parameters t and ε≥ t/ n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least εn contains a set in S. When t= 1 , this corresponds to the ε-net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an ε–t-net of size O((d(1 + log t) / ε) log (1 / ε)). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1 / ε) -sized ε–t-nets. We also present an explicit construction of ε–t-nets (including ε-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ε-nets (i.e., for t= 1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest. Finally, we use our techniques to generalize the notion of ε-approximation and to prove the existence of small-sized ε–t-approximations for sufficiently large hypergraphs with a bounded VC-dimension.