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 ε n 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.