The $ε$-$t$-Net Problem

Abstract

We study a natural generalization of the classical -net problem (Haussler-Welzl 1987), which we call the -t-net problem: Given a hypergraph on n vertices and parameters t and ≥ t n , find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least n contains a set in S. When t = 1, this corresponds to the -net problem.We prove that any sufficiently large hypergraph with VC-dimension d admits an -t-net of size O( (1+log t)d log 1 ). 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)-sized -t-nets.We also present an explicit construction of -t-nets (including -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of -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.