The ε-t-Net Problem
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We study a natural generalization of the classical ϵ-net problem (Haussler–Welzl 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( (1+log t)d/ϵlog1/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.
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