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/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.
@InProceedings{alon_et_al:LIPIcs.SoCG.2020.5, author = {Alon, Noga and Jartoux, Bruno and Keller, Chaya and Smorodinsky, Shakhar and Yuditsky, Yelena}, title = {{The \epsilon-t-Net Problem}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.5}, URN = {urn:nbn:de:0030-drops-121639}, doi = {10.4230/LIPIcs.SoCG.2020.5}, annote = {Keywords: epsilon-nets, geometric hypergraphs, VC-dimension, linear union complexity} }
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