The $ε$-$t$-Net Problem
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866909147022229504 |
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| author | Alon, Noga Jartoux, Bruno Keller, Chaya Smorodinsky, Shakhar Yuditsky, Yelena |
| author_facet | Alon, Noga Jartoux, Bruno Keller, Chaya Smorodinsky, Shakhar Yuditsky, Yelena |
| contents | 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 $ε\geq \frac t n$, 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 $O(\frac{ (1+\log t)d}ε \log \frac{1}ε)$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}ε)$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_07061 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The $ε$-$t$-Net Problem Alon, Noga Jartoux, Bruno Keller, Chaya Smorodinsky, Shakhar Yuditsky, Yelena Discrete Mathematics Computational Geometry Combinatorics 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 $ε\geq \frac t n$, 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 $O(\frac{ (1+\log t)d}ε \log \frac{1}ε)$. For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $O(\frac{1}ε)$-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. |
| title | The $ε$-$t$-Net Problem |
| topic | Discrete Mathematics Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/2003.07061 |