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The $ε$-$t$-Net Problem

Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, Yelena Yuditsky

Abstract

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.

The $ε$-$t$-Net Problem

Abstract

We study a natural generalization of the classical -net problem (Haussler--Welzl 1987), which we call the "--net problem": Given a hypergraph on vertices and parameters and , find a minimum-sized family of -element subsets of vertices such that each hyperedge of size at least contains a set in . When , this corresponds to the -net problem. We prove that any sufficiently large hypergraph with VC-dimension admits an --net of size . For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of -sized --nets. We also present an explicit construction of --nets (including -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of -nets (i.e., for ), 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.

Paper Structure

This paper contains 23 sections, 22 theorems, 6 equations, 2 figures.

Table of Contents

  1. Results
  2. Construction of Auxiliary Hypergraphs
  3. Some preparatory results
  4. A first hypergraph on t-subsets
  5. A smaller, well-behaved hypergraph on t-subsets
  6. Existence of Small -t-Nets
  7. Deterministic Construction of -t-Nets
  8. Deterministic construction of -nets
  9. Deterministic construction of -t-nets
  10. Extension of the Direct Construction of -2-Nets to -t-Nets
  11. t-VC-dimension versus classical VC-dimension
  12. Geometric -2-Nets
  13. Non-piercing regions
  14. Small union complexity
  15. More Geometric -2-Nets
  16. ...and 8 more sections

Key Result

Theorem 1

For every $\epsilon\in(0,1)$ and $t \in \mathbb{N}\setminus\{0\}$, every hypergraph on $\geq C_1 \left(\frac{t-1}{\epsilon}\right)^{d^*}$ vertices with VC-dimension $d$ and dual shatter function $\pi^*_H(m)\leq C m^{d^*}$ admits an $\epsilon$-$t$-net of size $O(\frac{d (1+\log t)}{\epsilon} \log \fr

Figures (2)

  • Figure 1: The binary entropy function.
  • Figure 2: A set $P$ of $n$ points such that in the hypergraph of $P$ with respect to axis-parallel rectangles, for $\epsilon=\frac{2}{n}$, each $\epsilon$-2-net is of size $\Omega(n^2)=\Omega(\frac{1}{\epsilon^2})$.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Lemma 3: VC71Sau72She72
  • Definition 4
  • Proposition 5
  • Corollary 6
  • Theorem 7: Welzl88
  • Definition 8
  • Remark 9
  • Theorem 9
  • ...and 25 more