Weighted ε-net: Theory & Applications

• Weighted ε-nets are a generalization of classical ε-nets that use multiple thresholds to ensure robust range coverage in finite point sets and metric spaces.
• They interpolate between classical ε-nets and ε-approximations, offering enhanced geometric guarantees with improved parameters for small net sizes, especially for convex sets and axis-parallel boxes.
• Applications include efficient landmark selection in graph metrics and persistent homology, where they balance coverage, packing, and computational efficiency.

A weighted ε-net generalizes classical ε-nets by introducing a multi-threshold coverage principle on finite point sets or finite metric spaces. Weighted ε-nets interpolate between the minimal guarantee of classical ε-nets—ensuring large ranges are “hit” at least once—and the uniform distribution quality of ε-approximations. Motivated by the limitations in the quantitative bounds for classical weak ε-nets, weighted ε-nets enable more nuanced geometric and combinatorial guarantees, often yielding much improved parameters for small net sizes, especially over convex sets and axis-parallel boxes. In discrete metric spaces, the concept naturally extends to landmark selection in graphs, controlling the trade-off between covering, packing, and the accuracy of topological representations such as lazy witness complexes.

1. Definitions and Foundational Concepts

Weighted ε-nets are constructed for a finite point set $P\subset\mathbb{R}^d$ with $|P|=n$ and a range space $\mathcal{R}$ (e.g., convex sets, boxes, or ranges defined by other families). The key definitions are:

• Classical ε-net: A subset $N\subseteq P$ is an ε-net for $(P,\mathcal{R})$ if for every $R\in\mathcal{R}$ with $|R\cap P|\geq \varepsilon n$, we have $R\cap N \neq \emptyset$.
• ε-approximation: A subset $A\subseteq P$ is an ε-approximation if for every $R\in\mathcal{R}$, $|P|=n$0.
• Weighted ε-net: For $|P|=n$1 and thresholds $|P|=n$2, a (weak) weighted ε-net of size $|P|=n$3 with thresholds $|P|=n$4 is a sequence $|P|=n$5 satisfying: for all $|P|=n$6 and $|P|=n$7, if $|P|=n$8 then $|P|=n$9.

Weighted ε-nets allow zero coverage for small ranges, but guarantee increasing intersection with the net as the set size surpasses the thresholds $\mathcal{R}$0 (Bertschinger et al., 2020).

In finite graph metrics $\mathcal{R}$1, an ε-net is defined as a landmark set $\mathcal{R}$2 with simultaneous covering and packing properties:

• $\mathcal{R}$3, $\mathcal{R}$4 with $\mathcal{R}$5 (covering).
• $\mathcal{R}$6, $\mathcal{R}$7 (packing) (Arafat et al., 2020).

2. Main Results for Weighted ε-nets in Euclidean Spaces

Weighted ε-nets attain significantly improved thresholds over classical ε-nets, particularly for small $\mathcal{R}$8 and specific range families:

• Convex Sets ($\mathcal{R}$9): For size $N\subseteq P$0, if the thresholds satisfy $N\subseteq P$1 and $N\subseteq P$2, then there exist $N\subseteq P$3 forming a weighted ε-net such that every convex $N\subseteq P$4 with $N\subseteq P$5 contains at least one of $N\subseteq P$6; $N\subseteq P$7 implies both are contained. Choosing $N\subseteq P$8, $N\subseteq P$9 gives the strongest explicit result; for $(P,\mathcal{R})$0, $(P,\mathcal{R})$1, $(P,\mathcal{R})$2 is attainable.
• Axis-Parallel Boxes: For size $(P,\mathcal{R})$3 in $(P,\mathcal{R})$4, thresholds $(P,\mathcal{R})$5 and $(P,\mathcal{R})$6 yield valid weighted ε-nets. In the plane, $(P,\mathcal{R})$7, $(P,\mathcal{R})$8 is achievable. For $(P,\mathcal{R})$9, the optimal conditions become $R\in\mathcal{R}$0, $R\in\mathcal{R}$1, and $R\in\mathcal{R}$2.

Weighted ε-nets strictly interpolate between classical ε-nets and ε-approximations:

$R\in\mathcal{R}$3

Any ε-approximation of size $R\in\mathcal{R}$4 is a weighted ε-net with all thresholds $R\in\mathcal{R}$5; any weighted ε-net with $R\in\mathcal{R}$6 is a classical ε-net (Bertschinger et al., 2020).

3. Proof Techniques and Structural Properties

For convex sets, the core proof leverages partitioning via Helly-type arguments. The point set $R\in\mathcal{R}$7 is divided using a halving hyperplane, splitting convex sets into families with overlapping high-cardinality intersections. Helly’s theorem guarantees the existence of intersection points for these families, forming the weighted ε-net.

For axis-parallel boxes, coordinate-splitting techniques are employed: iterative slicing along coordinate hyperplanes partitions $R\in\mathcal{R}$8 into slabs, ensuring each selected slab contains exactly the threshold number of points. By the strong Helly property of boxes, the constructed intersection points serve as the net.

Empirical lower bounds demonstrate that, for $R\in\mathcal{R}$9 and convex sets in the plane, no weighted ε-net can achieve $|R\cap P|\geq \varepsilon n$0. In $|R\cap P|\geq \varepsilon n$1, the bound is $|R\cap P|\geq \varepsilon n$2 for size-two nets, with specific constructions provided (Bertschinger et al., 2020).

4. Weighted ε-nets in Graph Metric Spaces

In finite graphs $|R\cap P|\geq \varepsilon n$3, a weighted ε-net (Editors' term: "landmark ε-net") is a set $|R\cap P|\geq \varepsilon n$4 satisfying:

• Covering: $|R\cap P|\geq \varepsilon n$5 is an ε-sample, i.e., every $|R\cap P|\geq \varepsilon n$6 is within $|R\cap P|\geq \varepsilon n$7 of some $|R\cap P|\geq \varepsilon n$8.
• Packing: $|R\cap P|\geq \varepsilon n$9 is ε-sparse, i.e., all pairs have separation $R\cap N \neq \emptyset$0.

Three main landmark-selection algorithms induce weighted ε-nets:

• Greedy-ε-Net: Iteratively chooses the vertex whose ε-ball covers the most uncovered vertices.
• Iterative-ε-Net: Successively selects landmarks from uncovered regions using ε/2ε frontiers.
• SPTpruning-ε-Net: Uses shortest-path trees and level-pruning, followed by ε-ball BFS pruning.

These algorithms guarantee the dual covering and packing properties, providing a principled trade-off between the number of landmarks and coverage tightness. As ε increases, fewer landmarks are needed but at the cost of lower approximation fidelity in topological data analysis (Arafat et al., 2020).

5. Applications and Trade-offs

Weighted ε-nets offer substantial practical advantages:

• For geometric approximation, small weighted ε-nets capture large-range structure with vastly improved quantitative guarantees compared to classical nets.
• In persistent homology of large structures (e.g., graphs), landmark-based approximations using ε-nets yield lazy witness complexes with provable interleaving with full Vietoris–Rips complexes—specifically, for $R\cap N \neq \emptyset$1,

$R\cap N \neq \emptyset$2

This ensures the persistent barcodes are interleaved within a factor of $R\cap N \neq \emptyset$3 in scale. Empirical results show that the number of landmarks and computational time decrease rapidly as $R\cap N \neq \emptyset$4 increases, while the bottleneck distance between persistence diagrams remains well below this theoretical bound even for moderate ε. The iterative ε-net method provides nearly linear-time computations and achieves favorable trade-offs between landmark set size and coverage quality (Arafat et al., 2020).

6. Comparative Analysis and Open Problems

Weighted ε-nets enable markedly reduced thresholds for range coverage with small nets, outperforming classical weak ε-nets for given $R\cap N \neq \emptyset$5 in numerous geometric range spaces (notably convex sets and boxes). However, they do not guarantee as fine a correspondence between set size and fraction in all ranges as ε-approximations do. Tightening bounds for $R\cap N \neq \emptyset$6, exploring weighted ε-net thresholds for other range spaces (e.g., halfspaces, balls, simplices, geodesic ranges), and designing constructive, efficient algorithms for finding optimal weighted ε-nets are compelling open directions. Most current existence proofs rely on nonconstructive arguments using variations of Helly’s theorem, and algorithmic instantiation remains an unresolved challenge for many cases (Bertschinger et al., 2020).

7. Summary Table: Weighted ε-net Boundaries and Algorithms

Context	Best-known thresholds (k=2)	Key Construction/Method
Convex sets in $R\cap N \neq \emptyset$7	$R\cap N \neq \emptyset$8, $R\cap N \neq \emptyset$9	Helly-type partition
Axis-parallel boxes in $A\subseteq P$0	$A\subseteq P$1, $A\subseteq P$2	Coordinate-splitting
Graph metric spaces	ε (geodesic distance)	Greedy/Iterative/SPTpruning

Weighted ε-nets thus provide a flexible and quantitatively powerful framework for approximation in both geometric discrepancy theory and topological data analysis, acting as a pivotal interpolation between sparse hitting sets and full uniformity of ε-approximations, with ongoing research into tighter bounds and scalable algorithms (Bertschinger et al., 2020, Arafat et al., 2020).