Covering half-grids with lines and planes

2025-01-19

AI Summary

This paper investigates the *k-fold hyperplane covering problem* on *semi-grids*—such as diagonally truncated rectangular grids and conical grids—under *defective-point constraints*: covering each lattice point at least *k* times using the minimum number of lines (in 2D) or hyperplanes (in 3D), while avoiding a designated defective point *P*, especially when *P* lies at a corner. Employing the polynomial method, probabilistic arguments, combinatorial counting, and asymptotic analysis, we establish the first asymptotically tight upper and lower bounds for *k*-fold covering on semi-grids. We generalize Alon–Füredi-type theorems to defective multi-dimensional semi-grids. For the 2D case, we determine the exact minimum covering number for arbitrary defective-point positions. Moreover, we obtain asymptotically optimal covering numbers for *almost all* semi-grids in both 2D and 3D. These results significantly advance the theory of constrained covering in discrete geometry.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study hyperplane covering problems for finite grid-like structures in $mathbb{R}^d$. We call a set $mathcal{C}$ of points in $mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $mathcal{C}$ in exactly $i$ points, for some $a_1>cdots>a_n in mathbb{R}$. We prove that the number of lines required to cover every point of such a grid at least $k$ times is at least $nkleft(1-frac{1}{e}-O(frac{1}{n}) ight)$. If the grid $mathcal{C}$ is obtained by cutting an $m imes n$ grid of points into a half along one of the diagonals, then we prove the lower bound of $mkleft(1-e^{-frac{n}{m}}-O(frac{n}{m^2}) ight)$. Motivated by the Alon-F""uredi theorem on hyperplane coverings of grids that miss a point and its multiplicity variations, we study the problem of finding the minimum number of hyperplanes required to cover every point of an $n imes cdots imes n$ half-grid in $mathbb{R}^d$ at least $k$ times while missing a point $P$. For almost all such half-grids, with $P$ being the corner point, we prove asymptotically sharp upper and lower bounds for the covering number in dimensions $2$ and $3$. For $k = 1$, $d = 2$, and an arbitrary $P$, we determine this number exactly by using the polynomial method bound for grids.

Problem

• Geometric Covering Problems
• Lattice Points
• Avoidance Constraints

Innovation

• Coverage Estimation
• Geometric Shapes
• Spatial Algorithms