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arXiv · 2305.00825

Covering grids with multiplicity

Anurag Bishnoi·Simona Boyadzhiyska·Shagnik Das·Yvonne den Bakker
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Abstract

Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We solve this problem for various types of grids, in particular showing the tightness of the Ball--Serra bound when one side is much larger than the other. In other cases, we prove new lower bounds that improve upon Ball--Serra and provide an asymptotic answer for almost all grids. For the standard grid $\{0,\ldots,n-1\} \times \{0,\ldots,n-1\}$, we prove nontrivial upper and lower bounds on the number of lines needed. To prove our results, we combine linear programming duality with some combinatorial arguments.

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BibTeXRIS

Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Yvonne den Bakker. 2023-05-01. Covering grids with multiplicity. https://arxiv.org/abs/2305.00825

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