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arXiv · 2611.02046

On Guarding Line Arrangements

Valentin Brimkov
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Abstract

We study the problem of guarding a set of straight lines in the plane by selecting a minimum-cardinality set of points such that every line contains at least one selected point. This problem is known to be strongly NP-hard \cite{megiddo}. We investigate approximation algorithms for the problem and analyze their performance guarantees under various structural assumptions on the underlying line arrangement. In particular, we present approximation schemes based on set-cover techniques, relaxation methods, and properties of maximal lines. We establish constant-factor bounds on the approximation ratio for several classes of arrangements and derive relationships between the guarding number and the incidence parameters of the arrangement. Connections with Dirac-type incidence results are also explored.

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Valentin Brimkov. 2026-11-01. On Guarding Line Arrangements. https://arxiv.org/abs/2611.02046

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