← 研踪 PaperTrailarXiv:2611.02046 · v1
On Guarding Line Arrangements
首次提交 2026-11-02 00:49
版本更新 2026-11-02 00:49
雷达观测 2026-11-04 23:43
摘要
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.
作者备注：21 pages, no figures
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v1 · 2026-11-02 00:49 · 最新观测版本
On Guarding Line Arrangements
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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