On Guarding Line Arrangements
Valentin Brimkov
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 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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