Abstract
We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the previous best approximation factor of 5 (see King in Proceedings of the 13th Latin American Symposium on Theoretical Informatics, pp. 629–640, 2006). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.
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This paper combines the results of [7] and [19], which were obtained independently: the first gave a 4-approximation for the weighted version of the guarding problem and an extension to partial covering, and the second gave a 4-approximation for the unweighted version. A preliminary version appeared in the Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS) 2009.
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Elbassioni, K., Krohn, E., Matijević, D. et al. Improved Approximations for Guarding 1.5-Dimensional Terrains. Algorithmica 60, 451–463 (2011). https://doi.org/10.1007/s00453-009-9358-4
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DOI: https://doi.org/10.1007/s00453-009-9358-4