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Improved Approximations for Guarding 1.5Dimensional Terrains
"... We present a 4approximation 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 currently best approximation factor of 5 (see [14]). Unlike most of the previous techniques, our method is base ..."
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We present a 4approximation 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 currently best approximation factor of 5 (see [14]). 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.
A new upper bound for the vcdimension of visibility regions
 In Proceedings of the 27th Symposium on Computational Geometry, SoCG ’11
, 2011
"... In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not “visually discernible”, that is, T 6 = vis(v)∩S holds for the visibility regions vis(v) of all points v in P. In ..."
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In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not “visually discernible”, that is, T 6 = vis(v)∩S holds for the visibility regions vis(v) of all points v in P. In other words, the VCdimension d of visibility regions in a simple polygon cannot exceed 14. Since Valtr [15] proved in 1998 that d ∈ [6, 23] holds, no progress has been made on this bound. By net theorems our reduction immediately implies a smaller upper bound to the number of guards needed to cover P. 1