Art gallery theorems and algorithms
Art gallery theorems and algorithms
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
Lectures on Discrete Geometry
VC-Dimension of Exterior Visibility
IEEE Transactions on Pattern Analysis and Machine Intelligence
New Results on Visibility in Simple Polygons
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
A new upper bound for the VC-dimension of visibility regions
Computational Geometry: Theory and Applications
Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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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 ≠ vis(v) ∩ S holds for the visibility regions vis(v) of all points v in P. In other words, the VC-dimension $d$ of visibility regions in a simple polygon cannot exceed 14. Since Valtr [v-ggwps-98] proved in 1998 that d ∈ [6,23] holds, no progress has been made on this bound. Our reduction immediately implies a smaller upper bound to the number of guards needed to cover P by ε-net theorems.