Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Fast vertex guarding for polygons with and without holes
Computational Geometry: Theory and Applications
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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We provide an O(log log opt)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building ε-nets of size $O(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon})$ for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Brönnimann and Goodrich to build an approximation algorithm from this ε-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon’s vertices. If a finite set of potential guard locations is not specified, e.g., when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm’s running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log opt)-approximation algorithms that use generic net finders for set systems of finite VC-dimension.