Computational complexity of art gallery problems
IEEE Transactions on Information Theory
Computing
Approximation algorithms for NP-hard problems
Inapproximability of finding maximum hidden sets on polygons and terrains
Computational Geometry: Theory and Applications
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Approximation algorithms for terrain guarding
Information Processing Letters
How Many People Can Hide in a Terrain?
ISAAC '99 Proceedings of the 10th International Symposium on Algorithms and Computation
Digital Elevation Models and TIN Algorithms
Algorithmic Foundations of Geographic Information Systems, this book originated from the CISM Advanced School on the Algorithmic Foundations of Geographic Information Systems
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We study the problem Minimum Hidden Guard Set, which consists of positioning a minimum number of guards in a given polygon or terrain such that no two guards see each other and such that every point in the polygon or on the terrain is visible from at least one guard. By constructing a gap-preserving reduction from Maximum 5-Ocurrence- 3-Satisfiability, we show that this problem cannot be approximated by a polynomial-time algorithm with an approximation ratio of n1-驴 for any 驴 0, unless NP = P, where n is the number of polygon or terrain vertices. The result even holds for input polygons without holes. This separates the problem from other visibility problems such as guarding and hiding, where strong inapproximability results only hold for polygons with holes. Furthermore, we show that an approximation algorithm achieves a matching approximation ratio of n.