Computational complexity of art gallery problems
IEEE Transactions on Information Theory
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Finding the visibility graph of a simple polygon in time proportional to its size
SCG '87 Proceedings of the third annual symposium on Computational geometry
Optimization, approximation, and complexity classes
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Computing
A note on the combinatorial structure of the visibility graph in simple polygons
Theoretical Computer Science - Special issue on design and analysis of geometrical algorithms for robot motion planning and vision
Negative results on characterizing visibility graphs
Computational Geometry: Theory and Applications
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Computing the largest empty convex subset of a set of points
SCG '85 Proceedings of the first annual symposium on Computational geometry
Inapproximability Results for Guarding Polygons without Holes
ISAAC '98 Proceedings of the 9th International Symposium on Algorithms and Computation
How Many People Can Hide in a Terrain?
ISAAC '99 Proceedings of the 10th International Symposium on Algorithms and Computation
Positioning Guards at Fixed Height Above a Terrain - An Optimum Inapproximability Result
ESA '98 Proceedings of the 6th Annual European Symposium on Algorithms
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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In an alternative approach to "characterizing" the graph class of visibility graphs of simple polygons, we study the problem of finding a maximum clique in the visibility graph of a simple polygon with n vertices. We show that this problem is very hard, if the input polygons are allowed to contain holes: a gap-preserving reduction from the maximum clique problem on general graphs implies that no polynomial time algorithm can achieve an approximation ratio of n1/8-Ɛ/4 for any Ɛ 0, unless NP = P. To demonstrate that allowing holes in the input polygons makes a major difference, we propose an O(n3) algorithm for the maximum clique problem on visibility graphs for polygons without holes (other O(n3) algorithms for this problem are already known [3,6,7]). Our algorithm also finds the maximum weight clique, if the polygon vertices are weighted. We then proceed to study the problem of partitioning the vertices of a visibility graph of a polygon into a minimum number of cliques. This problem is APX-hard for polygons without holes (i.e., there exists a constant γ 0 such that no polynomial time algorithm can achieve an approximation ratio of 1 + γ). We present an approximation algorithm for the problem that achieves a logarithmic approximation ratio by iteratively applying the algorithm for finding maximum weighted cliques. Finally, we show that the problem of partitioning the vertices of a visibility graph of a polygon with holes cannot be approximated with a ratio of n1/14-γ/4 for any γ 0 by proposing a gap-preserving reduction. Thus, the presence of holes in the input polygons makes this partitioning problem provably harder.