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Strategies for Optimal Placement of Surveillance Cameras in Art Galleries
 In Proceedings of 18th International Conference on Computer Graphics and Vision
"... The Art Gallery problem (AGP) consists of minimizing the number of cameras required to guard an art gallery whose boundary is an nvertex polygon P. In this paper, we report our ongoing work in exploring an exact algorithm for a few variants of AGP, which iteratively computes optimal solutions to S ..."
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The Art Gallery problem (AGP) consists of minimizing the number of cameras required to guard an art gallery whose boundary is an nvertex polygon P. In this paper, we report our ongoing work in exploring an exact algorithm for a few variants of AGP, which iteratively computes optimal solutions to Set Cover problems (SCPs) corresponding to discretizations of P. Besides having proven in [Couto et al. 2007] that this procedure always converges to an exact solution of the original continuous problem, we have evidence that, in practice, convergence is achieved after only a few iterations, even for random polygons of hundreds of vertices. Nonetheless, we observe that the number of iterations required is highly dependent on the way P is initially discretized. As each iteration involves the solution of an SCP, the strategy for discretizing P is of paramount importance. We present here some of the discretization strategies we have been working with and new ones that will be studied in the near future. In comparison to the current literature, our results show a significant improvement in the size of the instances that can be solved to optimality while maintaining low execution times: no more than 65 seconds for random polygons of up to one thousand vertices.
Optimizing the Minimum Vertex Guard Set on Simple Polygons via a Genetic Algorithm
"... Abstract: The problem of minimizing the number of vertexguards necessary to cover a given simple polygon (MINIMUM VERTEX GUARD (MVG) problem) is NPhard. This computational complexity opens two lines of investigation: the development of algorithms that establish approximate solutions and the deter ..."
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Abstract: The problem of minimizing the number of vertexguards necessary to cover a given simple polygon (MINIMUM VERTEX GUARD (MVG) problem) is NPhard. This computational complexity opens two lines of investigation: the development of algorithms that establish approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation and propose an approximation algorithm based on general metaheuristic genetic algorithms to solve the MVG problem. Based on our algorithm, we conclude that on average the minimum number of vertexguards needed to cover an arbitrary and an orthogonal polygon with n vertices is 38.6/n and 40.6/n, respectively. We also conclude that this result is very satisfactory in the sense that it is always close to optimal (with an approximation ratio of 2, for arbitrary polygons; and with an approximation ratio of 1.9, for orthogonal polygons).
Algorithms for Art Gallery Illumination
, 2014
"... We consider a variant of the Art Gallery Problem, where a polygonal region is to be covered with light sources, with light fading over distance. We describe two practical algorithms, one based on a discrete approximation, and another based on nonlinear programming by means of simplex partitioning s ..."
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We consider a variant of the Art Gallery Problem, where a polygonal region is to be covered with light sources, with light fading over distance. We describe two practical algorithms, one based on a discrete approximation, and another based on nonlinear programming by means of simplex partitioning strategies. For the case where the light positions are given, we describe a fully polynomialtime approximation scheme. For both algorithms we present an experimental evaluation.
Facets for Art Gallery Problems
"... Abstract The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NPhard, and its inherent continuous structure (with both the set of points that need to be guarded and the set of ..."
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Abstract The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NPhard, and its inherent continuous structure (with both the set of points that need to be guarded and the set of points that can be used for guarding being uncountably infinite) makes it difficult to apply a straightforward formulation as an Integer Linear Program. We use an iterative primaldual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate subset of primal covering and dual packing constraints and variables is considered; these correspond to possible guard positions and points that are to be guarded. Particularly useful are cutting planes for eliminating fractional solutions. We identify two classes of facets, based on Edge Cover and Set Cover (SC) inequalities. Solving the separation problem for the latter is NPcomplete, but exploiting the underlying geometric structure, we show that large subclasses of fractional SC solutions cannot occur for the AGP. This allows us to separate the relevant subset of facets in polynomial time. We also characterize all facets for finite AGP relaxations with coefficients in {0, 1, 2}. Finally, we demonstrate the practical usefulness of our approach with improved solution quality and speed for a wide array of large benchmark instances; as it turns out, our results yield a significant improvement. Partially supported by DFG project Kunst!, KR 3133/11.