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17
Exact Solutions and Bounds for General Art Gallery Problems
, 2010
"... The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NPhard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recent ..."
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The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NPhard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constantfactor approximation algorithms nor exact solution methods are known. We present a primaldual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and—in case of convergence and integrality—ends with an optimal solution. We describe our implementation and give results for an assortment of polygons, including nonorthogonal polygons with holes.
Efficient sensor placement for surveillance problems
"... Abstract. We study the problem of covering a twodimensional spatial region P, cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p, i.e., the segment px does not intersect any occluder. The goal ..."
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Abstract. We study the problem of covering a twodimensional spatial region P, cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p, i.e., the segment px does not intersect any occluder. The goal is to compute a placement of the minimum number of sensors that cover P. We propose a landmarkbased approach for covering P. Suppose P has ς holes, and it can be covered by h sensors. Given a small parameter ε> 0, let λ: = λ(h, ε) = (h/ε) log ς. We prove that one can compute a set L of O(λ log λ log (1/ε)) landmarks so that if a set S of sensors covers L, then S covers at least (1 − ε)fraction of P. It is surprising that so few landmarks are needed, and that the number does not depend on the number of vertices in P. We then present efficient randomized algorithms, based on the greedy approach, that, with high probability, compute O ( ˜ hlog λ) sensor locations to cover L; here ˜ h ≤ h is the number sensors needed to cover L. We propose various extensions of our approach, including: (i) a weight function over P is given and S should cover at least (1−ε) of the weighted area of P, and (ii) each point of P is covered by at least t sensors, for a given parameter t ≥ 1. 1
Guarding orthogonal art galleries using sliding cameras: algorithmic and hardness results
 In Proc. MFCS, volume 8087 of LNCS
, 2013
"... Abstract. Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s ⊆ P as its trajectory. The camera can see a point p ∈ P if there exists a point q ∈ s such that pq is a line segment normal to s that is completely contained in P. In th ..."
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Abstract. Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s ⊆ P as its trajectory. The camera can see a point p ∈ P if there exists a point q ∈ s such that pq is a line segment normal to s that is completely contained in P. In the minimumcardinality sliding cameras problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S) while in the minimumlength sliding cameras problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we first settle the complexity of the minimumlength sliding cameras problem by showing that it is polynomial tractable even for orthogonal polygons with holes, answering a question posed by Katz and Morgenstern [9]. Next we show that the minimumcardinality sliding cameras problem is NPhard when P is allowed to have holes, which partially answers another question posed by Katz and Morgenstern [9]. 1
The art gallery theorem for polyominoes
, 2012
"... We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. ..."
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We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that ⌊m+1 3 ⌋ point guards are always sufficient and sometimes necessary to cover an mpolyomino, possibly with holes. When m ≤ 3n 4 − 4, the sufficiency condition yields a strictly lower guard number than ⌊n 4⌋, given by the art gallery theorem for orthogonal polygons.
Optimal positioning of surveillance UGVs
 in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems
, 2008
"... with surveillance cameras present a flexible complement to the numerous stationary sensors being used in security applications today. However, to take full advantage of the flexibility and speed offered by a group of UGV platforms, a fast way to compute desired camera locations that cover or surroun ..."
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with surveillance cameras present a flexible complement to the numerous stationary sensors being used in security applications today. However, to take full advantage of the flexibility and speed offered by a group of UGV platforms, a fast way to compute desired camera locations that cover or surround a set of buildings e.g., in response to an alarm, is needed. In this paper we focus on two problems. The first is how to create a lineofsight perimeter around a given set of buildings with a minimal number of UGVs. The second problem is how to find UGV positions such that a given set of walls are covered by the cameras while taking constraints in terms of zoom, range, resolution and field of view into account. For the first problem we propose a polynomial time algorithm and for the second problem we extend our previous work to include zoom cameras and furthermore provide a theoretical analysis of the approach itself. A number of examples are presented to illustrate the two algorithms. I.
Graph Planning for Environmental Coverage
, 2011
"... Tasks such as street mapping and security surveillance seek a route that traverses a given space to perform a function. These task functions may involve mapping the space for accurate modeling, sensing the space for unusual activity, or searching the space for objects. When these tasks are performed ..."
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Tasks such as street mapping and security surveillance seek a route that traverses a given space to perform a function. These task functions may involve mapping the space for accurate modeling, sensing the space for unusual activity, or searching the space for objects. When these tasks are performed autonomously by robots, the constraints of the environment must be considered in order to generate more feasible paths. Additionally, performing these tasks in the real world presents the challenge of operating in dynamic, changing environments. This thesis addresses the problem of effective graph coverage with environmental constraints and incomplete prior map information. Prior information about the environment is assumed to be given in the form of a graph. We seek a solution that effectively covers the graph while accounting for space restrictions and online changes. For realtime applications, we seek a complete but efficient solution that has fast replanning capabilities. For this work, we model the set of coverage problems as arc routing problems. Although these routing problems are generally NPhard, our approach aims for optimal solutions through the use of lowcomplexity algorithms in a branchandbound framework when time permits and approximations when time restrictions apply. Additionally, we account for environmental constraints by embedding those constraints into the graph. In this thesis, we present algorithms that address the multidimensional routing problem and its subproblems and evaluate them on both computergenerated and physical road network data. viFunding Thiswork was partially sponsoredby theU.S. Army Research Laboratory contract
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.
Roadmapbased level clearing of buildings
 in Proc. of the 4th Intern. Conf. on Motion in Games, 2011, Lecture Notes in Computer Science (LNCS
, 2011
"... Abstract. In this paper we describe a roadmapbased approach for a multiagent search strategy to clear a building or multistory environment. This approach utilizes an encoding of the environment in the form of a graph (roadmap) that is used to encode feasible paths through the environment. The roa ..."
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Abstract. In this paper we describe a roadmapbased approach for a multiagent search strategy to clear a building or multistory environment. This approach utilizes an encoding of the environment in the form of a graph (roadmap) that is used to encode feasible paths through the environment. The roadmap is partitioned into regions, e.g., one per level, and we design regionbased search strategies to cover and clear the environment. We can provide certain guarantees within this roadmapbased framework on coverage and the number of agents needed. Our approach can handle complex and realistic environments where many approaches are restricted to simple 2D environments. 1
Simple Rectilinear Polygons are Perfect under Rectangular Vision
"... The Art Gallery problem (see O’Rourke [3] for an overview) asks for the minimum number, g(P), of guards (points) that are necessary to see a given polygon P. Computing g(P) is NPhard. The maximum number, w(P), of independent witness points within P is a lower bound on g(P) (w(P) ≤ g(P)); here, two ..."
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The Art Gallery problem (see O’Rourke [3] for an overview) asks for the minimum number, g(P), of guards (points) that are necessary to see a given polygon P. Computing g(P) is NPhard. The maximum number, w(P), of independent witness points within P is a lower bound on g(P) (w(P) ≤ g(P)); here, two points w1, w2 ∈ P are independent if their visibility regions are pairwisedisjoint (i.e., no single guard can see both) [1]. In this paper we consider a special kind of visibility, rectangular vision (or rvisibility [4]), in which p, q ∈ P see each other if the rectangle spanned by those two points, r(p, q), is fully contained in P. Worman and Keil [4] show that g(P) can be computed in polynomial time in rectilinear simple polygons P
THE THREEDIMENSIONAL ART GALLERY PROBLEM AND ITS SOLUTIONS
"... iii This thesis addressed the threedimensional Art Gallery Problem (3DAGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudopolyhedron as well as the placement of these guards. This study exclusively focused on the versi ..."
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iii This thesis addressed the threedimensional Art Gallery Problem (3DAGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudopolyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3DAGP in which the art gallery is modelled by an orthogonal pseudopolyhedron, instead of a pseudopolyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most reallife buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3DAGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudopolyhedron. This formulation of the art gallery