Approximation Algorithms for Edge-Covering Problem

The problem of minimizing the number of vertex-guards necessary to cover a given simple polygon (MINIMUM VERTEX GUARD (MVG) problem) is NP-hard. 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 vertex-guards 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).

Vertex Guarding in Weak Visibility Polygons

The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P , is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of O(log n), which was improved upto O(log log OPT) by King and Kirkpatrick in 2011. It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in O(n^2) time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than ((1 − \epsilon)/12) ln n for any \epsilon > 0, unless NP = P.

2013

This paper focuses on a variation of the Art Gallery problem that considers open edge guards. The “open” prefix means the endpoints of an edge where a guard is are not taken into account for visibility purposes. This paper studies the number of open edge guards that are sufficient and sometimes necessary to guard some classes of simple polygons.

The problem of minimizing the number of guards placed on vertices needed to guard a given simple polygon (MINIMUM VERTEX GUARD problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that determine approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation proposing an approximation algorithm based on the general metaheuristic Genetic Algorithms to solve the MINIMUM VERTEX GUARD problem.

Computational Geometry: Theory and Applications, 2009

1997

Abstract The rectilinear polygon cover problem is one in which a certain class of features of a rectilinear polygon of n vertices has to be covered with the minimum number of rectangles included in the polygon. In particular, one can consider covering the entire interior, the boundary and the set of corners of the polygon. These problems have important applications in, for example, storing images and in the manufacture of integrated circuits. In this paper we consider covering the corners of the polygons, also known as the corner-cover problem.

International Journal of Computational Geometry & Applications, 1993

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path (route) which provide a natural connection between the Art Gallery problem, the m-watchmen problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n 3 + n 2 m 2 )-time algorithm to compute the best set of m watchmen in a histogram.

Theoretical computer science, 2001

There are many di erent kinds of guards in a simple polygon that have been proposed and discussed. In this paper, we consider a new type of guard, boundary guard, which is a guard capable of moving along a boundary of a polygon and every interior point of the polygon can be seen by the mobile guard. We propose an algorithm to ÿnd the shortest boundary guard of a simple polygon P in O(n log n) time, where n is the number of vertices of P.

SIAM Journal on Discrete Mathematics, 1996

We prove the following graph coloring result: Let G be a 2{connected bipartite planar graph. Then one can triangulate G in such a w ay that the resulting graph is 3{colorable. This result implies several new upper bounds for guarding problems including the rst non{trivial upper bound for the rectilinear Prison Yard Problem: 1. n 3 vertex guards are su cient t o w atch the interior of a rectilinear polygon with holes. 2. 5n 12 + 3 v ertex guards resp. n+4 3 point guards are su cient t o w atch simultaneously both the interior and exterior of a rectilinear polygon. Moreover, we s h o w a new lower bound of 5n 16 vertex guards for the rectilinear Prison Yard Problem and prove it to be asymptotically tight for the class of orthoconvex polygons.

Clarity edge covering problem is a version of art gallery problem.In this problem the goal is finding the minimum number of guards which covers all edges.Here, meaning of visibility is different from art gallery problem and it is restricted. In this paper, a logarithmic approximation algorithm is presented for vertex guard version. It's time complexity is) (3 n O .

Computational Geometry, 1996

A Tk guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible or rectangularly visible from some point c o n tained in G. W e i n vestigate the function rn h k, which i s t h e largest number of Tk guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove n e w l o wer and upper bounds on parts of this function. In particular, we s h o w the following bounds: 1. rn 0 k n k+4 , with equality for even k 2. rn h 1 = 3n+4h+4 16 3. rn h 2 n 6. These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k 1. In the course of proving the upper bounds, new results on partitioning are obtained.

International Journal of Computational Geometry & Applications, 2010

We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the "art gallery problem", is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through experimentation, comparing the upper bounds on the optimal guard number given by our methods with computed lower bounds based on heuristics for placing a large number of visibility-independent "witness points". We give experimental evidence that our heuristics perform well in practice, on a large suite of input data; often the heuristics give a provably optimal result, while in other cases there is only a small gap between the computed upper and lower bounds on the optimal guard number.

Information sciences, 1994

We consider the problem of placing guards in a polygon so that (a) the area, or (b) the portion of the boundary visible to the guards is maximized. We show that finding optimum placements for k guards is NP-hard if k is a variable. We reduce the problem of optimally placing one guard to solving a high order equation, and give a polynomial time approximation scheme for placing one guard in a simple polygon. P (i.e., their union is P). It was shown in [12] that finding a minimum star cover is NP-hard [7]. This and many other results on the Art Gallery and related problems can be found in [13, 151. In this paper, we consider the problem of finding an optimum placement for a number of guards. An early discussion of this type of problem appears in . We consider two natural optimization criteria: (a) placing the guards so that the area inside the polygon that is visible to at least one guard is maximized, and (b) placing the guards so that the portion of the

2005

Tables 1 through 4 summarize the experiments for the VCP. The first two columns in these tables refer to the number of vertices and the number of edges in the input graph. The column Optimum has the optimum value and the cpu time to solve the instance. Tables 1 and 3 show results for the Greedy algorithm when this algorithm was executed 10 times. Recall that this algorithm is not deterministic.

2000

We consider the problem of locating a moving target using a group of guards cooperatively moving inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. We develop algorithms that sweep such a chain of guards through a polygon to locate the target. Our two main results are the following: