An Approximation Algorithm for Clarity Edge Covering Problem
Somaie Abdolzadeh
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Abstract
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 .
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On visibility problems in the plane-solving minimum vertex guard problems by successive approximations
Proc. of the 9th Int. Symp. on …, 2006
We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. It is known that this is an NP-hard Art Gallery Problem with relevant practical applications. In this paper we present an approximation method that solves the problem by successive approximations, which we introduced in [21]. We report on some results of its experimental evaluation and describe two algorithms for characterizing visibility from a point, that we designed for its implementation. Partially funded by LIACC through Programa de Financiamento Plurianual, Fundação para a Ciência e Tecnologia (FCT) and Programa POSI, and by CEOC (Univ. of Aveiro) through Programa POCTI, FCT, co-financed by EC fund FEDER.
Vertex Guarding in Weak Visibility 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.
Approximability of Guarding Weak Visibility Polygons
Approximability of Guarding Weak Visibility Polygons, 2017
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 up to O(log log OPT) by King and Kirkpatrick (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 − ϵ)/12) ln n for any ϵ > 0, unless NP = P. We also show that, for the special class of polygons without holes that are orthogonal as well as weakly visible from an edge, the approximation ratio can be improved to 3. Finally, we consider the point guard problem and show that it is NP-hard in the case of polygons weakly visible from an edge.
Locating Guards for Visibility Coverage of Polygons
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.
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ArXiv, 2017
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Edge covering with budget constrains
arXiv (Cornell University), 2013
We study two related problems: the Maximum weight m ′-edge cover (MWEC) problem and the Fixed cost minimum edge cover (FCEC) problem. In the MWEC problem, we are given an undirected simple graph G = (V, E) with integral vertex weights. The goal is to select a set U ⊆ V of maximum weight so that the number of edges with at least one endpoint in U is at most m ′. Goldschmidt and Hochbaum [7] show that the problem is NP-hard and they give a 3-approximation algorithm for the problem. We present an approximation algorithm that achieves a guarantee of 2, thereby improving the bound of 3 [7]. In the FCEC problem, we are given a vertex weighted graph, a bound k, and our goal is to find a subset of vertices U of total weight at least k such that the number of edges with at least one edges in in U is minimized. A 2(1 + ǫ)-approximation for the problem follows from the work of Carnes and Shmoys [4]. We improve the approximation ratio by giving a 2-approximation algorithm for the problem. Can we get better results using methods based on linear programming? We take a first step and show that the natural LP for FCEC has an
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