Approximation Algorithms to Minimum Vertex Cover Problems on Polygons and Terrains

Tomás, Ana Paula; Bajuelos, Antonio L.; Marques, Fábio

doi:10.1007/3-540-44860-8_90

Cited by 9 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Approximate solutions to the Orthogonal Art Gallery problem can be obtained by modeling it as a discrete combinatorial problem, the Minimum Set Cover problem, as shown by Erdem and Sclaroff in [8,9] and by Tomás et al in [23,24]. Both approaches discretize the polygon P in some way and then solve the corresponding optimization problem.…”

Section: The Discretized Orthogonal Art Gallery Problemmentioning

confidence: 99%

See 1 more Smart Citation

An Exact and Efficient Algorithm for the Orthogonal Art Gallery Problem

Couto

Souza

ezende

2007

XX Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI 2007)

View full text Add to dashboard Cite

In this paper, we propose an exact algorithm to solve the Orthogonal Art Gallery problem in which guards can only be placed on the vertices of the polygon P representing the gallery. Our approach is based on a discretization of P into a finite set of points in its interior. The algorithm repeatedly solves an instance of the Set Cover problem obtaining a minimum set Z of vertices of P that can view all points in the current discretization. Whenever P is completely visible from Z, the algorithm halts; otherwise, the discretization is refined and another iteration takes place. We establish that the algorithm always converges to an optimal solution by presenting a worst case analysis of the number of iterations that could be effected. Even though these could theoretically reach O(n 4 ), our computational experiments reveal that, in practice, they are linear in n and, for n ≤ 200, they actually remain less than three in almost all instances.Furthermore, the low number of points in the initial discretization, O(n 2 ), compared to the possible O(n 4 ) atomic visibility polygons, renders much shorter total execution times. Optimal solutions found for different classes of instances of polygons with up to 200 vertices are also described.

show abstract

Section: The Discretized Orthogonal Art Gallery Problemmentioning

confidence: 99%

“…Recently, Erdem and Sclaroff in [8,9] and Tomás et al in [23,24] modeled the problem as a discrete combinatorial problem and then solved the corresponding optimization problem. The results we present here follow this line.…”

Section: Introduction and Related Workmentioning

confidence: 99%

An Exact and Efficient Algorithm for the Orthogonal Art Gallery Problem

Couto

Souza

ezende

2007

XX Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI 2007)

View full text Add to dashboard Cite

show abstract

“…In [4] we showed that Π(P ) has at most 1+r +r 2 pieces. Later we noted that this upper bound is not sufficiently tightened.…”

Section: Lower and Upper Bounds On |π(P )|mentioning

confidence: 97%

“…In [4] we showed that |Π(P )| ≤ 1 + r + r 2 , where r is the number of reflex vertices of P . We shall now give sharper bounds both for maxP |Π(P )| and minP |Π(P )|.…”

mentioning

confidence: 99%

Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: Lower and Upper Bounds on the Number of Pieces

Bajuelos

Tomás

Marques

2004

Computational Science and Its Applications – ICCSA 2004

Self Cite

View full text Add to dashboard Cite

Given an orthogonal polygon P , let |Π(P )| be the number of rectangles that result when we partition P by extending the edges incident to reflex vertices towards INT(P ). In [4] we showed that |Π(P )| ≤ 1 + r + r 2 , where r is the number of reflex vertices of P . We shall now give sharper bounds both for maxP |Π(P )| and minP |Π(P )|. Moreover, we characterize the structure of orthogonal polygons in general position for which these new bounds are exact.

show abstract

“…This has also been a major motivation for us. In particular, we needed a sufficiently large number of varied orthogonal polygons to carry out an experimental evaluation of the algorithm proposed in [13] for solving the Minimum Vertex Guard problem for arbitrary polygons. This problem is that of finding a minimum set G of vertices of the given polygon P such that each point in the interior of P is visible from at least a vertex in G, belonging to the Art Gallery problems [8,16].…”

Section: Introductionmentioning

confidence: 99%

Generating Random Orthogonal Polygons

Tomás

Bajuelos

2004

Current Topics in Artificial Intelligence

Self Cite

View full text Add to dashboard Cite

Abstract. We propose two different methods for generating random orthogonal polygons with a given number of vertices. One is a polynomial time algorithm and it is supported by a technique we developed to obtain polygons with an increasing number of vertices starting from a unit square. The other follows a constraint programming approach and gives great control on the generated polygons. In particular, it may be used to find all n-vertex orthogonal polygons with no collinear edges that can be drawn in angrid, for small n, with symmetries broken.

show abstract

Approximation Algorithms to Minimum Vertex Cover Problems on Polygons and Terrains

Cited by 9 publications

References 18 publications

An Exact and Efficient Algorithm for the Orthogonal Art Gallery Problem

An Exact and Efficient Algorithm for the Orthogonal Art Gallery Problem

Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: Lower and Upper Bounds on the Number of Pieces

Generating Random Orthogonal Polygons

Contact Info

Product

Resources

About