Guarding a Terrain by Two Watchtowers

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.

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.

Algorithmica, 2011

We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 due to King (LATIN 2006, pages 629-640). Unlike previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.

We consider guarding classes of simple polygons using mobile guards (polygon edges and diagonals) under the constraint that no two guards may see each other. In contrast to most other art gallery problems, existence is the primary question: does a specific type of polygon admit some guard set? Types include simple polygons and the subclasses of orthogonal, monotone, and starshaped polygons. Additionally, guards may either exclude or include the endpoints (so-called open and closed guards). We provide a nearly complete set of answers to existence questions of open and closed edge, diagonal, and mobile guards in simple, orthogonal, monotone, and starshaped polygons, with some surprising results. For instance, every monotone or starshaped polygon can be guarded using hidden open mobile (edge or diagonal) guards, but not necessarily with hidden open edge or hidden open diagonal guards.

2017

Information Processing Letters, 2004

We observe that Matoušek's algorithm for computing all dominances for a set P of n points in R n can be employed for computing all pairs of points in such a set whose sum is greater or equal to a given point a ∈ R n. We apply this observation to the decision problem of the discrete planar two-watchtower problem and obtain an improved solution.

Algorithmica, 2013

We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT 2 ). Computational geometry Art gallery problems Monotone polygons Rectilinear polygons Approximation algorithms

ArXiv, 2017

We are interested in the problem of guarding simple orthogonal polygons with the minimum number of $ r $-guards. The interior point $ p $ belongs an orthogonal polygon $ P $ is visible from $ r $-guard $ g $, if the minimum area rectangle contained $ p $ and $ q $ lies within $ P $. A set of point guards in polygon $ P $ is named guard set (as denoted $ G $) if the union of visibility areas of these point guards be equal to polygon $ P $ i.e. every point in $ P $ be visible from at least one point guards in $ G $. For an orthogonal polygon, if dual graph of vertical decomposition is a path, it is named path polygon. In this paper, we show that the problem of finding the minimum number of $ r $-guards (minimum guard set) becomes linear-time solvable in orthogonal path polygons. The path polygon may have dent edges in every four orientations. For this class of orthogonal polygon, the problem has been considered by Worman and Keil who described an algorithm running in $ O(n^{17} poly\l...

Computational Geometry: Theory and Applications, 2009

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

Information Processing Letters, 2006

Let P be a polygon with n vertices. We say that two points of P see each other if the line segment connecting them lies inside (the closure of) P. In this paper we present efficient approximation algorithms for finding the smallest set G of points of P so that each point of P is seen by at least one point of G, and the points of G are constrained to be belong to the set of vertices of an arbitrarily dense grid. We also present similar algorithms for terrains and polygons with holes.

A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P. A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations.

The Computer Journal, 2014

This paper focuses on a variation of the Art Gallery problem that considers open edge guards and open mobile guards. A mobile guard can be placed on edges and diagonals of a polygon, and the "open" prefix means that the endpoints of such edge or diagonal are not taken into account for visibility purposes. This paper studies the number of guards that are sufficient and sometimes necessary to guard some classes of simple polygons for both open edge and open mobile guards. This problem is also considered for planar triangulation graphs using open edge guards.

2005

We present the first constant-factor approximation algorithm for a non-trivial instance of the optimal guarding (coverage) problem in polygons. In particular, we give an O(1)-approximation algorithm for placing the fewest point guards on a 1.5D terrain, so that every point of the terrain is seen by at least one guard. While polylogarithmic-factor approximations follow from set cover results, our new results exploit geometric structure of terrains to obtain a substantially improved approximation algorithm.

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.