Guard Problem Research Articles

On guarding real terrains: The terrain guarding and the blocking path problems

Locating a minimum number of guards on a terrain such that every point on the terrain is guarded by at least one of the guards is known as the Terrain Guarding Problem (TGP). In this paper, a realistic example of the terrain guarding problem is studied, involving the surveillance of a rugged geographical terrain by means of thermal cameras. A number of issues related to TGP are addressed with integer-programming models proposed to solve the problem. Also, a sensitivity analysis is carried out in which five fictitious terrains are created to see the effect of the resolution of the terrain, and of terrain characteristics, on coverage optimization and the required number of guards. Finally, a new problem, which is called the Blocking Path Problem (BPP), is introduced. BPP is about guarding a path on the terrain with a minimum number of guards such that the path blocks all possible infiltration routes. A discussion is provided about the relation of BPP to the Network Interdiction Problem (NIP), which has been studied extensively by the operations research community, and to the k-Barrier Coverage Problem, which has been studied under the Sensor Deployment Problem. BPP is solved via an integer-programming formulation based on a network paradigm.

Optimal Perimeter Guarding With Heterogeneous Robot Teams: Complexity Analysis and Effective Algorithms

We perform structural and algorithmic studies of significantly generalized versions of the optimal perimeter guarding (OPG) problem. As compared with the original OPG where robots are uniform, in this paper, many mobile robots with heterogeneous sensing capabilities are to be deployed to optimally guard a set of one-dimensional segments. Two complimentary formulations are investigated where one limits the number of available robots (OPG_LR) and the other seeks to minimize the total deployment cost (OPG_MC). In contrast to the original OPG which admits low-polynomial time solutions, both OPG_LR and OPG_MC are computationally intractable with OPG_LR being strongly NP-hard. Nevertheless, we develop fairly scalable pseudo-polynomial time algorithms for practical, fixed-parameter subcase of OPG_LR; we also develop pseudo-polynomial time algorithm for general OPG_MC and polynomial time algorithm for the fixed-parameter OPG_MC case. The applicability and effectiveness of selected algorithms are demonstrated through extensive numerical experiments.

Orthogonal Terrain Guarding is NP-complete

A terrain is an x-monotone polygonal curve, that is, every vertical line crosses the curve at most once. In the Terrain Guarding problem, a special case of the famous art gallery problem, one has to place at most $k$ guards on the vertices of a $n$-vertex terrain, in order to fully see it. In 2010, King and Krohn showed that Terrain Guarding is NP-hard [SODA '10, SIAM J. Comput. '11] thereby solving a long-standing open question. They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or orthogonal. Recently, Ashok et al. [SoCG'17] presented an FPT algorithm running in time $k^{O(k)}n^{O(1)}$ for Dominating Set in the visibility graphs of rectilinear terrains without 180-degree vertices. They ask if Orthogonal Terrain Guarding is in P or NP-hard. In the same paper, they give a subexponential-time algorithm running in $n^{O(\sqrt n)}$ (actually even $n^{O(\sqrt k)}$) for the general Terrain Guarding and notice that the hardness proof of King and Krohn only disproves a running time $2^{o(n^{1/4})}$ under the ETH. Hence, there is a significant gap between their $2^{O(n^{1/2} \log n)}$-algorithm and the no $2^{o(n^{1/4})}$ ETH-hardness implied by King and Krohn's result.In this paper, we adapt the gadgets of King and Krohn to rectilinear terrains in order to prove that even Orthogonal Terrain Guarding is NP-complete. Then, we show how to obtain an improved ETH lower bound of $2^{\Omega(n^{1/3})}$ by refining the quadratic reduction from Planar 3-SAT into a cubic reduction from 3-SAT. [In the conference version of this paper, we mistakenly claim a tighter lower bound.] This works for both Orthogonal Terrain Guarding and Terrain Guarding.

Universal Guard Problems

Given a set [Formula: see text] of [Formula: see text] points in the plane, how many universal guards are sometimes necessary and always sufficient to guard any simple polygon with vertex set [Formula: see text]? We call this problem a Universal Guard Problem and provide a spectrum of results. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of [Formula: see text] vertices, or to guard all polygons in a given set of [Formula: see text] polygons on an [Formula: see text]-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.

Approximability of guarding 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(logn), which was improved up to O(loglogOPT) 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(n2) 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)lnn 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.

Fast vertex guarding for polygons with and without holes

For a polygon P with n vertices, the vertex guarding problem asks for the minimum subset G of Pʼs vertices such that every point in P is seen by at least one point in G. This problem is NP-complete and APX-hard. The first approximation algorithm (Ghosh, 1987) involves decomposing P into O(n4) cells that are equivalence classes for visibility from the vertices of P. This discretized problem can then be treated as an instance of Set Cover and solved in O(n5) time with a greedy O(logn)-approximation algorithm. Ghosh (2010) recently revisited the algorithm, noting that minimum visibility decompositions for simple polygons (Bose et al., 2000) have only O(n3) cells, improving the running time of the algorithm to O(n4) for simple polygons.In this paper we observe that, since minimum visibility decompositions for simple polygons have only O(n2) sinks (i.e., cells of minimal visibility) (Bose et al., 2000), the running time of the algorithm can be further improved to O(n3). We then extend the result of Bose et al. to polygons with holes, showing that a minimum visibility decomposition of a polygon with h holes has only O((h+1)n3) cells and only O((h+1)2n2) cells of minimal visibility. We exploit this result to obtain a faster algorithm for vertex guarding polygons with holes. We then show that, in the same time complexity, we can attain approximation factors of O(loglogopt) for simple polygons and O((1+log(h+1))logopt) for polygons with holes.

Approximation algorithms for art gallery problems in polygons

In this paper, we present approximation algorithms for minimum vertex and edge guard problems for polygons with or without holes with a total of n vertices. For simple polygons, approximation algorithms for both problems run in O ( n 4 ) time and yield solutions that can be at most O ( log n ) times the optimal solution. For polygons with holes, approximation algorithms for both problems give the same approximation ratio of O ( log n ) , but the running time of the algorithms increases by a factor of n to O ( n 5 ) .

Note on covering monotone orthogonal polygons with star-shaped polygons

In 1986, Keil provided an O ( n 2 ) time algorithm for the problem of covering monotone orthogonal polygons with the minimum number of r-star-shaped orthogonal polygons. This was later improved to O ( n ) time and space by Gewali et al. in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45–63]. In this paper we simplify the latter algorithm—we show that with a little modification, the first step Sweep1 of the discussed algorithm—which computes the top ceilings of horizontal grid segments—can be omitted. In addition, for the minimum orthogonal guard problem in the considered class of polygons, our approach provides a linear time algorithm which uses O ( k ) additional space, where k is the size of the optimal solution—the algorithm in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45–63] uses both O ( n ) time and O ( n ) additional space.

Cooperative mobile guards in grids

A grid P is a connected union of vertical and horizontal segments. A mobile guard is a guard which is allowed to move along a grid segment, thus a point x is seen by a mobile guard g if either x is on the same segment as g or x is on a grid segment crossing g. A set of mobile guards is weakly cooperative if at any point on its patrol, every guard can be seen by at least one other guard. In this paper we discuss the classes of polygon-bounded grids and simple grids for which we propose a quadratic time algorithm for solving the problem of finding the minimum weakly cooperative guard set (MinWCMG). We also provide an O ( n log n ) time algorithm for the MinWCMG problem in horizontally or vertically unobstructed grids. Next, we investigate complete rectangular grids with obstacles. We show that as long as both dimensions of a grid are larger than the number of obstacles k, k + 2 weakly cooperative mobile guards always suffice to cover the grid. Finally, we prove that the MinWCMG problem is NP-hard even for grids in which every segment crosses at most three other segments. Consequently, the minimum k-periscope guard problem for 2D grids is NP-hard as well, and this answers the question posed by Gewali and Ntafos [L.P. Gewali, S. Ntafos, Covering grids and orthogonal polygons with periscope guards, Computational Geometry: Theory and Applications 2 (1993) 309–334].

A Constant‐Factor Approximation Algorithm for Optimal 1.5D Terrain Guarding

We present the first constant‐factor approximation algorithm for a nontrivial 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 the geometric structure of terrains to obtain a substantially improved approximation algorithm.

On Even Triangulations of 2-Connected Embedded Graphs

Recently, Hoffmann and Kriegel proved an important combinatorial theorem [SIAM J. Discrete Math., 9 (1996), pp. 210--224]: Every 2-connected bipartite plane multigraph G without 2-cycle faces has a triangulation in which all vertices have even degree (this is called an even triangulation). Combined with the classical Whitney's theorem, this result implies that every such graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2) time algorithm was obtained in [SIAM J. Discrete Math., 9 (1996), pp. 210--224] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2) time algorithm for solving this problem. In this paper, we develop a simple proof of the above theorem. Our proof reveals and relies on a natural correspondence between even triangulations of G and certain orientations of G. Based on this new proof, we obtain a very simple O(n) time algorithm for finding an even triangulation of G. We also extend our proof to show the existence of even triangulations for similar graphs on high genus surface.

K-guarding of polyhedral terrain

Site visibility analysis is an important research topic with many applications in Geographical Information Systems. This paper introduces a new paradigm in terrain guarding, called k-guarding. K-guarding is a generalization of the classic guarding problem where, instead of only one guard, each surface patch is guarded by at least k guards. Afterwards, two optimization problems based on k-guarding are defined: an optimum k-guarding, and a constrained k-guarding. There are three heuristic approaches—k-greedy add, k-stingy drop, and k-reverse greedy—that are proposed as a solution to the above-mentioned optimization problems. The first two are known approaches adapted to k-guarding, while k-reverse greedy is a new, original heuristic. The heuristics are compared using actual topographic surfaces. It is shown that our approach (k-reverse greedy) gives on average the best near optimum solutions. The most surprising finding of the experiments is that the combination of heuristics introduced here yields even better results.

Proper interval graphs and the guard problem

This paper is a study of the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. We prove that proper interval graphs with ⩾2 vertices have hamiltonian paths, those with ⩾3 vertices have hamiltonian cycles, and those with ⩾4 vertices are hamiltonian-connected if and only if they are, respectively, 1-, 2-, or 3-connected. We also study the guard problem in spiral polygons by connecting the class of nontrivial connected proper interval graphs with the class of stick-intersection graphs of spiral polygons.

Tight bounds for the number of edge guards for spiral polygons

In this paper we give tight lower and upper bounds for the number of edge guards required for covering spiral polygons. We have proved that [(n + 2)/5]edge guards are necessary and sufficient to cover a spiral polygon. It has been shown by Aggarwal [2] that [(n + 2)/5]diagonal guards are necessary and sufficient to cover a spiral polygon. Edge guards are more restrictive than diagonal guards. Hence the previous theorem can be got as a corollary using our theorem. The necessary condition of the edge guard problem for spiral polygons has not been investigated although the diagonal guard problem for the same has been solved [2]. The necessary proof of the edge guard problem follows from the necessary condition of the diagonal guard problem but we have given an alternate proof of necessity.

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

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 or route which provide a natural connection between the art gallery problem, the m-watchmen routes 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 routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.