Herein, we consider the continuous 1.5-dimensional(1.5D) terrain guarding problem with two-sided guarding. We provide an x-monotone chain T and determine the minimal number of vertex guards such that all points of T have been two-sided guarded. A point p is two-sided guarded if there exist two vertices vi (left of p) and (right of p) that both see p. A vertex vi sees a point p on T if the line segment connecting vi to p is on or above T. We demonstrate that the continuous 1.5D terrain guarding problem can be transformed to the discrete terrain guarding problem with a finite point set X and that if X is two-sided guarded, then T is also two-sided guarded. Through this transformation, we achieve an optimal algorithm that solves the continuous 1.5D terrain guarding problem under two-sided guarding.
We present an optimal, linear-time algorithm for the following version of terrain guarding: given a 1.5D terrain and a horizontal line, place the minimum number of guards on the line to see all of the terrain. We prove that the cardinality of the minimum guard set coincides with the cardinality of a maximum number of ``witnesses'', i.e., terrain points, no two of which can be seen by a single guard. We show that our results also apply to the Art Gallery problem in ``monotone mountains'', i.e., $x$-monotone polygons with a single edge as one of the boundary chains. This means that any monotone …
A terrain T is an x-monotone polygonal chain in the plane; T is orthogonal if each edge of T is either horizontal or vertical. In this paper, we give an exact algorithm for the problem of guarding the convex vertices of an orthogonal terrain with the minimum number of reflex vertices.
We consider a variant of the art gallery problem where all guards are limited to seeing to the right inside a monotone polygon. We call such guards: half-guards. We provide a polynomial-time approximation for point guarding the entire monotone polygon. We improve the best known approximation of 40 from [11], to 8. We also provide an NP-hardness reduction for point guarding a monotone polygon with half-guards.
There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see an entire rectangle, or along a staircase, or along an orthogonal path with at most $k$ bends. In this paper, we study all these guarding models in the special case of orthogonal polygons that have bounded treewidth in some sense. Exploiting algorithms for graphs of bounded treewidth, we show that the problem of finding the minimum number of guards in these models becomes linear-time solvable in orthogonal polygons of bounded treewidth.
In this paper, we consider the 1.5-dimensional orthogonal terrain guarding problem. In this problem, we assign an x-monotone chain T because each edge is either horizontal or vertical, and determine the minimal number of vertex guards for all vertices of T. A vertex vi sees a point p on T if the line segment connecting vi to p is on or above T. We provide an optimal algorithm with O(n) for a subproblem of the orthogonal terrain guarding problem. In this subproblem, we determine the minimal number of vertex guards for all right(left) convex verteices of T. Finally, we provide a 2-approximation …
Victor Klee introduce the art gallery problem during a conference in Stanford in August 1976 with that question: "How many guards are required to guard an art gallery?" In 1987, Ghosh provided an approximation algorithm for vertex guards problem that achieved $ O(\log n) $ approximation ratio. In 2017, Bhattacharya et. al presented an approximation algorithm for guarding weak visibility polygons. In our paper, we first implement these algorithms and then we test them for different types of polygons. We compare their performance in terms of number of guards used by them. In the last part, we ha…
A terrain is an x-monotone polygonal curve, i.e., successive vertices have increasing x-coordinates. Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most $k$ guards on a terrain made of $n$ vertices in order to fully see it. In 2010, King and Krohn showed that Terrain Guarding is NP-complete [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 ter…
In the continuous 1.5-dimensional terrain guarding problem we are given an $x$-monotone chain (the \emph{terrain} $T$) and ask for the minimum number of point guards (located anywhere on $T$), such that all points of $T$ are covered by at least one guard. It has been shown that the 1.5-dimensional terrain guarding problem is \NP-hard. The currently best known approximation algorithm achieves a factor of $4$. For the discrete problem version with a finite set of guard candidates and a finite set of points on the terrain that need to be monitored, a polynomial time approximation scheme (PTAS) ha…
Terrain Guarding Problem(TGP), which is known to be NP-complete, asks to find a smallest set of guard locations on a terrain $T$ such that every point on $T$ is visible by a guard. Here, we study this problem on 1.5D orthogonal terrains where the edges are bound to be horizontal or vertical. We propose a 2-approximation algorithm that runs in O($n \log m$) time, where $n$ and $m$ are the sizes of input and output, respectively. This is an improvement over the previous best algorithm, which is a 2-approximation with O($n^2$) running time.
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.
A set $G$ of points on a 1.5-dimensional terrain, also known as an $x$-monotone polygonal chain, is said to guard the terrain if any point on the terrain is 'seen' by a point in $G$. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. We prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. Our proo…
We provide an O(log log OPT)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building epsilon-nets of size O(1/epsilon log log 1/epsilon) for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this epsilon-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon's vertices. If a finite set of potentia…
A sliding camera inside an orthogonal polygon $P$ is a point guard that travels back and forth along an orthogonal line segment $γ$ in $P$. The sliding camera $g$ can see a point $p$ in $P$ if the perpendicular from $p$ onto $γ$ is inside $P$. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding $P$ with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. Finally, we study art gallery theorems for sliding cameras, th…
The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time $O(\log n)$-approximation algorithm for placing as few guards as possible at vertices of a simple $n$-gon $P$, such that every point in $P$ is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite som…
A hidden guard set $ G $ is a set of point guards in polygon $ P $ that all points of the polygon are visible from some guards in $ G $ under the constraint that no two guards may see each other. In this paper, we consider the problem for finding minimum hidden guard sets in histogram polygons under orthogonal visibility. Two points $ p $ and $ q $ are orthogonally visible if the orthogonal bounding rectangle for $ p $ and $ q $ lies within $ P $. It is known that the problem is NP-hard for simple polygon with general visibility and it is true for simple orthogonal polygon. We proposed a linea…
There is an old conjecture by Shermer \cite{sher} that in a polygon with $n$ vertices and $h$ holes, $\lfloor \dfrac{n+h}{3} \rfloor$ vertex guards are sufficient to guard the entire polygon. The conjecture is proved for $h=1$ by Shermer \cite{sher} and Aggarwal \cite{aga} seperately. In this paper, we prove a theorem similar to the Shermer's conjecture for a special case where the goal is to guard the vertices of the polygon (not the entire polygon) which is equivalent to finding a dominating set for the visibility graph of the polygon. Our proof also guarantees that the selected vertex guard…
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…
We present a 4-approximation algorithm for the problem of placing a 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. 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.
In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an $x$-monotone chain of line segments in $\mathbb{R}^2$ (the terrain $T$) and ask for the minimum number of guards (located anywhere on $T$) required to guard all of $T$. We construct guard candidate and witness sets $G, W \subset T$ of polynomial size such that any feasible (optimal) guard cover $G^* \subseteq G$ for $W$ is also feasible (optimal) for the continuous TGP. This discretization allows us to (1) settle NP-completeness for the continuous TGP, (2) provide a Polynomial Time Approximation Scheme (PTAS) for the…
One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our main focus is on polygons the edges of which are convex arcs pointing towards the exterior or interior of the polygon (but not both), named piecewise-convex and piecewise-concave polygons. We prove that, …
We tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra.
Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It w…
Drones and other small unmanned aerial vehicles are starting to get permission to fly within city limits. While video cameras are easily available in most cities, their purpose is to guard the streets at ground level. Guarding the aerial space of a city with video cameras is a problem that so far has been largely ignored. In this paper, we present bounds on the number of cameras needed to guard the city's aerial space (roofs, walls, and ground) using cameras with 180-degree range of vision (the region in front of the guard), which is common for most commercial cameras. We assume all buildings …
We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called $k$-transmitters, model a wireless device with a signal that can pass through at most $k$ walls. We show it is NP-hard to compute a minimum cover of point 2-transmitters, point $k$-transmitters, and edge 2-transmitters in a simple polygon. The point 2-transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons.
The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a simple polygon $P$, is fully guarded. Most standard versions of this problem are known to be NP-hard. In 1987, Ghosh provided a deterministic $\mathcal{O}(\log n)$-approximation algorithm for the case of vertex guards and edge guards in simple polygons. In the same paper, Ghosh also conjectured the existence of constant ratio approximation algorithms for these problems. We present here three polynomial-time algorithms with a constant approximation ratio for guarding an $…
We address recently proposed chromatic versions of the classic Art Gallery Problem. Assume a simple polygon $P$ is guarded by a finite set of point guards and each guard is assigned one of $t$ colors. Such a chromatic guarding is said to be conflict-free if each point $p\in P$ sees at least one guard with a unique color among all guards visible from $p$. The goal is to establish bounds on the function $χ_{cf}(n)$ of the number of colors sufficient to guarantee the existence of a conflict-free chromatic guarding for any $n$-vertex polygon. Bärtschi and Suri showed $χ_{cf}(n)\in O(\log n)$ (Algo…
We present two new versions of the chromatic art gallery problem that can improve upper bound of the required colors pretty well. In our version, we employ restricted angle guards so that these modern guards can visit $α$-degree of their surroundings. If $α$ is between $0$ and $180$ degree, we demonstrate that the strong chromatic guarding number is constant. Then we use orthogonal $90$-degree guards for guarding the polygons. We prove that the strong chromatic guarding number with orthogonal guards is the logarithmic order. First, we show that for the special cases of the orthogonal polygon s…
Consider a sliding camera that travels back and forth along an orthogonal line segment $s$ inside an orthogonal polygon $P$ with $n$ vertices. The camera can see a point $p$ inside $P$ if and only if there exists a line segment containing $p$ that crosses $s$ at a right angle and is completely contained in $P$. In the minimum sliding cameras (MSC) problem, the objective is to guard $P$ with the minimum number of sliding cameras. In this paper, we give an $O(n^{5/2})$-time $(7/2)$-approximation algorithm to the MSC problem on any simple orthogonal polygon with $n$ vertices, answering a question…
An "edge guard set" of a plane graph $G$ is a subset $Γ$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $Γ$. Such a set is said to guard $G$. We improve the known upper bounds on the number of edges required to guard any $n$-vertex embedded planar graph $G$: 1- We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that $G$ can be guarded with at most $ \frac{2n}{5}$ edges, then extend this approach with a deeper analysis to yield an improved bound of $\frac{3n}{8}$ edges for any plane graph. 2- We prove that there exists an ed…
Rutgers, the State University of New Jersey（罗格斯大学，新泽西州立大学）
We investigate the problem of using mobile robots equipped with 2D range sensors to optimally guard perimeters or regions, i.e., 1D or 2D sets. Given such a set of arbitrary shape to be guarded, and $k$ mobile sensors where the $i$-th sensor can guard a circular region with a variable radius $r_i$, we seek the optimal strategy to deploy the $k$ sensors to fully cover the set such that $\max r_i$ is minimized. On the side of computational complexity, we show that computing a $1.152$-optimal solution for guarding a perimeter or a region is NP-hard, i.e., the problem is hard to approximate. The h…
We devise an algorithm for surveying a dynamic orthogonal polygonal domain by placing one guard at each vertex in a subset of its vertices, i.e., whenever an orthogonal polygonal domain {\cal P'} is modified to result in another orthogonal polygonal domain {\cal P}, our algorithm updates the set of vertex guards surveying {\cal P'} so that the updated guard set surveys {\cal P}. Our algorithm modifies the guard placement in O(k \lg{(n+n')}) amortized time while ensuring the updated orthogonal polygonal domain with h holes and n vertices is guarded using at most \lfloor (n+2h)/4 \rfloor vertex …
We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices. We first study the scenario in which viewers can be all points of the polygon (such as a mobile robot which moves in the interior of the polygon). We efficiently solve the related problem of minimising the number of guards and approxi…
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 $\mathcal{O}(\log n)$, which was improved upto $\mathcal{O}(\log\log OPT)$ by King and Kirkpatrick in 2011. It has been conjectured that constan…
Rutgers University at New Brunswick（罗格斯大学新不伦瑞克分校）；University of Science and Technology of China（中国科学技术大学）
We investigate the problem of optimally assigning a large number of robots (or other types of autonomous agents) to guard the perimeters of closed 2D regions, where the perimeter of each region to be guarded may contain multiple disjoint polygonal chains. Each robot is responsible for guarding a subset of a perimeter and any point on a perimeter must be guarded by some robot. In allocating the robots, the main objective is to minimize the maximum 1D distance to be covered by any robot along the boundary of the regions. For this optimization problem which we call optimal perimeter guarding (OPG…
Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which guards are restricted to lie on a dense grid inside $\cal P$. In the general problem, guards can be anywhere inside or on the boundary of $\cal P$. The general problem is called the \emph{point} guarding problem. It was proved that the point guarding problem is APX-complete,…
Given an orthogonal connected arrangement of line-segments, Minimum Corridor Guarding(MCG) problem asks for an optimal tree/closed walk such that, if a guard moves through the tree/closed walk, all the line-segments are visited by the guard. This problem is referred to as Corridor-MST/Corridor-TSP (CMST/CTSP) for the cases when the guarding walk is a tree/closed walk, respectively. The corresponding decision version of MCG is shown to be NP-Complete[1]. The parameterized version of CMST/CTSP referred to as k-CMST/k-CTSP, asks for an optimal tree/closed walk on at most k vertices, that visits a…
Let $P$ be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment $s\in P$ as its \emph{trajectory}. The camera can see a point $p\in P$ if there exists a point $q\in s$ such that $pq$ is a line segment normal to $s$ that is completely inside $P$. In the \emph{minimum-cardinality sliding cameras problem}, the objective is to find a set $S$ of sliding cameras of minimum cardinality to guard $P$ (i.e., every point in $P$ can be seen by some sliding camera) while in the \emph{minimum-length sliding cameras problem} the goal is to find such a …
Let an orthogonal polyhedron be the union of a finite set of boxes in $\mathbb R^3$ (i.e., cuboids with edges parallel to the coordinate axes), whose surface is a connected 2-manifold. We study the NP-complete problem of guarding a non-convex orthogonal polyhedron having reflex edges in just two directions (as opposed to three, in the general case) by placing the minimum number of edge guards on reflex edges only. We show that $$\left\lfloor \frac{r-g}{2} \right\rfloor +1$$ reflex edge guards are sufficient, where $r$ is the number of reflex edges and $g$ is the polyhedron's genus. This bound …
We study the Cooperative Guarding problem for polygons with holes in a mobile multi-agents setting. Given a set of agents, initially deployed at a point in a polygon with $n$ vertices and $h$ holes, we require the agents to collaboratively explore and position themselves in such a way that every point in the polygon is visible to at least one agent and that the set of agents are visibly connected. We study the problem under two models of computation, one in which the agents can compute exact distances and angles between two points in its visibility, and one in which agents can only compare dis…
Given an arrangement of lines in the plane, what is the minimum number $c$ of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between $Ω(\log…
We consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that $\l…
We study the Art Gallery Problem for face guards in polyhedral environments. The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate its interior? We consider both closed and open face guards (i.e., faces with or without their boundary), and we study several classes of polyhedra, including orthogonal polyhedra, 4-oriented polyhedra, and 2-reflex orthostacks. We give upper and lower bounds on the minimum number of faces required to guard the interior of a given polyhedron in ea…
We present fixed parameter tractable algorithms for the conflict-free coloring problem on graphs. Given a graph $G=(V,E)$, \emph{conflict-free coloring} of $G$ refers to coloring a subset of $V$ such that for every vertex $v$, there is a color that is assigned to exactly one vertex in the closed neighborhood of $v$. The \emph{k-Conflict-free Coloring} problem is to decide whether $G$ can be conflict-free colored using at most $k$ colors. This problem is NP-hard even for $k=1$ and therefore under standard complexity theoretic assumptions, FPT algorithms do not exist when parameterised by the so…
Art Gallery is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above men…
We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.
We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with $m>1$ viewpoints (or guards) located on the terrain surface. A point on the terrain is considered \emph{visible} if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the \emph{colored} visibility map, which is a partition of the terrain into regions whose points have exactly the same…
The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NP-hard, and its inherent continuous structure (with both the set of points that need to be guarded and the set of points that can be used for guarding being uncountably infinite) makes it difficult to apply a straightforward formulation as an Integer Linear Program. We use an iterative primal-dual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate …
A point p is 1-well illuminated by a set F of n point lights if p lies in the interior of the convex hull of F. This concept corresponds to triangle-guarding or well-covering. In this paper we consider the illumination range of the light sources as a parameter to be optimized. First, we solve the problem of minimizing the light sources' illumination range to 1-well illuminate a given point p. We also compute a minimal set of light sources that 1-well illuminates p with minimum illumination range. Second, we solve the problem of minimizing the light sources' illumination range to 1-well illumin…
