Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards

The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery (represented by a polygon $P$) is fully guarded, assuming that a guard’s field of view covers 360° as well as unbounded distance. This problem was first posed by Victor Klee in a conference in 1973, and has become a well investigated problem in computational geometry.

A polygon $P$ is defined to be a closed region in the plane bounded by a finite set of line segments, called edges of $P$, such that between any two points of $P$, there exists a path which does not intersect any edge of $P$. If the boundary of a polygon $P$ consists of two or more cycles, then $P$ is called a polygon with holes (see Figure 2). Otherwise, $P$ is called a simple polygon or a polygon without holes (see Figure 2).

An art gallery can be viewed as an $n$-sided polygon $P$ (with or without holes) and guards as points inside $P$. Any point $z\in P$ is said to be visible from a guard $g$ if the line segment ${zg}$ does not intersect the exterior of $P$. In general, guards may be placed anywhere inside $P$. If the guards are allowed to be placed only on vertices of $P$, they are called vertex guards. If there is no such restriction, then they are called point guards. The point guards that are constrained to lie on the boundary of $P$, but not necessarily at the vertices, are referred to as perimeter guards. Point, vertex and perimeter guards together are also referred to as stationary guards. If guards are allowed to patrol along a line segment inside $P$, they are called mobile guards. If they are allowed to patrol only along the edges of $P$, they are called edge guards [13, 26].

In 1975, Chvátal [6] showed that $\lfloor\frac{n}{3}\rfloor$ stationary guards are sufficient and sometimes necessary (see Figure 4) for guarding a simple polygon. In 1978, Fisk [11] presented a simpler and more elegant proof of this result. For a simple orthogonal polygon, whose edges are either horizontal or vertical, Kahn et al. [18] and also O’Rourke [25] showed that $\lfloor\frac{n}{4}\rfloor$ stationary guards are sufficient and sometimes necessary (see Figure 4).

The decision version of the art gallery problem is to determine, given a polygon $P$ and a number $k$ as input, whether the polygon $P$ can be guarded with $k$ or fewer point guards. This problem was first shown to be NP-hard for polygons with holes by O’Rourke and Supowit [27]. This problem was also shown to be NP-hard for simple polygons for guarding using only vertex guards by Lee and Lin [24]. Their proof was generalized to work for point guards by Aggarwal [2]. The problem was shown to be NP-hard even for simple orthogonal polygons by Katz and Roisman [20] and Schuchardt and Hecker [28]. Abrahamsen, Adamaszek and Miltzow [1] have recently shown that the art gallery problem for point guards is ETR-complete.

In 1987, Ghosh [12, 14] provided a deterministic $\mathcal{O}(\log n)$-approximation algorithm for the case of vertex and edge guards by discretizing the input polygon $P$ and treating it as an instance of the Set Cover problem. As pointed out by King and Kirkpatrick [21], newer methods for improving the approximation ratio of the Set Cover problem itself have been developed in the time after Ghosh’s algorithm was published. By applying these methods, the approximation ratio of Ghosh’s algorithm becomes $\mathcal{O}(\log OPT)$ for guarding simple polygons and $\mathcal{O}(\log h\log OPT)$ for guarding a polygon with $h$ holes, where $OPT$ denotes the size of the smallest guard set for $P$. Deshpande et al. [7] obtained an approximation factor of $\mathcal{O}(\log OPT)$ for point guards or perimeter guards by developing a sophisticated discretization method that runs in pseudo-polynomial time. Efrat and Har-Peled [8] provided a randomized algorithm with the same approximation ratio that runs in fully polynomial expected time. Bonnet and Miltzow [5] obtained an approximation factor of $\mathcal{O}(\log OPT)$ for the point guard problem assuming integer coordinates and a specific general position. For guarding simple polygons using perimeter guards, King and Kirkpatrick [21] designed a deterministic $\mathcal{O}(\log\log OPT)$-approximation algorithm in 2011. The analysis of this result was simplified by Kirkpatrick [22].

In 1998, Eidenbenz, Stamm and Widmayer [9, 10] proved that the problem is APX-complete, implying that an approximation ratio better than a fixed constant cannot be achieved unless NP = P. They also proved that if the input polygon is allowed to contain holes, then there cannot exist a polynomial time algorithm for the problem with an approximation ratio better than $((1−\epsilon)/12)\ln n$ for any $\epsilon>0$, unless NP $\subseteq$ TIME($n^{\mathcal{O}(\log\log n)}$). Extending their method, Bhattacharya, Ghosh and Roy [3] proved that, even for the special subclass of polygons with holes that are weakly visible from an edge, there cannot exist a polynomial time algorithm for the problem with an approximation ratio better than $((1−\epsilon)/12)\ln n$ for any $\epsilon>0$, unless NP = P. These inapproximability results establish that the approximation ratio of $\mathcal{O}(\log n)$ obtained by Ghosh in 1987 is in fact the best possible for the case of polygons with holes. However, for simple polygons, the existence of a constant factor approximation algorithm for vertex and edge guards was conjectured by Ghosh [12, 15] in 1987.

Ghosh’s conjecture has been shown to be true for vertex guarding in two special sub-classes of simple polygons, viz. monotone polygons and polygons weakly visible from an edge. In 2012, Krohn and Nilsson [23] presented an approximation algorithm that computes in polynomial time a guard set for a monotone polygon $P$, such that the size of the guard set is at most $30\cdot OPT$. Bhattacharya, Ghosh and Roy [3, 4] presented a 6-approximation algorithm that runs in $\mathcal{O}(n^{2})$ time for vertex guarding simple polygons that are weakly visible from an edge. For vertex guarding this subclass of simple polygons that are weakly visible from an edge, a PTAS has recently been proposed by Katz [19].

In this paper, we present three polynomial-time algorithms with a constant approximation ratio for guarding an $n$-sided simple polygon $P$ using vertex guards. The first algorithm, that has an approximation ratio of 18, guards all vertices of $P$ in $\mathcal{O}(n^{4})$ time. The second algorithm, that has the same approximation ratio of 18, guards the entire boundary of $P$ in $\mathcal{O}(n^{5})$ time. The third algorithm, that has an approximation ratio of 27, guards all interior and boundary points of $P$ in $\mathcal{O}(n^{5})$ time. As an extension we show, using similar techniques, constant-factor approximation can also be achieved for guarding $P$ we also present identical algorithms, maintaining both the approximation bounds as well as the running times, can be obtained using edge guards. In particular, we show that the same approximation ratios of 18, 18 and 27 hold for guarding all vertices, the entire boundary, and the interior of $P$, with time complexities $\mathcal{O}(n^{4})$, $\mathcal{O}(n^{5})$ and $\mathcal{O}(n^{5})$ respectively. The significance of our results lies in the fact that these results settle the long-standing conjecture by Ghosh [12] regarding the existence of constant-factor approximation algorithms for these problem, which has been open since 1987 despite several attempts by researchers.

In each of our algorithms, $P$ is first partitioned into a hierarchy of weak visibility polygons according to the link distance from a starting vertex (see Figure 6). This partitioning is very similar to the window partitioning given by Suri [29, 30] in the context of computing minimum link paths. Then, starting with the farthest level in the hierarchy (i.e. the set of weak visibility polygons that are at the maximum link distance from the starting vertex), the entire hierarchy is traversed backward level by level, and at each level, vertex guards (of two types, viz. inside and outside) are placed for guarding every weak visibility polygon at that level of $P$. At every level, a novel procedure is used that has been developed for placing guards in (i) a simple polygon that is weakly visible from an internal chord, or (ii) a union of overlapping polygons that are weakly visible from multiple disjoint internal chords. Note that these chords are actually the constructed edges introduced during the hierarchical partitioning of $P$.

Due to partitioning according to link distances, guards can only see points within the adjacent weak visibility polygons in the hierarchical partitioning of $P$. This property locally restricts the visibility of the chosen guards, and thereby ensures that the approximation bound on the number of vertex guards placed by our algorithms at any level leads directly to overall approximation bounds for guarding $P$. Thus, a constant factor approximation bound on the overall number of guards placed by our algorithms is a direct consequence of choosing vertex guards in a judicious manner for guarding each collection of overlapping weak visibility polygons obtained from the hierarchical partitioning of $P$. Our algorithms exploit several deep visibility structures of simple polygons which are interesting in their own right.

In Section 2, we introduce some preliminary definitions and notations that are used throughout the rest of the paper. In Section 3, we present the hierarchical partitioning of a simple polygon $P$ into weak visibility polygons. Next, in Section 4, we describe how the algorithm traverses the hierarchy of visibility polygons, starting from the farthest level, and uses the procedures from Section 5 at each level as a sub-routine for guarding $P$. In Section 5, we present a novel procedure for placing vertex guards necessary for guarding a simple polygon $Q$ that is weakly visible from a single internal chord $uv$ or from multiple disjoint chords. In Section 6, we establish the overall approximation ratios for the three approximation algorithms. In Section 7, we show how these algorithms can be modified to obtain similar approximation bounds while using edge guards. Finally, in Section 8, we conclude the paper with a few remarks.

Suppose we wish to guard an arbitrary vertex $z$ of $Q_{U}$. Then, a guard must be placed at a vertex of $Q$ belonging to $\mathcal{VP}(z)$. Henceforth, let $\mathcal{VVP}(z)$ denote the set of all polygonal vertices of $\mathcal{VP}(z)$. Further, let us define the inward visible vertices and the outward visible vertices of $z$, denoted by $\mathcal{VVP}^{+}(z)$ and $\mathcal{VVP}^{-}(z)$ respectively, as follows.

$\mathcal{VVP}^{+}(z)=\{x\in\mathcal{VVP}(z):\mbox{the segment $zx$ does not intersect $uv$}\}$

$\mathcal{VVP}^{-}(z)=\{x\in\mathcal{VVP}(z):\mbox{the segment $zx$ intersects $uv$}\}$

We shall henceforth refer to the vertex guards belonging to $\mathcal{VVP}^{+}(z)$ and $\mathcal{VVP}^{-}(z)$ as inside guards and outside guards for $z$ respectively.

Consider the weakly visible polygon in Figure 9, where the three vertices $z_{1}$, $z_{2}$ and $z_{3}$ of $Q_{U}$ are such that their respective sets of outward visible vertices are pairwise disjoint, i.e. $\mathcal{VVP}^{-}(z_{1})\cap\mathcal{VVP}^{-}(z_{2})=\emptyset$, $\mathcal{VVP}^{-}(z_{2})\cap\mathcal{VVP}^{-}(z_{3})=\emptyset$, and $\mathcal{VVP}^{-}(z_{1})\cap\mathcal{VVP}^{-}(z_{3})=\emptyset$. If an algorithm chooses only outside guards, then three separate guards are required for guarding $z_{1}$, $z_{2}$ and $z_{3}$. However, an optimal solution may place a single guard on any one of the five vertices of $Q_{U}$ for guarding $z_{1}$, $z_{2}$ and $z_{3}$.

On the other hand, in Figure 9, the three vertices $z_{1}$, $z_{2}$ and $z_{3}$ of $Q_{U}$ are such that their respective sets of inward visible vertices are pairwise disjoint, i.e. $\mathcal{VVP}^{+}(z_{1})\cap\mathcal{VVP}^{+}(z_{2})=\emptyset$, $\mathcal{VVP}^{+}(z_{2})\cap\mathcal{VVP}^{+}(z_{3})=\emptyset$, and $\mathcal{VVP}^{+}(z_{1})\cap\mathcal{VVP}^{+}(z_{3})=\emptyset$. If an algorithm chooses only inside guards, then three separate guards are required for guarding $z_{1}$, $z_{2}$ and $z_{3}$. However, the outward visible vertices of $z_{1}$, $z_{2}$ and $z_{3}$ overlap; and moreover there exists a vertex $Y$ of $Q_{L}$ such that $G\in\mathcal{VVP}^{-}(z_{1})\cap\mathcal{VVP}^{-}(z_{2})\cap\mathcal{VVP}^{-}(z_{3})$. Thus, an optimal solution may choose $y$ as an outside guard for guarding $z_{1}$, $z_{2}$ and $z_{3}$ together.

The above discussion suggests that it is better to choose guards from both $\mathcal{VVP}^{+}(z)$ and $\mathcal{VVP}^{-}(z)$ for guarding the same vertex $z$ of $Q_{U}$, in order to prevent computing a guard $S$ that is arbitrarily large compared to $G_{opt}$. Therefore, our algorithm selects a subset $Z$ of vertices of $Q_{U}$, and places a fixed number of both inside and outside guards corresponding to each of them, so that these guards together see the entire $Q_{U}$. Moreover, the vertices in $Z$ (and also the guards corresponding to them) are selected in such a way that they correspond to the guards in $G_{opt}$, though the correspondence is not necessarily one-to-one, and this enables us to provide an approximation bound on our chosen set of guards. We henceforth refer to this subset $Z$ of selected vertices as primary vertices. Further, let $Z^{U}\subseteq Z$ be such that each primary vertex in $Z^{U}$ is visible from at least one guard in $G_{opt}^{U}$. Similarly, let $Z^{L}\subseteq Z$ be such that each primary vertex in $Z^{L}$ is visible from at least one guard in $G_{opt}^{L}$. Since any $z\in Z$ must be visible from at least one guard of $G_{opt}^{U}$ or $G_{opt}^{L}$, we have $Z=Z^{U}\cup Z^{L}$, and so we have:

The general strategy for placement of guards by our algorithm for guarding only the vertices of $Q_{U}$ aims to establish a constant-factor approximation bound on $S$ by separately proving the following three bounds.

The above inequalities (1), (2), (3) and (4) together imply the following conclusion.

Observe that, for any vertex $z_{k}\in Z$, both $\mathcal{VVP}^{+}(z_{k})$ and $\mathcal{VVP}^{-}(z_{k})$ may be considered to be ordered sets by taking into account the natural ordering of the visible vertices of $Q$ in clockwise order along $bd_{c}(u,v)$ and in counter-clockwise order along $bd_{cc}(u,v)$ respectively. Let us denote the first visible vertex and the last visible vertex belonging to the ordered set $\mathcal{VVP}^{-}(z_{k})$ by $f(z_{k})$ and $l(z_{k})$ respectively (see Figure 9). Also, we denote by $l^{\prime}(z_{k})$ (similarly, $f^{\prime}(z_{k})$) the last visible point (similarly, first visible point) of $\mathcal{VP}(z_{k})\cap bd_{cc}(u,v)$, which is obtained by extending the ray $\overrightarrow{z_{k}p(v,z_{k})}$ (respectively, $\overrightarrow{z_{k}p(u,z_{k})}$) till it touches $bd_{cc}(u,v)$, where $p(v,z_{k})$ (similarly, $p(u,z_{k})$) is the parent of $z_{k}$ in $SPT(v)$ (respectively, $SPT(u)$). Observe that $bd_{cc}(f^{\prime}(z_{k}),l^{\prime}(z_{k}))\cap\mathcal{VP}(z_{k})$ is the only portion of the boundary $bd_{cc}(u,v)$ that has vertices visible from $z_{k}$. Note that all vertices of $bd_{cc}(f^{\prime}(z_{k}),l^{\prime}(z_{k}))$ may not be visible from $z_{k}$, since some of them may lie inside left or right pockets of $\mathcal{VP}(z_{k})$. Similarly, observe that $bd_{c}(p(u,z_{k}),p(v,z_{k}))$ is the only portion of the boundary $bd_{c}(u,v)$ that has vertices visible from $z_{k}$. Note that all vertices of $bd_{cc}(p(u,z_{k}),p(v,z_{k}))$ may not be visible from $z_{k}$, since some of them may lie inside left or right pockets of $\mathcal{VP}(z_{k})$.

Let us discuss how primary vertices are selected by our algorithm. Initially, since all vertices of $Q_{U}$ are unguarded, they are considered to be unmarked. As vertex guards are placed over successive iterations, vertices of $Q_{U}$ are marked as soon as they become visible from some guard placed so far. In the $k$-th iteration, the next primary vertex $z_{k}\in Z$ chosen by our algorithm is an unmarked vertex such that $l^{\prime}(z_{k})$ precedes $l^{\prime}(x)$ (henceforth denoted notationally as $l^{\prime}(z_{k})\prec l^{\prime}(x)$) for any other unmarked vertex $x$ of $Q_{U}$. An immediate consequence of such choice of the primary vertex $z_{k}$ is that all vertices on $bd_{c}(z_{k},p(v,z_{k}))$ must already be marked.

In Section 5.3, we consider guarding vertices of $Q_{U}$ in a special scenario where $G_{opt}$ uses vertex guards only from $Q_{L}$. In Section 5.4, we consider guarding vertices of $Q_{U}$ in the general scenario where the guards of $G_{opt}$ are not restricted to $Q_{L}$. In Section 5.5, we enhance the procedure for guarding vertices to ensure that all interior points of $Q_{U}$ are guarded as well.

Let us consider a special scenario where $G_{opt}^{L}=G_{opt}$ and $G_{opt}^{U}=\emptyset$, i.e. $G_{opt}$ uses only vertex guards from $Q_{L}$. In such a scenario, for every vertex $z_{k}\in Z$, $\mathcal{VVP}^{-}(z_{k})$ must contain a guard from $G_{opt}$. So, a natural idea is to place outside guards in a greedy manner so that they lie in the common intersection of outward visible vertices of as many vertices of $Q_{U}$ as possible. For any primary vertex $z_{k}$, let us denote by $\mathcal{OVV}^{-}(z_{k})$ the set of unmarked vertices of $Q_{U}$ whose outward visible vertices overlap with those of $z_{k}$. In other words,

So, each vertex of $Q_{U}$ belonging to $\mathcal{OVV}^{-}(z_{k})$ is visible from at least one vertex of $\mathcal{VVP}^{-}(z_{k})$. Further, $\mathcal{OVV}^{-}(z_{k})$ can be considered to be an ordered set, where for any pair of elements $x_{1},x_{2}\in\mathcal{OVV}^{-}(z_{k})$, we define $x_{1}\prec x_{2}$ if and only if $l^{\prime}(x_{1})$ precedes $l^{\prime}(x_{2})$ in counter-clockwise order on $bd_{cc}(u,v)$. For the current primary vertex $z_{k}$, let us assume without loss of generality that $\mathcal{OVV}^{-}(z_{k})=\{x^{k}_{1},x^{k}_{2},x^{k}_{3},\dots\,x^{k}_{m(k)}\}$ such that $l^{\prime}(x^{k}_{1})\prec l^{\prime}(x^{k}_{2})\prec\dots\prec l^{\prime}(x^{k}_{m(k)})$ in counter-clockwise order on $bd_{cc}(u,v)$.

Let us partition the vertices belonging to $\mathcal{OVV}^{-}(z_{k})$ into 3 sets, viz. $A^{k}$, $B^{k}$ and $C^{k}$ in the following manner. Every vertex of $\mathcal{OVV}^{-}(z_{k})$ visible from $l(z_{k})$ is included in $B^{k}$. Observe that, by definition $z^{k}\in B^{k}$. Obviously, for each vertex $x^{k}_{i}\in\mathcal{OVV}^{-}(z_{k})\setminus B^{k}$, $x^{k}_{i}$ is not visible from $l(z_{k})$ due to the presence of some constructed edge. The vertices of $\mathcal{OVV}^{-}(z_{k})\setminus B^{k}$ are categorized into $A^{k}$ and $C^{k}$ based on whether this constructed edge creates a right pocket or a left pocket. Suppose $x^{k}_{i}\in\mathcal{OVV}^{-}(z_{k})\setminus B^{k}$ is a vertex such that $\mathcal{VP}(x^{k}_{i})$ creates a constructed edge $t(x^{k}_{i})t^{\prime}(x^{k}_{i})$, where $t(x^{k}_{i})\in\mathcal{V}(Q_{L})$ is a polygonal vertex and $t^{\prime}(x^{k}_{i})$ is the point where $\overrightarrow{x^{k}_{i}t(x^{k}_{i})}$ first intersects $bd_{cc}(u,v)$. If $t(x^{k}_{i})$ lies on $bd_{cc}(f^{\prime}(z_{k}),l^{\prime}(z_{k}))$ and $t^{\prime}(x^{k}_{i})$ lies on $bd_{cc}(l(z_{k}),v)$, i.e. if $f(z_{k})\prec t(x^{k}_{i})\prec l^{\prime}(z_{k})$ and $l(z_{k})\prec t^{\prime}(x^{k}_{i})\prec v$, then $x^{k}_{i}$ is included in $A^{k}$. For instance, in Figure 11, $x_{1}^{2}\in A^{2}$ due to the constructed edge $t(x_{1}^{2})t^{\prime}(x_{1}^{2})$. On the other hand, if $t(x^{k}_{i})$ lies on $bd_{cc}(l^{\prime}(z_{k}),v)$ and $t^{\prime}(x^{k}_{i})$ lies on $bd_{cc}(f(z_{k}),l^{\prime}(z_{k}))$, i.e. if $l^{\prime}(z_{k})\prec t(x^{k}_{i})\prec v$ and $f(z_{k})\prec t^{\prime}(x^{k}_{i})\prec l^{\prime}(z_{k})$, then $x^{k}_{i}$ is included in $C^{k}$. For instance, in Figure 11, $x_{3}^{2}\in C^{2}$ due to the constructed edge $t(x_{3}^{2})t^{\prime}(x_{3}^{2})$. Observe that, all vertices of $A^{k}$ must lie on $bd_{c}(u,z_{k})$, whereas all vertices of $C^{k}$ must lie on $bd_{c}(z_{k},v)$.