An Approximation Algorithm for the Art Gallery Problem

In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bartschi and Suri (Algorithmica 2013): A covering of the polygon is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon. It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons. The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.

Computing maximum independent set on outerstring graphs and their relatives

A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.

A 2-approximation algorithm for the zookeeper's problem

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

Abstract. We present a new line sweep algorithm, HEAPSWEEP, for reporting bichromatic (``purple'') intersections between a red and a blue family of line segments. If the union of the segments in each family is connected as a point set, HEAPSWEEP reports all k purple intersections in time O((n+k) α(n) log 3 n) , where n is the total number of input segments and α(n) is the nearly constant inverse Ackermann function. To achieve these bounds, the algorithm maintains only partial information about the vertical ordering between curves of the same color, using a new data structure called a kinetic queue . In order to analyze the running time of HEAPSWEEP, we also show that a simple polygon containing a set of n line segments can be partitioned into monotone regions by a set of vertical threads cutting these segments O(n log n) times.

A 2-approximation algorithm for the zookeeper's problem

The range searching (or windowing) problem asks for an accommodation of a set of objects such that those objects that lie (partially) in a given axis-parallel rectangle can be reported efficiently. We solve the range searching problem for a set of n non-intersecting, but possibly touching, line segments in the plane and give a data structure that allows for range queries in Ο(k+ log2 n) time, where k is the number of reported answers. The structure is dynamic and allows for insertions and deletions of line segments in Ο(log2 n) time. The structure uses Ο(n log n) storage. The related problem of moving the window (range) parallel to one of the coordinate-axes, determining the first line segment that will become visible or stops being visible, is treated as well and similar bounds are obtained.

The potato-peeling problem asks for the largest convex polygon contained inside a given simple polygon. We give anO(n 7) time algorithm to this problem, answering a question of Goodman. We also give anO(n 6) time algorithm if the desired polygon is maximized with respect to perimeter.

This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomial-time and finds a factor-O(log OPT) approximation, where OPT is the treewidth of the graph. This is the first polynomial-time algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OPT⋅log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Real-world problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large graphs associated with real-world problems.

The circle formation is a good starting point for many symmetric formations like an approximation of a circle, a simple polygon, or a line segment etc in the plane. A distributed manner is presented to fulfill the coordinated task of N mobile agents swarm (which could be swarm robotics, unmanned air/ocean vehicles namely UAV, satellites, mobile sensors, aircraft and spacecraft etc.) being forced to achieve a circle formation while avoiding collision between themselves. The control development method is based on artificial potential function namely APF and Newton-Raphson iteration. Finally, the performance of the method is examined by numerical simulations, the results further indicated that the individual members living in swarm during the course of coordinative motion can realize the stabilization of particular predefined a desired circle or a convex polygon formation.

A sliding camera inside an orthogonal polygon P is a point guard that travels back and forth along an orthogonal line segment \(\gamma \) in P. The sliding camera g can see a point p in P if the perpendicular from p onto \(\gamma \) 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. On the other hand, we show that the problem is NP-hard on orthogonal polygons with holes even if only horizontal cameras are allowed. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of sliding cameras needed relative to the number of vertices n.

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

Fast vertex guarding for polygons with and without holes

Line segment visibility with sidedness constraints