The Art Gallery Problem is ∃ℝ-complete

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP 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. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

Query-Points Visibility Constraint Minimum Link Paths in Simple Polygons

We study the query version of constrained minimum link paths between two points inside a simple polygon P with n vertices such that there is at least one point on the path, visible from a query point. The method is based on partitioning P into a number of faces of equal link distance from a point, called a link-based shortest path map (SPM). Initially, we solve this problem for two given points s, t and a query point q. Then, the proposed solution is extended to a general case for three arbitrary query points s, t and q. In the former, we propose an algorithm with O(n) preprocessing time. Extending this approach for the latter case, we develop an algorithm with O(n3) preprocessing time. The link distance of a q-visible path between s, t as well as the path are provided in time O(log n) and O(m + log n), respectively, for the above two cases, where m is the number of links.

How to Keep an Eye on Small Things

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

BEAM or MFA I inspired Nv Neurons using op amps for line and line based polygon detection.

Tensor network topologies for function, first class or MFA I as BEAM circuits are described within the framework of complexity theory using Lie Computability definitions. An example of the design of opamp based Nv Neurons for the perception of shape from line detection Nv neurons is described, with circuits that detect number of lines and concavity and closure of lines in a finite region of interest. The possible role of BEAM robotics in 3R’s is described in nature inspired intent transcription, in multi -functionality and functionality driven evolution and transcription. Keywords: BEAM, MFA I, MFA II, Lie Computability, op amp circuits, Nv Neurons, Two Port Systems, large signal analysis, feedback principles, solitons, neuro-modulation. What: Nv Neurons are built from opamp based circuits, for line detection using an array or grid of inexpensive photo detectors, using an opamp based positive feedback loop and negative feedback loops for synergy principles. Story: The author first worked on this problem in his undergraduate senior year, when his advisor advised a bottom-up approach to BEAM based machine vision, biomimetics in synthetic neurons from discrete components and opamps. The problem was to differentiate a simple polygonal shape from the background. How: A simple polygon is found in traffic sign posts , creating the need for a hard wired circuit to recognize an octahedral stop sign and several triangular signs. We use line decomposition with a circuitry to compose the lines into polyhedral shapes.(Bheemaiah, n.d.) Why: BEAM is functional art, and forms the predecessor to MFA II or completely multi functional architecture of a broad umbrella of value addition in multi functionality, functoid and HOF based algebraic frameworks for MaC based definitions of architecture and design in code.(Autores and International Workshop on Higher-Order Algebra, Logic and Term Rewriting 1994; Kirchner and Wechler 1990; Dowek et al. 1996)

BEAM or MFA I inspired Nv Neurons using op amps for line and line based polygon detection.

Tensor network topologies for function, first class or MFA I as BEAM circuits are described within the framework of complexity theory using Lie Computability definitions. An example of the design of opamp based Nv Neurons for the perception of shape from line detection Nv neurons is described, with circuits that detect number of lines and concavity and closure of lines in a finite region of interest. The possible role of BEAM robotics in 3R’s is described in nature inspired intent transcription, in multi -functionality and functionality driven evolution and transcription. Keywords: BEAM, MFA I, MFA II, Lie Computability, op amp circuits, Nv Neurons, Two Port Systems, large signal analysis, feedback principles, solitons, neuro-modulation. What: Nv Neurons are built from opamp based circuits, for line detection using an array or grid of inexpensive photo detectors, using an opamp based positive feedback loop and negative feedback loops for synergy principles. Story: The author first worked on this problem in his undergraduate senior year, when his advisor advised a bottom-up approach to BEAM based machine vision, biomimetics in synthetic neurons from discrete components and opamps. The problem was to differentiate a simple polygonal shape from the background. How: A simple polygon is found in traffic sign posts , creating the need for a hard wired circuit to recognize an octahedral stop sign and several triangular signs. We use line decomposition with a circuitry to compose the lines into polyhedral shapes.(Bheemaiah, n.d.) Why: BEAM is functional art, and forms the predecessor to MFA II or completely multi functional architecture of a broad umbrella of value addition in multi functionality, functoid and HOF based algebraic frameworks for MaC based definitions of architecture and design in code.(Autores and International Workshop on Higher-Order Algebra, Logic and Term Rewriting 1994; Kirchner and Wechler 1990; Dowek et al. 1996)

Dynamic Algorithms for Visibility Polygons in Simple Polygons

We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon. A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point [Formula: see text] due to the insertion (resp. deletion) of vertex [Formula: see text] to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of [Formula: see text] due to the insertion (resp. deletion) of [Formula: see text]. An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point [Formula: see text] in [Formula: see text] amid vertex insertions and deletions to the simple polygon. If [Formula: see text] is not exterior to the current simple polygon, then the visibility polygon of [Formula: see text] is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of [Formula: see text]. An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment [Formula: see text] due to the insertion of vertex [Formula: see text] to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at [Formula: see text] and [Formula: see text] due to the insertion of [Formula: see text]. An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon. And, the algorithms that maintain the visibility polygon (resp. weak visibility polygon) compute the visibility polygon (resp. weak visibility polygon) with respect to the initial simple polygon during the preprocessing phase.

Approximate Shortest Paths in Polygons with Violations

We study the shortest [Formula: see text]-violation path problem in a simple polygon. Let [Formula: see text] be a simple polygon in [Formula: see text] with [Formula: see text] vertices and let [Formula: see text] be a pair of points in [Formula: see text]. Let [Formula: see text] represent the interior of [Formula: see text]. Let [Formula: see text] represent the exterior of [Formula: see text]. For an integer [Formula: see text], the shortest [Formula: see text]-violation path problem in [Formula: see text] is the problem of computing the shortest path from [Formula: see text] to [Formula: see text] in [Formula: see text], such that at most [Formula: see text] path segments are allowed to be in [Formula: see text]. The subpaths of a [Formula: see text]-violation path are not allowed to bend in [Formula: see text]. For any [Formula: see text], we present a [Formula: see text] factor approximation algorithm for the problem that runs in [Formula: see text] time. Here [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] are geometric parameters.

Walking an Unknown Street with Limited Sensing

This paper studies a searching problem in an unknown street. A simple polygon [Formula: see text] with two distinguished vertices, [Formula: see text] and [Formula: see text], is called a street if the two boundary chains from [Formula: see text] to [Formula: see text] are mutually weakly visible. We use a mobile robot to locate [Formula: see text] starting from [Formula: see text]. Assume that the robot has a limited sensing capability that can only detect the constructed edges (also called gaps) on the boundary of its visible region, but cannot measure any angle or distance. The robot does not have knowledge of the street in advance. We present a new competitive strategy for this problem and prove that the length of the path generated by the robot is at most 9-times longer than the shortest path. We also propose a matching lower bound to show that our strategy is optimal. Compared with the previous strategy, we further relaxed the restriction that the robot should take a marking device and use the data structure S-GNT. The analysis of our strategy is tight.