Polygonal Problem Research Articles

On the Two-Dimensional Knapsack Problem for Convex Polygons

We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O (1)-approximation algorithm for the general case and a pseudopolynomial time O (1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Additionally, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation—that is, we allow to increase the size of the knapsack by a factor of 1+δ for some δ > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles.

Some extremal problems for polygons in the Euclidean plane

Some extremal problems for polygons in the Euclidean plane

The nonlocal isoperimetric problem for polygons: Hardy–Littlewood and Riesz inequalities

The nonlocal isoperimetric problem for polygons: Hardy–Littlewood and Riesz inequalities

Self-avoiding walks contained within a square

We have studied self-avoiding walks contained within an L × L square whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and north-west corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least one edge on all four boundaries of the square. We provide compelling evidence that these two classes of walks grow identically. From our analysis we conjecture that the number of such walks C L , for both problems, behaves as CL∼λL2+bL+c⋅Lg, where (Guttmann and Jensen 2022 J. Phys. A: Math. Theor.) λ = 1.744 5498 ± 0.000 0012, b = −0.043 54 ± 0.0005, c = −1.35 ± 0.45, and g = 3.9 ± 0.1. Finally, we also studied the equivalent problem for self-avoiding polygons, also known as cycles in a square grid. The asymptotic behaviour of cycles has the same form as walks, but with different values of the parameters c, and g. Our numerical analysis shows that λ and b have the same values as for WCAS and that c = 1.776 ± 0.002 while g = −0.500 ± 0.005 and hence probably equals .

A robust, fast, and accurate algorithm for point in spherical polygon classification with applications in geoscience and remote sensing

The point in spherical polygon problem addresses the task of classifying an arbitrary query point as inside, outside, or on the boundary of a polygon drawn using great circle segments on the spherical surface. Point in spherical polygon has numerous applications in the geosciences and remote-sensing, where the problem of sorting point position data onto geographic regions on the surface of the earth or sensor field-of-view often arises. By resorting to a direct solution on the spherical surface, distortions arising from projection can be avoided. We examine the problem, with and without preprocessing, where preprocessing is a step applied when a large number of classifications are evaluated against the same spherical polygon. Several improvements are introduced to the existing point in spherical polygon algorithm (which does not include preprocessing), resulting in significantly faster inclusion queries and better handling of edge cases. Furthermore, we introduce a preprocessing algorithm by means of recursive, nonuniform spatial subdivision of longitudinal regions (“lunes”) on the sphere. With an O(nlogn) preprocessing step and O(n) space requirement, the algorithm decreases query time from O(n) to O(logn) for most “realistic” polygons found in geoscience and remote sensing applications. Several empirical tests demonstrate that the algorithm performs to theoretical expectations.

106.26 The nested polygons problem revisited

106.26 The nested polygons problem revisited

Extremal problems for spherical convex polygons

Extremal problems for spherical convex polygons

Repulsion-Based p-Dispersion with Distance Constraints in Non-Convex Polygons.

Motivated by the question of optimal facility placement, the classical p-dispersion problem seeks to place a fixed number of equally sized non-overlapping circles of maximal possible radius into a subset of the plane. While exact solutions to this problem may be found for placement into particular sets, the problem is provably NP-complete for general sets, and existing work is largely restricted to geometrically simple sets. This paper makes two contributions to the theory of p-dispersion. First, we propose a computationally feasible suboptimal approach to the p-dispersion problem for all non-convex polygons. The proposed method, motivated by the mechanics of the p-body problem, considers circle centers as continuously moving objects in the plane and assigns repulsive forces between different circles, as well as circles and polygon boundaries, with magnitudes inversely proportional to the corresponding distances. Additionally, following the motivating application of optimal facility placement, we consider existence of additional hard upper or lower distance bounds on pairs of circle centers, and adapt the proposed method to provide a p-dispersion solution that provably respects such constraints. We validate our proposed method by comparing it with previous exact and approximate methods for p-dispersion. The method quickly produces near-optimal results for a number of containers.

New bounds on guarding problems for orthogonal polygons in the plane using vertex guards with halfplane vision

Given a set F of k disjoint monotone orthogonal polygons with a total of m vertices, we present bounds on the number of vertex guards required to guard the free space and the boundaries of the polygons in F when the range of vision of each guard is bounded by 180∘ (the region in front of the guard). When the orthogonal polygons are axis aligned we prove that m2+⌊k4⌋+4 vertex guards are always sufficient. When the orthogonal polygons are arbitrary oriented, we show that m2+k+1 vertex guards are sometimes necessary and conjecture the bound is tight.

Extremal Problems for Convex Curves with a Given Self Chebyshev Radius

The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the relative Chebyshev radius. In particular, we determine the relative Chebyshev radius for an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and convex n-gons of a given relative Chebyshev radius.

The touring rays and related problems

The touring rays problem, which is also known as the traveling salesman problem for rays in the plane, asks to compute the shortest (closed) route that tours or intersects n given rays. We show that it can be reduced to the problem of computing a shortest route that intersects a set of ray-segments, inside a circle; at least one endpoint of every ray-segment is on the circle. Moreover, computing the shortest route intersecting all ray-segments in the circle is related to the solution of the well-known watchman route problem. Our method is further extended to solve the minimum-perimeter intersecting polygon problem, which asks for a (convex) polygon P of minimum perimeter such that for a given set of line segments, P contains at least one point of every line segment. Both of our algorithms run in O(n5) time, and they solve two long-standing open problems in computational geometry.

Decomposition in decision and objective space for multi-modal multi-objective optimization

Multi-modal multi-objective optimization problems (MMMOPs) have multiple subsets within the Pareto-optimal Set, each independently mapping to the same Pareto-Front. Prevalent multi-objective evolutionary algorithms are not purely designed to search for multiple solution subsets, whereas, algorithms designed for MMMOPs demonstrate degraded performance in the objective space. This motivates the design of better algorithms for addressing MMMOPs. The present work identifies the crowding illusion problem originating from using crowding distance globally over the entire decision space. Subsequently, an evolutionary framework, called graph Laplacian based Optimization using Reference vector assisted Decomposition (LORD), is proposed, which uses decomposition in both objective and decision space for dealing with MMMOPs. Its filtering step is further extended to present LORD-II algorithm, which demonstrates its dynamics on multi-modal many-objective problems. The efficacies of the frameworks are established by comparing their performance on test instances from the CEC 2019 multi-modal multi-objective test suite and polygon problems with the state-of-the-art algorithms for MMMOPs and other multi- and many-objective evolutionary algorithms. The manuscript is concluded by mentioning the limitations of the proposed frameworks and future directions to design still better algorithms for MMMOPs. The source code is available at https://worksupplements.droppages.com/lord.

The analysis of basins of convergence in the regular polygon problem of [formula omitted] bodies system with spheroidal primaries

In the present manuscript, we unveil the topology of the basins of convergence in the regular polygon problem of (N+1)-bodies in two different cases, i.e., in case-I, only the central primary creates the Manev-type quasi-homogeneous potential, and in case-II, the peripheral primaries create the Manev-type quasi-homogeneous potential. The regular polygon problem of (N+1)-bodies describes the motion of the test particle moving in the force field of N primaries, the ν=N−1 peripheral primaries of equal masses situated at the vertices of the imaginary regular ν-gon and the Nth primary with different mass i.e., the central primary situated at the centre of mass of the system. In this model, we assume that the primaries create quasi-homogeneous potentials instead of Newtonian potentials and forces. In order to approximate various phenomena due to the irregular shape of the primaries or due to emitting radiation, an inverse cube corrective term is inserted to the inverse square Newtonian law of gravitation. We, numerically, investigated the evolution of the positions of the libration points and their linear stability for different values of ν in both the cases. Further, the multivariate version of Newton-Raphson iterative scheme is applied to unveil the topology of the basins of convergence. Moreover, the “basin entropy” is also computed to analyse the basins of convergence quantitatively.

Inverse Steklov Spectral Problem for Curvilinear Polygons

Abstract This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $\pi $, we prove that the asymptotics of Steklov eigenvalues obtained in [ 20] determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard–Weierstrass factorization theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.

Size constrained k simple polygons

Given a geometric space and a set of weighted spatial points, the Size Constrained k Simple Polygons (SCkSP) problem identifies k simple polygons that maximize the total weights of the spatial points covered by the polygons and meet the polygon size constraint. The SCkSP problem is important for many societal applications including hotspot area detection and resource allocation. The problem is NP-hard; it is computationally challenging because of the large number of spatial points and the polygon size constraint. Our preliminary work introduced the Nearest Neighbor Triangulation and Merging (NNTM) algorithm for SCkSP to meet the size constraint while maximizing the total weights of the spatial points. However, we find that the performance of the NNTM algorithm is dependent on the t-nearest graph. In this paper, we extend our previous work and propose a novel approach that outperforms our prior work. Experiments using Chicago crime and U.S. Federal wildfire datasets demonstrate that the proposed algorithm significantly reduces the computational cost of our prior work and produces a better solution.

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.

Windowing queries using Minkowski sum and their extension to MapReduce

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal $$O(n^2)$$ time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with $$O(\log n+k)$$ query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of trajectories where the center of a query is a popular place if the length of the curves inside that query is at least f and use our data structure to solve the original problem as well as this new version. Other than length queries, we solve reporting queries that return the set of intersected segments. For disk queries, we design a point-location data structure for congruent disks with $$O(\log n)$$ query time and $$O(n^3\log n)$$ preprocessing. We also give algorithms for computing the length query for c-packed curves, which are a class of curves for which the length of the curve inside a disk of radius r is upper-bounded by cr, where c is a constant. Also, we use length queries for polygons to approximate the minimum value c for which a curve is c-packed, if such a c exists. Our results extend to MRC and MPC models for MapReduce, where we address these problems on a set of x-monotone curves. The round complexities of our MapReduce algorithms are constant. In addition, we also implemented our popular places algorithms on trajectories on inputs as big as 15K points to evaluate the efficiency of our algorithms in practice.

Localized Analysis of Signals on the Sphere Over Polygon Regions

To support localized signal analysis on the sphere for applications in geophysics, cosmology, acoustics and beyond, we develop a framework for the analytic evaluation of the integral of spherical signals and for the analytic solution of the Slepian spatial-spectral concentration problem over simple spherical polygons. We propose a polygon right angle triangulation method for the division of a simple spherical polygon into spherical right angle triangles, which allows us to decompose the problem of integrating signals or solving the spatial-spectral concentration problem over the polygon region into sub-problems that require the evaluation of the integral of complex exponential functions over spherical right angle triangles of arbitrary orientation and position. We derive closed-form expressions for the evaluation of such integrals by appropriately choosing the rotations and using Wigner- $D$ functions. The proposed framework enables us to solve the Slepian spatial-spectral concentration problem for simple spherical polygons, resulting in bandlimited, spatially optimally concentrated basis functions for signal representation and reconstruction, localized analysis and signal modeling on the sphere. We also present convergence criterion for the infinite series expansions involved in the evaluation of the integral of complex exponential functions, and establish the validity of the proposed developments by evaluating the integral and computing the Slepian basis functions over the geographical region of Australia and the volcanic plateau of Tharsis, using the Earth and Mars topography maps respectively.

On Fredholm Eigenvalues of Unbounded Polygons

An important open problem in geometric complex analysis is to establish some algorithms for explicit determination of the basic functionals intrinsically connected with conformal and quasiconformal mappings such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This problem has not been solved even for generic quadrilaterals. We provide a restricted solution of the problem for unbounded rectilinear polygons.

Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

Given a rectilinear domain \(\mathcal {P}\) of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in \(\mathcal {P}\). Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for the minimum-link shortest paths, we obtain the following results. Our algorithm computes a minimum-link shortest \(s\)-\(t\) path in \(O(n+h\log ^{3/2} h)\) time. For the one-point queries, we build a data structure of size \(O(n+ h\log h)\) in \(O(n+h\log ^{3/2} h)\) time for a source point s, such that given any query point t, a minimum-link shortest \(s\)-\(t\) path can be computed in \(O(\log n)\) time. For the two-point queries, with \(O(n+h^2\log ^2 h)\) time and space preprocessing, a minimum-link shortest \(s\)-\(t\) path can be computed in \(O(\log n+\log ^2 h)\) time for any two query points s and t; alternatively, with \(O(n+h^2\cdot \log ^{2} h \cdot 4^{\sqrt{\log h}})\) time and \(O(n+h^2\cdot \log h \cdot 4^{\sqrt{\log h}})\) space preprocessing, we can answer each two-point query in \(O(\log n)\) time. Note that \(h^2\cdot \log ^{2} h \cdot 4^{\sqrt{\log h}}=O(h^{2+\epsilon })\) for any \(\epsilon >0\). These results are particularly interesting when h is relatively small. For example, if \(h=O(n^{1/2-\epsilon })\) for any \(\epsilon >0\), then all above results match the best results for the problems in simple rectilinear polygons, which are optimal. The complexities for the other two types of paths are slightly worse, but still linearly depend on n (in addition to g(h) for some functions g(h) of h).