Problems In Computational Geometry Research Articles

Вложение гомотета в выпуклый компакт: алгоритм и его сходимость

The problem of covering of a given convex compact set by a homothetic image of another convex compact set with a given homothety center is considered, the coefficient of homothety is calculated. The problem has an old history and is closely related to questions about the Chebyshev center, problems about translates, and other problems of computational geometry. Polyhedral approximation methods and other approximation methods do not work in a space of already moderate dimension (more than 5 on a PC). We propose an approach based on the application of the gradient projection method, which is much less sensitive to dimension than the approximation methods. We select classes of sets for which we can prove the linear convergence rate of the gradient method, i. e. convergence with the rate of a geometric progression with a positive ratio strictly less than 1. These sets must be strongly convex and have, in a certain sense, smoothness of the boundary.

Visibility Graphs of Anchor Polygons

The visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given graph. Another important problem is to reconstruct such a polygon if there is any. These problems are well known and well-studied, but yet open problems in geometric graphs and computational geometry. However, they have been solved efficiently for special cases where the target polygon is known to be a tower or a spiral polygon. In this paper, we propose a linear time algorithm to solve these recognizing and reconstruction problems for another type of polygons, named anchor polygons.

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.

Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem

The cavity decomposition problem is a computational geometry problem, arising in the context of modern electronic CAD systems, that concerns detecting the generation and propagation of electromagnetic noise into multi-layer printed circuit boards. Algorithmically speaking, the problem can be formulated so as to contain, as sub-problems, the well-known polygon schematization and polygon decomposition problems. Given a polygon P and a finite set C of given directions, polygon schematization asks for computing a C -oriented polygon P ′ with “low complexity” and “high resemblance” to P , whereas polygon decomposition asks for partitioning P into a set of basic polygonal elements (e.g., triangles) whose size is as small as possible. In this article, we present three different solutions for the cavity decomposition problem, which are obtained by suitably combining existing algorithms for polygon schematization and decomposition, by considering different input parameters, and by addressing both methodological and implementation issues. Since it is difficult to compare the three solutions on a theoretical basis, we present an extensive experimental study, employing both real-world and random data, conducted to assess their performance. We rank the proposed solutions according to the results of the experimental evaluation, and provide insights on natural candidates to be adopted, in practice, as modules of modern printed circuit board design software tools, depending on the observed performance and on the different constraints on the desired output.

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.

Well-testing based turbidite lobes modeling using the ensemble smoother with multiple data assimilation

The representation of geological bodies is a difficult task, which involves a large number of parameters and assumptions that are commonly simplified in object-based modeling. A famous method that has been extensively applied for modeling geological bodies is the use of non-uniform rational B-Spline curves (NURBS) to delimit the boundaries of an object. Although NURBS provides highly detailed models, it only considers geological observations and assumptions. Moreover, the use of NURBS neglects information obtained from well-testing. This method also requires the complex and time-consuming process of determining the interior of the object in the parametric space. This is a classic problem in computational geometry, known as point location. To address these problems, this study proposes a well-testing-based object-based model of turbidite lobes using the ensemble smoother with multiple data assimilation. To escape the point location problem, we use single-valued B-Spline curves (SVBS) to build the turbidite system model. These curves are planar type B-Spline curves but they are defined as functions. The use of SVBS avoids the use of all complex algorithms and structures for solving the point location problem in the parametric space, if it is straightforward to decide if a point is in or out of the object. Consequently, it is possible to use the ensemble smoother with multiple data assimilation method to estimate the geometric parameters of the object, where a large number of realizations is required.

Symbolic-numeric methods oncomputing isolated singular solutions of algebraic systems

Solving systems of algebraic equations is one of the most fundamental problems in computational algebraic geometry. It is ubiquitous and widely applied across the engineering and sciences, such as in robotics, computer vision, machine learning, artificial intelligence, cryptography, optimization, control theory and etc. One main challenge is to compute isolated singular solutions, which plays an important rule in geometric modelings.Based on recent research results of the authors and their collaborators,a survey for symbolic-numeric methods on computing isolated singular solutions of algebraic systems is conducted, especially for refining and certifying approximate solutions.Some directions for future studies on the topic are discussed as well.

Eliminating Depth Cycles Among Triangles in Three Dimensions

The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any \(\varepsilon >0\), the triangles can be cut into \(O(n^{3/2+\varepsilon })\) connected semialgebraic pieces, whose description complexity depends only on the choice of \(\varepsilon \), such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. The pieces can be constructed efficiently. This work extends the recent study by two of the authors (Discrete Comput. Geom. 59(3), 725–741 (2018)) on eliminating depth cycles among lines in 3-space. Our approach is again algebraic, and makes use of a recent variant of the polynomial partitioning technique, due to Guth (Math. Proc. Camb. Philos. Soc. 159(3), 459–469 (2015)), which leads to a recursive algorithm for cutting the triangles. In contrast to the case of lines, our analysis here is considerably more involved, due to the two-dimensional nature of the objects being cut, so additional tools, from topology and algebra, need to be brought to bear. Our result makes significant progress towards resolving a decades-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics. In addition, we generalize our bound to well-behaved patches of two-dimensional algebraic surfaces of constant degree.

The distance between convex sets with Minkowski sum structure: application to collision detection

The distance between sets is a long-standing computational geometry problem. In robotics, the distance between convex sets with Minkowski sum structure plays a fundamental role in collision detection. However, it is typically nontrivial to be computed, even if the projection onto each component set admits explicit formula. In this paper, we explore the problem of calculating the distance between convex sets arising from robotics. Upon the recent progress in convex optimization community, the proposed model can be efficiently solved by the recent hot-investigated first-order methods, e.g., alternating direction method of multipliers or primal-dual hybrid gradient method. Preliminary numerical results demonstrate that those first-order methods are fairly efficient in solving distance problems in robotics.

The Parallel Seeding Algorithm for k-Means Problem with Penalties

As a classic NP-hard problem in machine learning and computational geometry, the [Formula: see text]-means problem aims to partition a data point set into [Formula: see text] clusters such that the sum of the squared distance from each point to its nearest center is minimized. The [Formula: see text]-means problem with penalties, denoted by [Formula: see text]-MPWP, generalizing the [Formula: see text]-means problem, allows that some points can be paid some penalties instead of being clustered. In this paper, we study the seeding algorithm of [Formula: see text]-MPWP and propose a parallel seeding algorithm for [Formula: see text]-MPWP along with the corresponding theoretical analysis.

The provably good parallel seeding algorithms for the k‐means problem with penalties

AbstractAs a classic NP‐hard problem in machine learning and computational geometry, the k‐means problem aims to partition the given points into k sets to minimize the within‐cluster sum of squares. The k‐means problem with penalties (k‐MPWP), as a generalizing problem of the k‐means problem, allows a point that can be either clustered or penalized with some positive cost. In this paper, we mainly apply the parallel seeding algorithm to the k‐MPWP, and show sufficient analysis to bound the expected solution quality in the case where both the number of iterations and the number of points sampled in each iteration can be given arbitrarily. The approximate guarantee can be obtained as , where is a polynomial function involving the maximal ratio M between the penalties. On one hand, this result can be viewed as a further improvement on the parallel algorithm for k‐MPWP given by Li et al., where the number of iterations depends on other factors. On the other hand, our result also generalizes the one solving the k‐means problem presented by Bachem et al., because k‐MPWP is a variant of the k‐means problem. Moreover, we present a numerical experiment to show the effectiveness of the parallel algorithm for k‐means with penalties.

On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Given a set of n points in ℝd, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the lp-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d = ω(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [6], Williams [48], David-Karthik-Laekhanukit [17]). In this paper, we show that for every p ∈ ℝ≥1 ∪ {0}, under the Strong Exponential Time Hypothesis (SETH), for every e > 0, the following holds: In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to ne factor in the running time) for $$d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ dimensions. Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in no(1)-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product. At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n2−e edges whose vertices can be realized as points in a $${\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}$$ -dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Micciancio-Sudan [18].

Robust vertex enumeration for convex hulls in high dimensions

Computation of the vertices of the convex hull of a set $S$ of $n$ points in $\mathbb{R} ^m$ is a fundamental problem in computational geometry, optimization, machine learning and more. We present "All Vertex Triangle Algorithm" (AVTA), a robust and efficient algorithm for computing the subset $\overline S$ of all $K$ vertices of $conv(S)$, the convex hull of $S$. If $\Gamma_*$ is the minimum of the distances from each vertex to the convex hull of the remaining vertices, given any $\gamma \leq \gamma_* = \Gamma_*/R$, $R$ the diameter of $S$, $AVTA$ computes $\overline S$ in $O(nK(m+ \gamma^{-2}))$ operations. If $\gamma_*$ is unknown but $K$ is known, AVTA computes $\overline S$ in $O(nK(m+ \gamma_*^{-2})) \log(\gamma_*^{-1})$ operations. More generally, given $t \in (0,1)$, AVTA computes a subset $\overline S^t$ of $\overline S$ in $O(n |\overline S^t|(m+ t^{-2}))$ operations, where the distance between any $p \in conv(S)$ to $conv(\overline S^t)$ is at most $t R$. Next we consider AVTA where input is $S_\varepsilon$, an $\varepsilon$ perturbation of $S$. Assuming a bound on $\varepsilon$ in terms of the minimum of the distances of vertices of $conv(S)$ to the convex hull of the remaining point of $S$, we derive analogous complexity bounds for computing $\overline S_\varepsilon$. We also analyze AVTA under random projections of $S$ or $S_\varepsilon$. Finally, via AVTA we design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models AVTA leads to significantly better reconstruction of the topic-word matrix than state of the art approaches~\cite{arora2013practical, bansal2014provable}. For non-negative matrix AVTA is competitive with existing methods~\cite{arora2012computing}. Empirically AVTA is robust and can handle larger amounts of noise than existing methods.

Modifications of the Charged Balls Method

AbstractThe Charged Balls Method is based on physical ideas. It allows one to solve problem of finding the minimum distance from a point to a convex closed set with a smooth boundary, finding the minimum distance between two such sets and other problems of computational geometry. This paper proposes several new quick modifications of the method. These modifications are compared with the original Charged Ball Method as well as other optimization methods on a large number of randomly generated model problems.We consider the problem of orthogonal projection of the origin onto an ellipsoid. The main aim is to illustrate the results of numerical experiments of Charged Balls Method and its modifications in comparison with other classical and special methods for the studied problem.

On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product

In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a · b. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact e2-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three main results regarding this problem.• Characterization of Multiplicative Approximation. First, we study the best multiplicative approximation ratio for Boolean Max-IP in sub-quadratic time. We show that, for Max-IP with two sets of n vectors from {0, 1}d, there is an n2−ω(1) time (d/logn)ω(1) - multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/logn)o(1)-approximating algorithm would refute SETH. Similar characterization is also achieved for additive approximation for Max-IP.• 2O(log* n)-dimensional Hardness for Exact Max-IP Over The Integers. Second, we revisit the hardness of solving Max-IP exactly for vectors with integer entries. We show that, under SETH, for Max-IP with sets of n vectors from Zd for some d = 2O(log* n), every exact algorithm requires n2−o(1) time. With the reduction from [Williams, SODA 2018], it follows that e2-Furthest Pair and Bichromatic e2-Closest Pair in 2O(log* n) dimensions require n2−o(1) time.• Connection with NP · UPP Communication Protocols. Last, We establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP · UPP communication protocols for Set-Disjointness, parallel to the connection between conditional lower bounds for approximating Max-IP and MA communication protocols for Set-Disjointness.The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018], and our algorithms utilize the polynomial method and simple random sampling. Our second result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for n vectors from {0, 1}d to n vectors from Ze where e = 2O(log* d), dramatically improving the previous reduction in [Williams, SODA 2018]. The key technical ingredient is a recursive application of Chinese Remainder Theorem.As a side product, we obtain an MA communication protocol for Set-Disjointness with complexity [EQUATION], slightly improving the [EQUATION] log n) bound [Aaronson and Wigderson, TOCT 2009], and approaching the [EQUATION] lower bound [Klauck, CCC 2003].Moreover, we show that (under SETH) one can apply the [EQUATION] BQP communication protocol for Set-Disjointness to prove near-optimal hardness for approximation to Max-IP with vectors in {− 1, 1}d. This answers a question from [Abboud et al., FOCS 2017] in the affirmative.

Comparison of two modified meta-heuristic soft algorithms for solving a bi-objective facility layout problem

Facility layout problem (FLP) is one of the most important location analysis problems in operation research and computational geometry. This paper presents a novel optimisation model-based two meta-heuristics of particle swarm optimisation (PSO) and parallel simulated annealing (SA). Then, the problem is solved using the modified algorithms to determine the total material handling cost and dead space. The results obtained from the two proposed models are then compared and analysed by referring to two well-known previous approaches to find the optimal solution for the problem. The obtained results show that the modified PSO and SA algorithms outperform the previous two methods and therefore can be effectively as an optimal solution for certain bi-objective facility layout problems.

Comparison of two modified meta-heuristic soft algorithms for solving a bi-objective Facility Layout Problem (FLP)

Facility layout problem (FLP) is one of the most important location analysis problems in operation research and computational geometry. This paper presents a novel optimisation model-based two meta-heuristics of particle swarm optimisation (PSO) and parallel simulated annealing (SA). Then, the problem is solved using the modified algorithms to determine the total material handling cost and dead space. The results obtained from the two proposed models are then compared and analysed by referring to two well-known previous approaches to find the optimal solution for the problem. The obtained results show that the modified PSO and SA algorithms outperform the previous two methods and therefore can be effectively as an optimal solution for certain bi-objective facility layout problems.

Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

We introduce a novel Conceptual Framework for finding approximations to both Minimum Weight Triangulation (MWT) and optimal Traveling Salesman Problem (TSP) of planar point sets. MWT is a classical problem of Computational Geometry with various applications, whereas TSP is perhaps the most researched problem in Combinatorial Optimization. We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality principle, measured via Compactness Index, as a key link between our two stated problems. Our experiments show that the proposed framework yields tight approximations for both problems.

A polynomial time algorithm for big data in a special case of minimum constraint removal problem

The minimum constraint removal (MCR) is one of the most important problems in motion planning and computational geometry. In this problem, there is no feasible path for a robot to move from the starting point towards the goal. Therefore, in order to find a collision-free path, the minimum constraints should be removed. The MCR problem is $$NP-hard$$ when constraints have arbitrary shapes or are in shape of convex polygons. Since it is not possible to solve the problem by deterministic algorithms in an acceptable time for the problem with big data, scientists have tried to solve it by approximation methods. In this paper, we present a deterministic (not approximation) algorithm that solve the problem in a polynomial time. The simplification we used here is about the type of data (neither about the accuracy of the solution, nor about the size of data set). In this paper, a special case of this problem is presented, in which all the constraints are axis-aligned-unit squares and the obstacles have only local effects. Local effect means there are no two cells which have the same label sets. We propose an algorithm for this variant of the problem which can be used for big data. The proposed algorithm has $$O(n^{3})$$ time complexity in the worst case.

More Logarithmic-factor Speedups for 3SUM, (median,+)-convolution, and Some Geometric 3SUM-hard Problems

This article presents an algorithm that solves the 3SUM problem for n real numbers in O (( n 2 / log 2 n )(log log n ) O (1) ) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. This work also obtains the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.