Exact and approximate solutions of the perspective-three-pointproblem

Model-based pose estimation techniques that match image and model triangles require large numbers of matching operations in real-world applications. The authors show that by using approximations to perspective, 2D lookup tables can be built for each of the triangles of the models. An approximation called `weak perspective' has been applied previously to this problem; the authors consider two other perspective approximations: paraperspective and orthoperspective. These approximations produce lower errors for off-center image features than weak perspective     

The singular control problem and approximate solutions

A method is presented which permits the solution of singular control problems by approximating the nonnormal system with a normal system. It is shown that the approximation can be made arbitrarily accurate. The optimal control of the approximate system is then determined and applied to the actual system. Bounds are obtained on the errors in boundary conditions and the degree of suboptimality brought about by the technique. The method is valid for minimum time and minimum fuel problems with linear constant plants. Examples are presented.     

Exact and approximate algorithms for the index selection problem inphysical database design

The index selection problem (ISP) is an important optimization problem in the physical design of databases. The aim of this paper is to show that ISP, although NP-hard, can in practice be solved effectively through well-designed algorithms. We formulate ISP as a 0-1 integer linear program and describe an exact branch-and-bound algorithm based on the linear programming relaxation of the model. The performance of the algorithm is enhanced by means of procedures to reduce the size of the candidate index set. We also describe heuristic algorithms based on the solution of a suitably defined knapsack subproblem and on Lagrangian decomposition. Finally, computational results on several classes of test problems are given. We report the exact solution of large-scale ISP instances involving several hundred indexes and queries. We also evaluate one of the heuristic algorithms we propose on very large-scale instances involving several thousand indexes and queries and show that it consistently produces very tight approximate (and sometimes provably optimal) solutions. Finally, we discuss possible extensions and future directions of research     

Integer programming problems with inexact data: Exact and approximate solutions

The study was partially supported by the Joint Foundation of the Government of Ukraine and International Scientific Foundation (grant No. K4C100).     

Exact,analytic, and locally approximate solutions to discrete event-simulation problems

We demonstrate how discrete event simulation problems can be encoded within a probabilistic proc algebra. From this encoding we can exploit vavious solution techniques to derive exact numerical solutions, analytic solutions and piecewise polynomial approximations for system properties. Our approach allows us to combine the advantages of simulation, viz problem decomposition and an executable representation, with standard analytic and numerical computation methods for calculating system properties.     

Pareto clustering search applied for 3D container ship loading plan problem

The 3D Container ship Loading Plan Problem (CLPP) is an important problem that appears in seaport container terminal operations. This problem consists of determining how to organize the containers in a ship in order to minimize the number of movements necessary to load and unload the container ship and the instability of the ship in each port. The CLPP is well known to be NP-hard. In this paper, the hybrid method Pareto Clustering Search (PCS) is proposed to solve the CLPP and obtain a good approximation to the Pareto Front. The PCS aims to combine metaheuristics and local search heuristics, and the intensification is performed only in promising regions. Computational results considering instances available in the literature are presented to show that PCS provides better solutions for the CLPP than a mono-objective Simulated Annealing.     

Exact and approximate rhythm matching algorithms

An interesting problem in music information retrieval is to classify songs according to rhythms. A rhythm is represented by a sequence of “Quick” (Q) and “Slow” (S) symbols, which correspond to the (relative) duration of notes, such that S?=?2Q. Christodoulakis et?al. presented an efficient algorithm that can be used to classify musical sequences according to rhythms. In this article, the above algorithm is implemented, along with a naive brute force algorithm to solve the same problem. The theoretical time complexity bounds are analyzed with the actual running times achieved by the experiments, and the results of the two algorithms are compared. Furthermore, new efficient algorithms are presented that take temporal errors into account. This, the approximate pattern matching version, could not be handled by the algorithms previously presented. The running times of two algorithmic variants are analyzed and compared and examples of their implementation are shown.     

Exact and approximate construction of offset polygons

The Minkowski sum of two sets $A,B\in {\mathbb{R}}^2$, denoted $A\oplus B$, is defined as $\{a+b\mid a\in A,b\in B\}$. We describe an efficient and robust implementation of the construction of the Minkowski sum of a polygon in ${\mathbb{R}}^2$ with a disc, an operation known as offsetting the polygon. Our software package includes a procedure for computing the exact offset of a straight-edge polygon, based on the arrangement of conic arcs computed using exact algebraic number-types. We also present a conservative approximation algorithm for offset computation that uses only rational arithmetic and decreases the running times by an order of magnitude in some cases, while having a guarantee on the quality of the result. The package will be included in the next public release of the Computational Geometry Algorithms Library, Cgal Version 3.3. It also integrates well with other Cgal packages; in particular, it is possible to perform regularized Boolean set-operations on the polygons the offset procedures generate.     

Exact and approximate flexible aggregate similarity search

Aggregate similarity search, also known as aggregate nearest-neighbor (Ann) query, finds many useful applications in spatial and multimedia databases. Given a group Q of M query objects, it retrieves from a database the objects most similar to Q, where the similarity is an aggregation (e.g., \({{\mathrm{sum}}}\), \(\max \)) of the distances between each retrieved object p and all the objects in Q. In this paper, we propose an added flexibility to the query definition, where the similarity is an aggregation over the distances between p and any subset of \(\phi M\) objects in Q for some support \(0< \phi \le 1\). We call this new definition flexible aggregate similarity search and accordingly refer to a query as a flexible aggregate nearest-neighbor ( Fann ) query. We present algorithms for answering Fann queries exactly and approximately. Our approximation algorithms are especially appealing, which are simple, highly efficient, and work well in both low and high dimensions. They also return near-optimal answers with guaranteed constant-factor approximations in any dimensions. Extensive experiments on large real and synthetic datasets from 2 to 74 dimensions have demonstrated their superior efficiency and high quality.     

Exact and approximate response of non-linear stochastic systems

The method of finding the exact solution and a new approximation method for a class of non-linear stochastic systems is presented. In both methods a non-linear transformation of the original system is applied. The idea of this method is presented for a non-linear system with stochastic parametric excitations. If the transformed system is a linear one then one can find an exact solution, otherwise the cumulant neglect closure technique is applied to this new system. The resuts obtained are illustrated by a numerical example.     

港航多视角下内河集装箱船舶配载决策

为满足内河集装箱运输中船舶航线配载实际决策需求,从港方和船方多视角出发,提出港航多视角下船舶航线配载决策方法。基于问题分析与特征提取构建考虑港方和船方双方利益的港航多视角下的船舶配载决策模型。考虑到问题的多目标优化特性,设计一种带模糊关联熵的启发式算法进行多目标并行寻优。通过算例实验验证了模型与算法的可行性与有效性。     

Exact and greedy solutions of the knapsack problem: the ratio of values of objective functions

Ratios δ of the values of objective functions of optimal Boolean (or integer) to the values of greedy solutions for the knapsack problem are considered. The relationship of the parameter δ with the ratio Δ of the values of objective functions for the optimal solution of linear relaxation to the values of optimal integer solution was found. Two-sided estimates for δ and Δ were obtained. A computational experiment was conducted to investigate the ratio of δ of problems of one- and two-dimensional knapsack problems with Boolean variables. A hypothesis on asymptotic behavior of the ratio δ with growth of the number of problem variables was formulated.     

Exact and approximate reasoning about temporal relations1

Allen gives an algebra for representing qualitative temporal information about the relationships between pairs of intervals. In this paper, we address a fundamental reasoning task that arises in applications of the algebra: Given (possibly indefinite) knowledge about the relationships between intervals, find all feasible relationships between two intervals. We call this the minimal labels problem. Finding the minimal labels can be viewed as computing the deductive consequences of our knowledge. Determining exact solutions to this problem has been shown to be (almost assuredly) intractable. Allen gives an approximation algorithm based on constraint propagation. We present new approximation algorithms; determine analytically under what conditions the algorithms are exact; and examine, through some computational experiments, the quality of the approximate solutions produced by the algorithms. We also give a simple test for predicting when the approximation algorithms will and will not produce good quality approximations. Finally, we survey three example applications of the interval algebra chosen from the literature to show where the results of this paper could be useful.     

Exact and approximate graph matching using random walks

In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection.     

Exact and approximate feedback linearization without the linear controllability assumption

The feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable. In this paper, the assumption of linear controllability is removed; necessary and sufficient conditions are given through Lie orbital symmetries, thus giving a geometric characterization of the problem in the analytic case. Both the exact and the approximate linearization problems are considered in the analytic case.     

Exact and approximate computation of B-spline curves on surfaces

Curves on surfaces are important elements in computer aided geometric design. After presenting a method to explicitly compute these curves in three-dimensions, practical algorithmic issues are discussed concerning the efficiency of the implementation. Good approximations are important because of the quite high degree of exact curves on surfaces. We present two approximate solutions to the problem. The first is derived from the exact representation, while the second extends conventional least-squares approximation by incorporating the geometry of the surface as well. The efficiency and behaviour of the algorithms are evaluated by means of examples.     

Exact and approximate approaches to the identification of stochastic max-plus-linear systems

Stochastic max-plus linear systems, i.e., perturbed systems that are linear in the max-plus algebra, belong to a special class of discrete-event systems that consists of systems with synchronization but no choice. In this paper, we study the identification problem for such systems, considering two different approaches. One approach is based on exact computation of the expected values and consists in recasting the identification problem as an optimization problem that can be solved using gradient-based algorithms. However, due to the structure of stochastic max-plus linear systems, this method results in a complex optimization problem. The alternative approach discussed in this paper, is an approximation method based on the higher-order moments of a random variable. This approach decreases the required computation time significantly while still guaranteeing a performance that is comparable to the one of the exact solution.     

Exact and approximate area-proportional circular Venn and Euler diagrams

Scientists conducting microarray and other experiments use circular Venn and Euler diagrams to analyze and illustrate their results. As one solution to this problem, this paper introduces a statistical model for fitting area-proportional Venn and Euler diagrams to observed data. The statistical model outlined in this paper includes a statistical loss function and a minimization procedure that enables formal estimation of the Venn/Euler area-proportional model for the first time. A significance test of the null hypothesis is computed for the solution. Residuals from the model are available for inspection. As a result, this algorithm can be used for both exploration and inference on real data sets. A Java program implementing this algorithm is available under the Mozilla Public License. An R function venneuler() is available as a package in CRAN and a plugin is available in Cytoscape.