Permutation Problem Research Articles

On a Nonstationary Route Problem with Constraints

The extremal route problem of permutations under constraints in the form of preceding conditions is investigated. It is supposed that an executer leaves the initial point (the base) after which he visits a system of megalopolises (finite goal sets) and performs some work on each megalopolis. The cost functions for executor permutations and interior works depend on the “visiting moment” that can correspond to the real time or can also correspond to the natural regular succession (the first visiting, the second visiting, and so on). An economic variant of the widely interpreted dynamic programming method (DPM) is constructed. On this basis an optimal computer realized algorithm is constructed. A variant of a greed algorithm is proposed.

English

English

Approximate Predictive Control of a Distillation Column Using an Evolving Artificial Neural Network Coupled with a Genetic Algorithm

Conventional gradient-based techniques are prone to getting trapped into local optimum and convergence is slow. To overcome these drawbacks, this study attempts to combine genetic algorithm, avoiding local minima and achieving global convergence quickly and correctly by searching in several regions simultaneously. The authors’ methodology utilizes a hybrid genetic algorithm-back propagation strategy. The proposed algorithm combines the local searching ability of the gradient-based back-propagation strategy with the global searching capability of the genetic algorithm. The genetic algorithm is used to decide the initial weights of the gradient decent methods so that all of the initial weights can be searched intelligently. The genetic operators and parameters are carefully designed to avoid premature convergence and permutation problems. For an evaluation purpose, the performance and generalization capabilities of genetic algorithm-back propagation are compared with those of models developed with the commonly used technique of back propagation (back propagation-artificial neural network). This strategy is applied to a highly nonlinear system of distillation column. The results demonstrate that a carefully designed hybrid genetic algorithm-back propagation neural network outperforms the gradient descent-based neural network in identification and, consequently, approximate model predictive control of the distillation column. Since the derived genetic algorithm-back propagation model is more accurate than a back propagation-artificial neural network model, an approximate model predictive control controller reveals a better performance.

Parallelization of genetic algorithms for sorting permutations by reversals over biological data

Reversals are operations of great biological significance for the analysis of the evolutionary distance between organisms. Genome rearrangement through reversals, consists in finding the shortest sequence of reversals to transform one genome represented as a signed or unsigned permutation into another. When genes are non oriented and correspondingly permutations are unsigned, sorting by reversals came arise as a challenging problem in combinatorics of permutations. In fact, this problem is known to be NP-hard, but the question whether it is NP-complete remains open for more than twenty years. When permutations are signed and correspondingly genes are oriented, the problem is known to be in P. A parallelization of a standard GA Genetic Algorithm is proposed for the problem of sorting unsigned permutations. This GA was previously reported in the literature as the most competitive regarding precision for which as control mechanism an 1.5-approximation algorithm was used. For the parallelization, the MPI Library of the C language was used and experiments were performed for calculating the execution time and precision. By increasing the number of individuals, experiment showed improvement in relation to previous approaches. Additionally, a virtualization of the GA using a MicroBlaze processor from Xilinx was performed on OVP for which the average number of executed instructions was of approximately 1.40 Giga instruction per second. In this extended version of this works originally presented in NaBIC 2013 biological data was generated and it was shown how the parallelization can be applied for their analysis. Specifically, the evolutionary distances between different pairs of organism were computed based on the set of non common genes in their mitochondrial DNA genome and the reversal distance between the sequences of common genes.

About a Routing Problem of the Tool Motion on Sheet Cutting

For the routing problem of tool permutations under the thermal cutting of parts from sheet material realized on CNC machines, questions connected with constructing precise (optimal) and heuristic algorithms used on the stage of mathematical simulation of route elements under sequential megalopolises circuit are investigated. Cutting points and points of tool cut-off are items (cities) of the above-mentioned megalopolises. In each megalopolis, interior works are provided. These works are connected with motion to the equidistant curve of the cut contour of a part from the cutting point and (with cutting completed) with motion from the equidistant curve to the tool cut-off (we keep in mind a working run). The problem about the time-optimal process of cutting which is a special variant of the generalized courier problem is investigated (the problem of the routing on the megalopolises with precedence conditions). An optimal procedure based on the dynamic programming and an effective heuristic algorithm realized on a multicore computer are proposed. A dynamic programming based procedure uses a special extension of the main problem. This extension provides the replacement of admissibility by precedence with the admissibility by deletion (from the list of tasks). Precedence conditions are used for decreasing computational complexity: it excludes the building of the whole array of the Bellman function values (this function is replaced by the layers system).

Convex Relaxations for Permutation Problems

Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing, for example. We write seriation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case, and we derive several convex relaxations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution and hence solve semisupervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains, and DNA assembly from shotgun gene sequencing data.

On the Computational Intractability of Exact and Approximate Dictionary Learning

The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete dictionary from training data sets of certain signal classes for use in sparse representations has turned out to be of particular importance regarding practical signal processing applications. Most popular dictionary learning algorithms involve NP-hard sparse recovery problems in each iteration, which may give some indication about the complexity of dictionary learning but does not constitute an actual proof of computational intractability. In this technical note, we show that learning a dictionary with which a given set of training signals can be represented as sparsely as possible is indeed NP-hard. Moreover, we also establish hardness of approximating the solution to within large factors of the optimal sparsity level. Furthermore, we give NP-hardness and non-approximability results for a recent dictionary learning variation called the sensor permutation problem. Along the way, we also obtain a new non-approximability result for the classical sparse recovery problem from compressed sensing.

Quantum Cryptanalysis of Multivariate Permutation Problem

Quantum computation is a new computational model based on quantum mechanical principle. Shor invented the polynomial time algorithms for the prime factorization and discrete logarithm problem, which indicated that the cryptosystems based on them are totally unsafe in the quantum world. Grover constructed an algorithm that finds a solution in only O( 2 n )steps whereas the exhaustive search algorithm needs O(2 n ) steps on average. In this paper we investigate the cryptanalysis of a new cryptography problem----multivariate permutation problem (MPP), which could be used to design public-key cryptosystem, with the help of the two quantum algorithms. Specially, we discuss the strength of a private key of the REESSE1+ public-key cryptosystem, whose security is based on the hardness of MPP. Besides, some suggestions are also given about the implementation of the REESSE1+.

An efficient algorithm based on sparse optimization for the aircraft departure scheduling problem

The aircraft departure scheduling problem (ASP) is a salient problem in the airports’ runway control system, which proves to be nondeterministic polynomial hard. This paper formulates the ASP in the form of a permutation problem and proposes a new approximation algorithm based on sparse optimization to solve it. Then the numerical study is conducted, which validates that this new approximation algorithm has much better performance than ant colony and CPLEX. After that, we consider the ASP in real-time scenario, where the performance of this new algorithm is also good. Finally, some conclusions are summarized, and the future research directions are pointed out.

Application of Genetic Algorithm on Travelling Salesman Person

GA (Genetic algorithm) is an optimization method based on operators (mutation and crossover) utilizing a survival of the fittest idea. They are utilized favorably in various problems. (TSP) Travelling salesman problem is one of the famous studied. TSP is a permutation problem in which the aim is to determine the shortest tour between n different points (cities), otherwise, the problem aims to find a route covering all cities where that the total distance is minimal. In this study a single salesman travels to each of the cities and close the loop by returning to the city he started, the aim of this study is to determine the minimum number of generations in which salesman does the minimum path, cities are chosen at random as initial population. The new generations are then created iteratively till the proper path is attained.

다채널 주파수영역 독립성분분석에서 분리된 신호 전력비의 공분산을 이용한 주파수 빈 정렬

In frequency domain ICA, the frequency bin permutation problem falls off the quality of separated signals. In this paper, we propose a new algorithm to solve the frequency bin permutation problem using the covariance of power ratio of separated signals in multi-channel FD-ICA. It makes use of the continuity of the spectrum of speech signals to check if frequency bin permutation occurs in the separated signal using the power ratio of adjacent frequency bins. Experimental results have shown that the proposed method could fix the frequency bin permutation problem in the multi-channel FD-ICA.

Joint Mixing Vector and Binaural Model Based Stereo Source Separation

In this paper the mixing vector (MV) in the statistical mixing model is compared to the binaural cues represented by interaural level and phase differences (ILD and IPD). It is shown that the MV distributions are quite distinct while binaural models overlap when the sources are close to each other. On the other hand, the binaural cues are more robust to high reverberation than MV models. According to this complementary behavior we introduce a new robust algorithm for stereo speech separation which considers both additive and convolutive noise signals to model the MV and binaural cues in parallel and estimate probabilistic time-frequency masks. The contribution of each cue to the final decision is also adjusted by weighting the log-likelihoods of the cues empirically. Furthermore, the permutation problem of the frequency domain blind source separation (BSS) is addressed by initializing the MVs based on binaural cues. Experiments are performed systematically on determined and underdetermined speech mixtures in five rooms with various acoustic properties including anechoic, highly reverberant, and spatially-diffuse noise conditions. The results in terms of signal-to-distortion-ratio (SDR) confirm the benefits of integrating the MV and binaural cues, as compared with two state-of-the-art baseline algorithms which only use MV or the binaural cues.

Independent vector analysis with a generalized multivariate Gaussian source prior for frequency domain blind source separation

Independent vector analysis (IVA) is designed to retain the dependency within individual source vectors, while removing the dependency between different source vectors. It can theoretically avoid the permutation problem inherent to independent component analysis (ICA). The dependency in each source vector is retained by adopting a multivariate source prior instead of a univariate source prior. In this paper, a multivariate generalized Gaussian distribution is adopted as the source prior which can exploit frequency domain energy correlation within each source vector. As such, it can utilize more information describing the dependency structure and provide improved source separation performance. This proposed source prior is suitable for the whole family of IVA algorithms and found to be more robust in applications where non-stationary signals are separated than the one preferred by Lee. Experimental results on real speech signals confirm the advantage of adopting the proposed source prior on three types of IVA algorithm.

An Efficient Separation for Convolutive Mixtures

This paper describes a new efficient blind source separation method; in this method we uses a non-uniform filter bank and a new structure with different sub-bands. This method provides a reduced permutation and increased convergence speed comparing to the full-band algorithm. Recently, some structures have been suggested to deal with two problems: reducing permutation and increasing the speed of convergence of the adaptive algorithm for correlated input signals. The permutation problem is avoided with the use of adaptive filters of orders less than the full-band adaptive filter, which operate at a sampling rate lower than the sampling rate of the input signal. The decomposed signals by analysis bank filter are less correlated in each sub-band than the input signal at full-band, and can promote better rates of convergence.

Universal aspects of curved, flat, and stationary-state Kardar-Parisi-Zhang statistics

Motivated by the recent exact solution of the stationary-state Kardar-Parisi-Zhang (KPZ) statistics by Imamura and Sasamoto [ Phys. Rev. Lett. 108 190603 (2012)], as well as a precursor experimental signature unearthed by Takeuchi [ Phys. Rev. Lett. 110 210604 (2013)], we establish here the universality of these phenomena, examining scaling behaviors of directed polymers in a random medium, the stochastic heat equation with multiplicative noise, and kinetically roughened KPZ growth models. We emphasize the value of cross KPZ-class universalities, revealing crossover effects of experimental relevance. Finally, we illustrate the great utility of KPZ scaling theory by an optimized numerical analysis of the Ulam problem of random permutations.

Research on Routing Algorithm Based on Limitation Arrangement Principle in Mathematics

Since the research on information consistency of the whole network under OSPF protocol has been insufficient in recent years, an algorithm based on limitation arrangement principle for routing decision is proposed and it is a permutation and combination problem in mathematical area. The most fundamental function of this algorithm is to accomplish the information consistency of the whole network at a relatively fast speed. Firstly, limitation arrangement principle algorithm is proposed and proved. Secondly, LAP routing algorithm in single link network and LAP routing algorithm in single link network with multiloops are designed. Finally, simulation experiments are worked by VC6.0 and NS2, which proves that LAPSN algorithm and LAPSNM algorithm can solve the problem of information consistency of the whole network under OSPF protocol and LAPSNM algorithm is superior to Dijkstra algorithm.

An Efficient Approximation Algorithm for Aircraft Arrival Sequencing and Scheduling Problem

The aircraft arrival sequencing and scheduling (ASS) problem is a salient problem in airports' runway scheduling system, which proves to be nondeterministic polynomial (NP) hard. This paper formulates the ASS in the form of a constrained permutation problem and designs a new approximation algorithm to solve it. Then the numerical study is conducted, which validates that this new algorithm has much better performance than ant colony (AC) algorithm and CPLEX, especially when the aircraft types are not too many. In the end, some conclusions are summarized.

Solution Classification for Perspective-Three-Point Problem Base on PST Method

The perspective-n-point (PnP) problem is originated from camera calibration. It is to determine the position and orientation of the camera with respect to a scene object from n correspondent points. And a new stable algorithm by using a geometric constraint called perspective similar triangle (PST) can give new equations to solve P3P. The PST method achieves high stability in the permutation problem and in presence of image noise. Using the complete discrimination system, we obtain the solution classification of the new equation for the P3P problem. The solution classification gives a set of formulas to determine the number of real solutions to the P3P problem. Based on the formulas, we may know whether the parameters give multiple solutions or not and are critical or not which is very important to present robust algorithm.

Graphics processing unit‐accelerated bounding for branch‐and‐bound applied to a permutation problem using data access optimization

SUMMARYBranch‐and‐bound (B&B) algorithms are attractive methods for solving to optimality combinatorial optimization problems using an implicit enumeration of a dynamically built tree‐based search space. Nevertheless, they are time‐consuming when dealing with large problem instances. Therefore, pruning tree nodes (subproblems) is traditionally used as a powerful mechanism to reduce the size of the explored search space. Pruning requires to perform the bounding operation, which consists of applying a lower bound function to the subproblems generated during the exploration process. Preliminary experiments performed on the Flow‐Shop scheduling problem (FSP) have shown that the bounding operation consumes over 98% of the execution time of the B&B algorithm. In this paper, we investigate the use of graphics processing unit (GPU) computing as a major complementary way to speed up the search. We revisit the design and implementation of the parallel bounding model on GPU accelerators. The proposed approach enables data access optimization. Extensive experiments have been carried out on well‐known FSP benchmarks using an Nvidia Tesla C2050 GPU card. Compared to a CPU‐based single core execution using an Intel Core i7‐970 processor without GPU, speedups higher than 100 times faster are achieved for large problem instances. At an equivalent peak performance, GPU‐accelerated B&B is twice faster than its multi‐core counterpart. Copyright © 2013 John Wiley & Sons, Ltd.

A block based estimation of distribution algorithm using bivariate model for scheduling problems

Recently, estimation of distribution algorithms (EDAs) have gradually attracted a lot of attention and have emerged as a prominent alternative to traditional evolutionary algorithms. In this paper, a block-based EDA using bivariate model is developed to solve combinatorial problems. Instead of generating a set of chromosomes, our approach generates a set of promising blocks using bivariate model and these blocks are reserved in an archive for future use. These blocks will be updated every other k generation. Then, two rules, i.e., AC1 and AC2, are developed to generate a new chromosome by combining the set of selected blocks and rest of genes. This block based approach is very efficient and effective when compared with the traditional EDAs. According to the experimental results, the block based EDA outperforms EDA, GA, ACO and other evolutionary approaches in solving benchmark permutation problems. The block based approach is a new concept and has a very promising result for other applications.