Polynomial Problems Research Articles

Joint decision-making of virtual module formation and scheduling considering queuing time

Formation and scheduling are the most important decisions in the virtual modular manufacturing system; however, the global performance optimization of the system may be sacrificed via the superposition of two independent decision-making results. The joint decision of formation and scheduling is very important for system design. Complex and discrete manufacturing enterprises such as shipbuilding and aerospace often comprise multiple tasks, processes, and parallel machines, resulting in complex routes. The queuing time of parts in front of machines may account for 90% of the production cycle time. This study established a weighted allocation model of a formation-scheduling joint decision problem considering queuing time in system. To solve this nondeterministic polynomial (NP) problem, an adaptive differential evolution-simulated annealing (ADE-SA) algorithm is proposed. Compared with the standard differential evolution (DE) algorithm, the adaptive mutation factor overcomes the disadvantage that the scale of DE’s differential vector is difficult to control. The selection strategy of the SA algorithm compensates for the deficiency that DE’s greedy strategy may fall into a local optimal solution. The comparison results of four algorithms of a series of random examples demonstrate that the overall performance of ADE-SA is superior to the genetic algorithm, and average iteration, maximum completion time, and move time are 24%, 11%, and 7% lower than the average of other three algorithms, respectively. The method can generate the joint decision-making scheme with better overall performance, and effectively identify production bottlenecks through quantitative analysis of queuing time.

On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

Online Deployment Algorithms for Microservice Systems With Complex Dependencies

Cloud and edge computing have been widely adopted in many application scenarios. With the increasing demand of fast iteration and complexity of business logic, it is challenging to achieve rapid development and continuous delivery in such highly distributed cloud and edge computing environment. At present, microservice-based architecture has been the dominant deployment style, and a microservice system has to evolve agilely to offer stable Quality of Service (QoS) in the situation where user requirement changes frequently. Many research have been conducted to optimally re-deploy microservices to adapt to changing requirements. Nevertheless, complex dependencies between microservices and the existence of multiple instances of one single microservice in a microservice system have not been fully considered in existing works. This paper defines SPPMS, the Service Placement Problem in Microservice Systems that feature complex dependencies and multiple instances, as a Fractional Polynomial Problem (FPP) . Considering the high computation complexity of FPP, it is then transformed into a Quadratic Sum-of-Ratios Fractional Problem (QSRFP) which is further solved by the proposed greedy-based algorithms. Experiments demonstrate that our models and algorithms outperform existing approaches in both quality and computation speed.

Matrix approach to solve polynomial equations

Polynomials are widely employed to represent numbers derived from mathematical operations in nearly all areas of mathematics. The ability to factor polynomials entirely into linear components allows for a wide range of problem simplifications. This paper presents and demonstrates a novel straightforward approach to solving polynomial problems by converting them to matrix equations. Each polynomial of degree n can be decomposed into a sum of degree ⌈n2⌉ polynomials squared, i.e., ∑i=0naixi=∑i=1⌈n2⌉+1∑j=0j=⌈n2⌉bi,jxj2. It follows that the complexity of factorizing a polynomial of degree 2n is equivalent to that of factorizing polynomial of degree 2n−1. The proposed method for solving fourth-degree polynomials will be a valuable contribution to linear algebra due to its simplicity compared to the current method. This work presents a unique approach to solving polynomials of four or fewer degrees and presents new possibilities for tackling larger degrees. Additionally, our methodology can also be used for educational purposes.

A (t,n)- Secret image sharing with steganography based on Rook polynomial and LWE problem

A (t,n)- Secret image sharing with steganography based on Rook polynomial and LWE problem

Conic Relaxations with Stable Exactness Conditions for Parametric Robust Convex Polynomial Problems

Conic Relaxations with Stable Exactness Conditions for Parametric Robust Convex Polynomial Problems

A deep reinforcement learning algorithm for the rectangular strip packing problem.

As a branch of the two-dimensional (2D) optimal blanking problem, rectangular strip packing is a typical non-deterministic polynomial (NP-hard) problem. The classical packing solution method relies on heuristic and metaheuristic algorithms. Usually, it needs to be designed with manual decisions to guide the solution, resulting in a small solution scale, weak generalization, and low solution efficiency. Inspired by deep learning and reinforcement learning, combined with the characteristics of rectangular piece packing, a novel algorithm based on deep reinforcement learning is proposed in this work to solve the rectangular strip packing problem. The pointer network with an encoder and decoder structure is taken as the basic network for the deep reinforcement learning algorithm. A model-free reinforcement learning algorithm is designed to train network parameters to optimize the packing sequence. This design can not only avoid designing heuristic rules separately for different problems but also use the deep networks with self-learning characteristics to solve different instances more widely. At the same time, a piece positioning algorithm based on the maximum rectangles bottom-left (Maxrects-BL) is designed to determine the placement position of pieces on the plate and calculate model rewards and packing parameters. Finally, instances are used to analyze the optimization effect of the algorithm. The experimental results show that the proposed algorithm can produce three better and five comparable results compared with some classical heuristic algorithms. In addition, the calculation time of the proposed algorithm is less than 1 second in all test instances, which shows a good generalization, solution efficiency, and practical application potential.

Microservice combination optimisation based on improved gray wolf algorithm

Microservices architecture is a new paradigm for application development. The problem of optimising the performance of microservice architectures from a non-functional perspective is a typical Nondeterministic Polynomial (NP) problem. Therefore, aiming to quantify the non-functional requirements of computing microservice systems, while solving the problem of latency in computing the best combination of services with the maximum QoS objective function value, this paper proposes a microservice combination approach based on the QoS model and a CGWO algorithm for optimisation computation for this model. The experimental results verify that the error rate of the method is only 0.528% on the non-functional combination optimisation problem, and the computational efficiency of the algorithm increases by 97.29% when the complexity of the problem search space increases, while CGWO improves 65.97% and 81.25% respectively in the accuracy of optimisation compared to the prototype of the algorithm (GWO), and has a stable optimisation performance, aspect. It proves that the research in this paper has a high advantage in automatically searching for the best QoS for the microservice combination problem.

Diversity-driven automated web API recommendation based on implicit requirements

With the rapid growth of the web API sharing community, Mashup development has become a popular way for software developers to quickly create Mashup applications. Mashup development allows developers to quickly implement the functionality they want by combining a chosen set of web APIs, which greatly improves development efficiency. However, the existence of numerous web API candidates make it difficult to select the appropriate API quickly. Most existing automated web API recommendation approaches focus on the developer’s description of requirements but typically overestimate the developer’s ability to fully describe their application’s requirements, thus ignoring their implicit and diversified functional requirements. In this paper, we propose a novel automated approach called DI-RAR to retrieve and recommend web API groups for Mashup creation. Specifically, we formulate an automated web API recommendation task as a nondeterministic polynomial problem. First, a self-attention model assigns weights to query to distinguish the core and non-core requirements. then Dynamic planning retrieval generates steiner trees to retrieve API groups and uncovers strongly related implicit requirements to enrich the mashup’s functions. Finally we apply simhash technique to filter similar results and finally provide diverse API groups. experiments based on a real-world dataset are performed to demonstrate the feasibility and efficiency of DI-RAR.

On the Polyhedral Homotopy Method for Solving Generalized Nash Equilibrium Problems of Polynomials

The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player’s objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method.

The electrostatic equilibrium problem for classical discrete orthogonal polynomials

We present two types of generic difference characterizations of zeros of the classical discrete orthogonal polynomials associated with a discrete weight function satisfying a difference equation of the Pearson type. These characterizations use either the total forward or modified partial central difference operators of discrete discriminants, which approximate the total electrostatic energy of the system of n n unit charges moving freely on an interval in presence of external potential given by the iterated discrete weight.

Non-Deterministic and Polynomial Time Problem Simulator

The Non-Deterministic and Polynomial Time Problem is a problem in combinatorial optimization. Finding the quickest route for an object to travel through a list of cities and return to the starting city is the goal of this problem. Cities are listed, along with the distance between each pair. It belongs to the category of computer problems known as NP-complete problems, for which no effective algorithmic solution has yet been discovered; at this time, there is no polynomial solution. In order to discover a near-optimal solution as quickly as possible, we attempted to tackle this extremely challenging problem in this study utilizing a variety of heuristics, including Simulated Annealing and Genetic Algorithm. Using these sophisticated heuristic techniques, we attempt to depart from the local optimum.

Delegating signing rights in a multivariate proxy signature scheme

In the context of digital signatures, the proxy signature holds a significant role of enabling an original signer to delegate its signing ability to another party (i.e., proxy signer). It has significant practical applications. Particularly it is useful in distributed systems, where delegation of authentication rights is quite common. For example, key sharing protocol, grid computing, and mobile communications. Currently, a large portion of existing proxy signature schemes are based on the hardness of problems like integer factoring, discrete logarithms, and/or elliptic curve discrete logarithms. However, with the rising of quantum computers, the problem of prime factorization and discrete logarithm will be solvable in polynomial-time, due to Shor's algorithm, which dilutes the security features of existing ElGamal, RSA, ECC, and the proxy signature schemes based on these problems. As a consequence, construction of secure and efficient post-quantum proxy signature becomes necessary. In this work, we develop a post-quantum proxy signature scheme Mult-proxy, relying on multivariate public key cryptography (MPKC), which is one of the most promising candidates of post-quantum cryptography. We employ a 5-pass identification protocol to design our proxy signature scheme. Our work attains the usual proxy criterion and a one-more-unforgeability criterion under the hardness of the Multivariate Quadratic polynomial (MQ) problem. It produces optimal size proxy signatures and optimal size proxy shares in the field of MPKC.

Multi-objective optimization of simultaneous buffer and service rate allocation in manufacturing systems based on a data-driven hybrid approach

The challenge presented by simultaneous buffer and service rate allocation in manufacturing systems represents a difficult non-deterministic polynomial problem. Previous studies solved this problem by iteratively utilizing a generative method and an evaluative method. However, it typically takes a long computation time for the evaluative method to achieve high evaluation accuracy, while the satisfactory solution quality realized by the generative method requires a certain number of iterations. In this study, a data-driven hybrid approach is developed by integrating a tabu search–non-dominated sorting genetic algorithm II with a whale optimization algorithm–gradient boosting regression tree to maximize the throughput and minimize the average buffer level of a manufacturing system subject to a total buffer capacity and total service rate. The former algorithm effectively searches for candidate simultaneous allocation solutions by integrating global and local search strategies. The prediction models built by the latter algorithm efficiently evaluate the candidate solutions. Numerical examples demonstrate the efficacy of the proposed approach. The proposed approach improves the solution efficiency of simultaneous allocation, contributing to dynamic production resource reconfiguration of manufacturing systems.

A Self-Adjusting Search Domain Method-Based Genetic Algorithm for Solving Flexible Job Shop Scheduling Problem.

As a nondeterministic polynomial (NP) problem, the flexible job shop scheduling problem (FJSP) is a difficult problem to be solved in terms of finding an acceptable solution. In last decades, genetic algorithm (GA) displays very promising performance in the field. In this article, a hybrid algorithm combining global and local search with reinitialization (GLRe)-based GA is proposed to minimize makespan for FJSP. The solution of FJSP is conveniently represented by a double-layer chromosome representation method, which is convenient for subsequent genetic operations, that is, sorting of operations and selection of machines. Two strategies of choosing the job with the most remaining operations (CRO) and 6-dimensional variable determined search position (6D-VSP) are proposed as two components for GA, which are applied to generate a population with superior quality and reduce the global search space during the initialization stage. At the same time, in order to prevent the loss of diversity during evolution, a reinitialization strategy is introduced in the later stage of evolution to adaptively adjust the search domain of the problem. Finally, two sets of benchmark data are tested. The experimental results demonstrate the accuracy and effectiveness of the GLRe proposed in this article for solving FJSP.

Energy efficient MIMO–NOMA aided IoT network in B5G communications

To accelerate future intelligent wireless systems , we designed an energy-efficient Massive multiple-input-multiple-output (MIMO)- non-orthogonal multiple access (NOMA) aided internet of things (IoT) network in this paper to support the massive number of distributed users and IoT devices with seamless data transfer and maintain connectivity between them. Massive MIMO has been identified as a suitable technology to implement the energy efficient IoT network in beyond 5G (B5G) communications due to its distinct characteristics with large number of antennas. However, to provide fast data transfer and maintain hyper connectivity between the IoT devices in B5G communications will bring the challenge of energy deficiency. Hence, we considered a massive MIMO–NOMA aided IoT network considering imperfect channel state information and practical power consumption at the transmitter. The far users of the base stations are selected to investigate the power consumption and quality of service. Then, calculate the power consumption which is non-convex function and non-deterministic polynomial problem. To solve the above problem, fractional programming properties are applied which converted polynomial problem into the difference of convex function . And then we employed the successive convex approximation technique to represent the non-convex to convex function. Effective iterative based branch and the reduced bound process are utilized to solve the problem. Numerical results observe that our implemented approach surpasses previous standard algorithms on the basis of convergence, energy-efficiency and user fairness.

An Urban Bus Network Generation Algorithm Based on Particle Swarm Optimization and Force Field Properties

Due to continuous urban sprawl, large-scale bus network design has become a major challenge in urban transport planning. The continuous increase in urban population and scale makes the factors considered in the urban route network design increasingly complex. Contemporary public transportation network design problems are based more on efficiency goals such as the accessibility and comfort of the transportation network, which increases the difficulty of analyzing the problem. Bus network design is not only an NP-hard (nondeterministic polynomial) problem but also a multivariable and multiobjective problem. This paper focuses on the bivariate and multiobjective bus network design problem of route generation and station selection. This paper proposes an algorithm called the Pseudo Force Field. By combining the idea of Particle Swarm Optimization (PSO) and the properties of the force field, a feasible route generation scheme is provided for the design of the bus network. The algorithm does not need to determine the end station and has a high degree of completion of the demand. This solves the problem of the selection of terminal stations in large-scale road network design. On this basis, the article combines Genetic Algorithm (GA) and Pareto frontier to provide a new route optimization algorithm and proves the effectiveness of the algorithm. The model has achieved theoretical results in the design of the bus route network in the megacity of Shenzhen, China.

Recent progress in the security evaluation of multivariate public‐key cryptography

AbstractMultivariate public‐key cryptography (MPKC) is considered a leading candidate for post‐quantum cryptography (PQC). It is based on the hardness of the multivariate quadratic polynomial (MQ) problem, which is a problem of finding a solution to a system of quadratic equations over a finite field. In this paper, we survey some recent progress in the security analysis of MPKC. Among various existing multivariate schemes, the most important one is the Rainbow signature scheme proposed by Ding et al. in 2005, which was later selected as a finalist in the third round of the PQC standardization project by the National Institute of Standards and Technology. Under the circumstances, some recent research studies in MPKC have focussed on the security analysis of the Rainbow scheme. In this paper, the authors first explain efficient algorithms for solving the MQ problem and the research methodology for estimating their complexity in MPKC. Then, the authors survey some recent results related to the security analysis of the Rainbow scheme. In particular, the authors provide a detailed description of the complexity analysis for solving the bi‐graded polynomial systems studied independently by Nakamura et al. and Smith‐Tone et al., and then expound the rectangular MinRank attack against Rainbow proposed by Beullens.

Power constraint waveform design based on ADMM algorithm for MIMO radar transmit beampattern

To address the problems of high sidelobe and large fluctuation amplitude of the mainside on matching the radar beampattern of multiple-input multiple-output (MIMO) radar, we proposed a transmission beampattern design method based on alternating direction method of multipliers (ADMM) under power limitation and peak-to-average ratio (PAR) constraint. The method used the Least-Most-Rule for the transmitting beamform design, i.e., the ratio of the maximum value of sidelobe to the minimum value of mainlobe as objective function. Meanwhile, the sum of power over the mainside and sidelobe is constrained to match the desired pattern. Then, the log-sum-exp function and Taylor's formula are used to approximate the optimization problem to produce a fourth-order polynomial on the beamform vector; finally, we introduce a dual variable and equation constraints and apply ADMM method to convert the fourth-order polynomial problem into the second-order polynomial problem for solution. In addition, the computational complexity of the proposed method is also analyzed. Two sets of experiments verify the effectiveness of the method in the simulation part.

Multi-scale thermal radiation effects correction via a fast surface fitting with Chebyshev polynomials.

In an uncooled infrared imaging system, thermal radiation effects are caused by the heat source from the target or the detection window, which affects the ability of target detection, tracking, and recognition seriously. To address this problem, a multi-scale correction method via a fast surface fitting with Chebyshev polynomials is proposed. A high-precision Chebyshev polynomial surface fitting is introduced into thermal radiation bias field estimation for the first time, to the best of our knowledge. The surface fitting in the gradient domain is added to the thermal radiation effects correction model as a regularization term, which overcomes the ill-posed matrix problem of high-order bivariate polynomials surface fitting, and achieves higher accuracy under the same order. Additionally, a multi-scale iterative strategy and vector representation are adopted to speed up the iterative optimization and surface fitting, respectively. Vector representation greatly reduces the number of basis function calls and achieves fast surface fitting. In addition, split Bregman optimization is used to solve the minimization problem of the correction model, which decomposes the multivariable optimization problem into multiple univariate optimization sub-problems. The experimental results of simulated and real degraded images demonstrate that our proposed method performs favorably against the state of the art in thermal radiation effects correction.