NPC Problem Research Articles

Application of ant algorithm in solving TSP problems

Ant colony system algorithms, as a new class of global search algorithms, can solve the TSP (Traveling Salesman Problem). At the same time, the TSP problem is a classical NP-C problem, the solution of TSP problem involves many fields, such as network routing, vehicle routing, logistics and transportation, so it is very important to solve TSP problem effectively. In this experiment, the ant colony optimization algorithm is simulated and experimented using Matlab software on bayg29 dataset and the algorithm strategy is improved by controlling different parameters. Our main goal is to improve the performance of the algorithm through meticulous parameter fine-tuning and policy improvement. We explored a range of parameter adjustments, including variables such as pheromone evaporation rate, ant colony size, etc. By undertaking this comprehensive investigation, we aspire to push the boundaries of ant colony optimization, unlocking its potential to deliver highly efficient solutions for complex TSP instances. This research holds promise for revolutionizing the efficiency and cost-effectiveness of solutions across a wide range of practical applications.

A hybrid genetic algorithm with variable neighborhood search for batch dispersion problem to improve traceability

Batch dispersion problem (BDP) restricts batch traceability in large-scale discrete production and negatively impacts batch recall costs. However, previous research has ignored the complexity of the BDP in their analyses. This paper investigates the BDP under the composed bill of materials (BOM) and develops a mathematical model for the BDP with the goal of minimizing the total batch dispersion by utilizing the batch dispersion as a measure of the degree of dispersed usage of part batches. BDP-GAVNS, a hybrid genetic algorithm with variable neighborhood search, is devised for the BDP based on the demonstration that the BDP is an NPC problem. In BDP-GAVNS, memory banks were introduced to increase the diversity of individuals performing crossover operations. Additionally, the encoding method and infeasible solution repair program are designed according to the characteristics of BDP. Numerical experiments validate the viability and effectiveness of BDP-GAVNS in solving BDP. They demonstrate that (1) the optimal combination occurs when the ratio of individuals produced by the three types of population initialization methods, namely global selection (GS), local selection (LS), and random selection (RS), to the population takes values of 0.30, 0.10, and 0.60, respectively; (2) The memory bank enriches the source of individuals required for crossover operations and improves the performance of crossover operations; and (3) The BDP-GAVNS is more effective than the other five heuristic algorithms including genetic algorithms in seeking the optimal solution of BDP.

High capacity NP-Complete problems solver based on dual-comb asynchronous optical sampling

Photonic schemes to solve the Nondeterministic Polynomial-Complete (NP–C) problems could offer an interesting alternative to such computationally intensive tasks. Previous studies had shown that they can be solved by measuring the propagation delays through an optical oracle that corresponds to the NP-C problem. However, to tackle a problem with a large instance, the previous schemes face significant challenges to achieve a large measurement range, fast speed and high resolution, simultaneously. In this work, we propose an optical solver based on asynchronous optical sampling driven by a simple fiber dual-comb laser. It is experimentally demonstrated that it can realize picosecond resolution and tens of nanosecond range with a fiber oracle network consisting of 15 couplers and sections of fibers that represents a typical Traveling Salesman problem. With a width of the pulse interferogram of less than 1.5 ps, i.e. an optical path resolution of ∼0.3 mm in fiber, the output from an oracle with 58 likely paths is measured accurately, from which the Hamilton path is easily identified in ∼0.7 ms. It is expected that further optimizing the parameters of the laser as well as the oracle would enable the solving larger scale NP-C problems.

Minimum Adversarial Examples.

Deep neural networks in the area of information security are facing a severe threat from adversarial examples (AEs). Existing methods of AE generation use two optimization models: (1) taking the successful attack as the objective function and limiting perturbations as the constraint; (2) taking the minimum of adversarial perturbations as the target and the successful attack as the constraint. These all involve two fundamental problems of AEs: the minimum boundary of constructing the AEs and whether that boundary is reachable. The reachability means whether the AEs of successful attack models exist equal to that boundary. Previous optimization models have no complete answer to the problems. Therefore, in this paper, for the first problem, we propose the definition of the minimum AEs and give the theoretical lower bound of the amplitude of the minimum AEs. For the second problem, we prove that solving the generation of the minimum AEs is an NPC problem, and then based on its computational inaccessibility, we establish a new third optimization model. This model is general and can adapt to any constraint. To verify the model, we devise two specific methods for generating controllable AEs under the widely used distance evaluation standard of adversarial perturbations, namely constraint and constraint (structural similarity). This model limits the amplitude of the AEs, reduces the solution space’s search cost, and is further improved in efficiency. In theory, those AEs generated by the new model which are closer to the actual minimum adversarial boundary overcome the blindness of the adversarial amplitude setting of the existing methods and further improve the attack success rate. In addition, this model can generate accurate AEs with controllable amplitude under different constraints, which is suitable for different application scenarios. In addition, through extensive experiments, they demonstrate a better attack ability under the same constraints as other baseline attacks. For all the datasets we test in the experiment, compared with other baseline methods, the attack success rate of our method is improved by approximately 10%.

Generative trapdoors for public key cryptography based on automatic entropy optimization

Trapdoor is a key component of public key cryptography design which is the essential security foundation of modern cryptography. Normally, the traditional way in designing a trapdoor is to identify a computationally hard problem, such as the NPC problems. So the trapdoor in a public key encryption mechanism turns out to be a type of limited resource. In this paper, we generalize the methodology of adversarial learning model in artificial intelligence and introduce a novel way to conveniently obtain sub-optimal and computationally hard trapdoors based on the automatic information theoretic search technique. The basic routine is constructing a generative architecture to search and discover a probabilistic reversible generator which can correctly encoding and decoding any input messages. The architecture includes a trapdoor generator built on a variational autoencoder (VAE) responsible for searching the appropriate trapdoors satisfying a maximum of entropy, a random message generator yielding random noise, and a dynamic classifier taking the results of the two generator. The evaluation of our construction shows the architecture satisfying basic indistinguishability of outputs under chosen-plaintext attack model (CPA) and high efficiency in generating cheap trapdoors.

A Note of Independent Number and Domination Number of Qn,k,m-Graph

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

广义<i>b</i>-基超立方体网络的控制数

控制数是刻画容错网络中资源共享可靠性的一个参数。确定网络的控制数是NPC问题。Lakshmivarahan, Dhall提出了著名的互连网络—广义b-基超立方体网络。该文给出了当b=3,n=2,3,4 时广义b-基超立方体网络控制数的具体值，以及当5≤n≤8 时控制数的界；进一步提出了两个问题和与该问题相对应的两个猜想。 Domination number is a parameter to describe the reliability of resource sharing in a fault-tole- rant network. Determining the domination numbers of a graph is a NPC problem. Generalized base-b hypercube has been put forward by Lakshmivardhan and Dhall, which is a famous inter-connection network. In this paper, we study the exact values of the domination numbers of generalized base-b hypercube for b=3,n=2,3,4 and the bounds of the domination numbers for 5≤n≤8. Furthermore, two problems and two conjectures with the problems are proposed.

The application of Coarse-Grained Parallel Genetic Algorithm with Hadoop in University Intelligent Course-Timetabling System

The university course-timetabling problem is a NP-C problem. The traditional method of arranging course is inefficient, causes a high conflict rate of teacher resource or classroom resource, and is poor satisfaction in students. So it does not meet the requirements of modern university educational administration management. However, parallel genetic algorithm (PGA) not only have the advantages of the traditional genetic algorithm(GA), but also take full advantage of the computing power of parallel computing. It can improve the quality and speed of solving effectively, and have a broad application prospect in solving the problem of university course-timetabling problem. In this paper, based on the cloud computing platform of Hadoop, an improved method of fusing coarse-grained parallel genetic algorithm (CGPGA) and Map/Reduce programming model is deeply researched, and which is used to solve the problem of university intelligent courses arrangement. The simulation experiment results show that, compared with the traditional genetic algorithm, the coarse-grained parallel genetic algorithm not only improves the efficiency of the course arrangement and the success rate of the course, but also reduces the conflict rate of the course. At the same time, this research makes full use of the high parallelism of Map/Reduce to improve the efficiency of the algorithm, and also solves the problem of university scheduling problem more effectively.

A Network Coding Based Energy Efficient Data Backup in Survivability-Heterogeneous Sensor Networks

Sensor nodes deployed outdoors are subject to environmental detriments and often need to cache data for an extended period of time. This paper introduces sensor nodes which are robust to environmental damages, and proposes to utilize Network Coding to back up data in the robust sensors for future data retrieval in an energy efficient way. Our goal is to help regular sensors select robust sensors to back up their data with low energy consumption, such that when needed, all the data can be retrieved by querying only a subset of robust sensors. We formally formulate this backup problem, theoretically prove its NP-Completeness, discover two novel theoretical guidelines for problem solving, and propose two algorithms accordingly to tackle this NP-C problem. The guidelines are based on random linear network coding and provide lower bounds of the number of robust sensors that each regular sensor should choose for data backup, such that the required fault tolerance is provided. A centralized algorithm and a distributed algorithm are developed based on the guidelines such that regular sensors can back up their data efficiently. Both analysis and simulation show our algorithms are effective in achieving fault tolerance, low energy consumption, and high retrieval efficiency.

The Improved McEliece PKC and its Security Analysis

Because the decoding problem of general linear codes is an NPC problem, at the same time Goppa codes have fast decoding algorithm, McEliece first proposed a public-key cryptosystem based on error-correcting codes. This paper constructs the largest distance codes based on metric, then improves the McEliece cryptosystem based on the largest distance codes. We've proved that the cryptosystem is able to resist the attack of getting the private key by a known public key. Finally, through the analysis of existing viable attack method, we proved the feasibility and security of the improved McEliece cryptosystem.

Low energy WSN data aggregation algorithm based on improved aggregation tree model

To keep the data synchronisation and fidelity of each node in wireless sensor network (WSN), we propose a router-combined data fusion algorithm based on broadcast tree. The approximation algorithm is used to transform the optimum aggregation tree into NP-C problem to solve the smallest Steiner tree. Then the cluster-based optimising directed diffusion routing is adopted to construct aggregation tree for data fusion. At the fusion phase, we introduce consensus factor to support degree matrix, to replace original constant factor and provide more weight effect for reliable sensor. The improved support degree matrix will measure the support degree of each sensor to form a novel weighted fusion algorithm. The analyses on simulation show that our fusion algorithm can make data fusion effectively, which also has higher performance in delay, energy and accuracy.

Application of an Improved Genetic Algorithm with the Search Space Compression in TSP

The Traveling Salesman Problem is a combinatorial optimization problem, the problem has been shown to belong to the NPC problem, the possible solution of Traveling Salesman Problem and the scale of the cities have the exponential relation, so the more bigger of the scale. In this paper, improve the search process of the genetic algorithm by introducing the idea is to compress the search space. The simulation results show that for solving the TSP, the algorithm can quickly obtain multiple high-quality solutions. It can reduce the blindness of random search and accelerate convergence of the algorithm.

Genetic algorithm for pareto optimum-based route selection

A quality of service (QoS) or constraint-based routing selection needs to find a path subject to multiple constraints through a network. The problem of finding such a path is known as the multi-constrained path (MCP) problem, and has been proven to be NP-complete that cannot be exactly solved in a polynomial time. The NPC problem is converted into a multiobjective optimization problem with constraints to be solved with a genetic algorithm. Based on the Pareto optimum, a constrained routing computation method is proposed to generate a set of nondominated optimal routes with the genetic algorithm mechanism. The convergence and time complexity of the novel algorithm is analyzed. Experimental results show that multiobjective evolution is highly responsive and competent for the Pareto optimum-based route selection. When this method is applied to a MPLS and metropolitan-area network, it will be capable of optimizing the transmission performance.

Several key problems in automatic layout design of spacecraft modules *

Abstract Computer-aided layout design of spacecraft modules, such as satellite modules and manned spaceship modules, is of great significance and egregious complexity. It is known as a combinatorial optimization and NPC problem in mathematics a conceptual design and complex system in engineering. The main difficulties include representation and formulation of the problem in mathematics and the solution strategy and pragmatic approaches in engineering practice. After a brief survey of the state-of-the-art in relevant fields, this paper summarizes the research work of the authors' group on automatic layout design of spacecraft modules in the last 15 years, mainly focusing on 5 key problems. They are modeling and problem-solving algorithms, interference calculation, theory and applications of layout topological pattern, decision-making in layout design, and their pragmatic approaches in engineering practice.

A Note on Testing the Resultant

Shub (in "From Topology to Computation: Proceedings of the Smalefest." (M. Hirsch, J. Marsden, and M. Shub, Eds.), pp. 281-301, Springer-Verlag, New York/Berlin, 1993) proposes to attack the complex PC versus the NPC problem by focussing on lower bounds on testing the resultant of quadratic forms for zero. Taking up this question we show in the present paper a lower bound of order n3 for the resultant.

Computational complexity to verify the unstability of effectivity function

Ann-person game is considered where each player has a preference order over a finite setA of possible alternatives and a rule for social choice is given in the form of an effectivity functionE. The effectivity function is called stable if for any combination of individual preference orders there exists a subset ofA called a core such that any alternative in the core cannot be ‘dominated’ by such individual preferences. It has been shown by Keiding (1985) that the effectivity functionE is stable if and only ifE does not generate any ‘Cycle’. This paper is concerned with the computational complexity of the problem (CYCLE) for determining whether or not a given effectivity function has a Cycle, We show that a familiar NPC problem SATISFIABILITY can be transformed into CYCLE through a polynomial time procedure. This, combined with the fact that CYCLE is an NP problem, implies NP-completeness of CYCLE, and therefore that of verifying the unstability of the effectivity function, thereby formally proving a previously unanswered conjecture.