A study on the impact of heuristic initialization functions in a genetic algorithm solving the N-queens problem

The N-Queen problem is a classical combinatorial optimization and constraint satisfaction problem that involves placing N queens on an N × N chessboard such that no two queens threaten each other. Due to its exponential search space and increasing computational complexity with larger board sizes, the problem serves as an important benchmark for evaluating the efficiency of heuristic and metaheuristic optimization techniques. This study presents a comprehensive performance analysis of the Simulated Annealing (SA) optimization approach for solving the N-Queen problem across varying board dimensions. The proposed methodology models the N-Queen problem as an optimization task by defining an objective function that minimizes the number of conflicting queen pairs. Simulated Annealing is employed to iteratively explore the solution space, allowing probabilistic acceptance of suboptimal solutions to escape local minima. Key parameters, including initial temperature, cooling schedule, and termination criteria, are systematically tuned to evaluate their impact on convergence speed and solution quality. Experimental results demonstrate that the SA-based approach effectively finds valid solutions for large values of N with high success rates and reduced computational effort compared to exhaustive and deterministic methods. Performance metrics such as solution accuracy, convergence time, number of iterations, and conflict reduction rate are analyzed and discussed. The results indicate that Simulated Annealing offers a robust and scalable solution for the N-Queen problem, particularly for higher-dimensional instances where traditional methods become impractical. Overall, this study highlights the suitability of Simulated Annealing as an efficient metaheuristic for solving complex combinatorial optimization problems and provides insights into its performance characteristics when applied to the N-Queen problem.

N-Queen problem is a problem which a N-Queen pawn in is place chess with n x n size. N-Queen pawn is a put in such away in chess board with under condition that the queen pawns do not attack each other. The attacking movement of N-Queen problem is similar to the way of the queen pawn attacking in chess. Commonly the queen pawn moves horizontally to left and right, forward and backward vertically and also diagonally, so there are no queen pawns in a line of horizontal, vertical and diagonal.Heuristical searching is one of the method which can be used to solve the game of N-Queen problem selectively, by giving solution of the shortest time channel efficiently in order to able the user to solve this game well, fast and relevantly. Some algorithms that use heuristic is Iterative Deepening algorithm A* (IDA*) and Branch and Bound (B&B) algorithm. The used heuristic function is by seeing the numbers of boxes which are empty and the number of queen which is not be put in board yet.The aim of making this final project to implement the solving of N-Queen problem using heuristic searching (B&B and IDA*). From this implementation could be seen that IDA* and B&B algorithm is able to give channel in solving N- Queen problem. After a repetition of test by using 19 sheet of data, it is shown the comparison of result between IDA* algorithm and B&B algorithm which IDA* algorithm result the shorter channel in solving N-Queen problem based on the node 61%, and time 41% which better than B&B algorithm.

Traditional Hopfield networking has been widely used to solve combinatorial optimization problems. However, high order Hopfiled networks, as an expansion of traditional Hopfield networks, are seldom used to solve combinatorial optimization problems. In theory, compared with low order networks, high order networks have better properties, such as stronger approximations and faster convergence rates. In this chapter, the authors focus on how to use high order networks to model combinatorial optimization problems. Firstly, the high order discrete Hopfield Network is introduced, then the authors discuss how to find the high order inputs of a neuron. Finally, the construction method of energy function and the neural computing algorithm are presented. In this chapter, the N queens problem and the crossbar switch problem, which are NP-complete problems, are used as examples to illustrate how to model practical problems using high order neural networks. The authors also discuss the performance of high order networks for modeling the two combinatorial optimization problems.

The Differential Evolution (DE) algorithm was initially proposed for continuous numerical optimization, but it has been applied with success in many combinatorial optimization problems, particularly permutation-based integer combinatorial problems. In this paper, a new and general approach for combinatorial optimization is proposed using the Differential Evolution algorithm. The proposed approach aims at preserving its interesting search mechanism for discrete domains, by defining the difference between two candidate solutions as a differential list of movements in the search space. Thus, a more meaningful and general differential mutation operator for the context of combinatorial optimization problems can be produced. We discuss three alternatives for using the differential list of movements within the differential mutation operation. We present results on instances of the Traveling Salesman Problem (TSP) and the N-Queen Problem (NQP) to illustrate the adequacy of the proposed approach for combinatorial optimization.

Dual-mode dynamics neural network for combinatorial optimization

The Traveling Salesman Problem (TSP) is a benchmark NP-hard problem in combinatorial optimization[14]. This paper presents a comparative study of three Genetic Algorithm (GA) variants: (i) Serial GA, (ii) Parallel-Isolated GA (independent subpopulations without migration), and (iii) Parallel-Migrating GA (island model with periodic elite exchange). Experiments on 50-, 100-, and 200-city Euclidean datasets demonstrate that the migrating island GA achieves superior scalability, faster convergence, and improved solution consistency. While the serial GA performs adequately on smaller datasets, its performance degrades significantly at scale, whereas the migrating parallel GA maintains high-quality solutions and reduced runtime. We expand the classical distance-only objective to a practical multi-objective (distance, traffic, safety, cost) weighted fitness and implement the solver with parallel islands using Python multiprocessing (code and datasets available in the project repository).

Niching ethods are useful extension of evolutionary algorithm that can permit population to search many peaks in parallel in multimodal domains. Traditional niching methods that can effectively be used to solve function optimization problems (FOPs) are not suitable for solving combinatorial optimization problems (COPs). We propose a new genetic algorithm (GA) for forming niches that can be efficiently applied to COPs. For this, we propose four requirements that are needed for an ideal method of forming niches for COPs. We then discuss a niching method we designed that satisfies these requirements. The proposed GA was applied to job shop scheduling problems (JSP) to demonstrate its effectiveness.

The performance and efficiency of A* search algorithm heavily depends on the quality of the heuristic function. Therefore, designing an optimal heuristic function becomes the primary goal of developing a search algorithm for specific domains in artificial intelligence. However, it is difficult to design a well-constructed heuristic function without careful consideration and trial-and-error, especially for complex pathfinding problems. The complexity of a heuristic function increases and becomes unmanageable to design when an increasing number of parameters are involved. Existing approaches often avoid complex heuristic function design: they either trade-off the accuracy for faster computation or taking advantage of the parallelism for better performance. The objective of this paper is to reduce the difficulty of complex heuristic function design for A* search algorithm. We aim to design an algorithm that can be automatically optimized to achieve rapid search with high accuracy and low computational cost. In this paper, we present a novel design and optimization method for a Multi-Weighted-Heuristics function (MWH) named Evolutionary Heuristic A* search (EHA*) to: (1) minimize the effort on heuristic function design via Genetic Algorithm (GA), (2) optimize the performance of A* search and its variants including but not limited to WA* and MHA*, and (3) guarantee the completeness and optimality. EHA* algorithm enables high performance searches and significantly simplifies the processing of heuristic design. We apply EHA* to multiple grid-based pathfinding benchmarks to evaluate the performance. Our experiment result shows that EHA* (1) is capable of choosing an accurate heuristic function that provides an optimal solution, (2) can identify and eliminate inefficient heuristics, (3) is able to automatically design multi-heuristics function, and (4) minimizes both the time and space complexity.

Editorial

In this paper, we propose a saturation computation method of neural networks for efficiently solving combinatorial optimization problems. In this computation method, once the neuron is in excitatory state, then its input potential is considered to be in positive saturation where the input potential can only be reduced but cannot be increased, and once the neuron is in inhibitory state, then its input potential is considered to be in negative saturation where the input potential can only be increased but cannot be reduced. The proposed method is applied to N-Queens problem. The performance is evaluated through simulations where the results show that the saturation method improves the searching capability of neural networks and shortens the computation time. Particularly, the simulation results show that the performance of the proposed method surpasses the exiting methods for N-queens problem in synchronous parallel model.

A new search method for combinatorial optimization problem inspired by the spin glass system

This paper presents a new multi-objective optimization method, which is inspired from the idea of non-dominated sorting genetic algorithm (NSGA) and genetic quantum algorithm (GQA). The GQA has been tested on well known test beds in single objective optimization and compared with the genetic algorithm (GA) in the lead author’s previous work [22]. This paper aims to apply the idea of GQA to multi-objective optimization (MOO). The developed method is called non-dominated sorting genetic quantum algorithm (NSGQA). The developed method is tested with benchmark problems collected from literature, which have characteristics representing various aspects of a MOO problem. Test results show that NSGQA has better performance on most benchmark problems than currently popular MOO methods such as the NSGA. The integration of GQA with MOO, and the systematic comparison with other MOO methods on benchmark problems, should be of general interest to researchers on MOO and to practitioners using MOO methods in design.

In the nqueensnomut, nqueensconstmut, and nqueenssigmoid tabs, the end heuristic (number of queens attacking each other on the board, with 0 m) found by that genetic algorithm is displayed along with the time the algorithm took to converge to that solution (in generations). In the stats tabs, t-tests are run to compare the accuracies and convergence generations among the different mutation rate genetic algorithms.<br>Key: mut rate: mutation rate, converg: convergence (generations), end h (end heuristic) nomut: no mutation, constmut, constant mutation.

Genetic algorithm is heuristic searching algorithm which based on nature selection of mechanism and nature genetic. The basic concept that inspires the genetic algorithm is that evolution theory. In the process of searching the best solution on classic genetic algorithm often occurs optimum local. Optimum local is a common problem which is often occurs in genetic algorithm, and one of the reasons is that because of the population diversity. If population diversity is too low is led to optimum local, and if it is too high caused more times to look for the best solution. Mutation operator plays an important role in the process of genetic algorithm to manage that population diversity and an important element of mutation operator is mutation rate. On classic genetic algorithm that mutation rate is set in the beginning while the process of genetic algorithm depends on how many generations are. Therefore is needed to control mutation rate in generation. Controlling mutation operator, especially mutation rate based on Fuzzy Logic Controller (FLC) to manage population diversity, not too high or too low, in order to get optimal result. Evaluation is done 10 times execution by comparing the performance of standard genetic algorithm (GA), IAG and genetic algorithm based on Fuzzy Logic Controller (FLC) and experimental results show that there is an improvement on genetic algorithm performance based on FLC. Keywords: Genetic Algorithm, Fuzzy Logic Controller (FLC), Mutation Rate

Researchers have utilised nature-inspired metaheuristic algorithms to find solutions for complex combinatorial optimisation problems and Non-deterministic polynomial (NP) problems. The N-queens problem is categorised as an NP-Hard problem as it becomes insurmountable for large N. This paper proposed a solution for the N-queens problem based on the Apiary Organisational-Based Optimisation Algorithm (AOOA). AOOA is a nature-inspired metaheuristic optimisation algorithm for NP-Hard problems comprised of multiple beehives inside the apiary, each with its population. In contrast to the backtracking method, AOOA employs a fitness function and randomisation techniques through seven stages to approach the optimal distributions of queens so they don’t attack each other. Experiments were carried out for different values of N ranging from 4 to 10. All solutions were found for Ns (4–8), which comprised 100% of the total solutions. At the same time, 97.7272% of total solutions were found for N = 9 and 98.6187% for N = 10. Moreover, several metaheuristic algorithms have been implemented to solve the N-queens problem, and the average number of 100 iterations has been compared. The results reflect the superiority of AOOA over the competing algorithms in finding the possible number of solutions in earlier iterations and consequently reducing the computational cost of further iterations.