Geometric particle swarm optimization for the sudoku puzzle

Geometric particle swarm optimization (GPSO) is a recently introduced generalization of traditional particle swarm optimization (PSO) that applies to all combinatorial spaces. The aim of this paper is to demonstrate the applicability of GPSO to non-trivial combinatorial spaces. The Sudoku puzzle is a perfect candidate to test new algorithmic ideas because it is entertaining and instructive as well as a nontrivial constrained combinatorial problem. We apply GPSO to solve the sudoku puzzle

Stochastic Optimization Approaches for Solving Sudoku

In this paper the Sudoku problem is solved using stochastic search techniques and these are: Cultural Genetic Algorithm (CGA), Repulsive Particle Swarm Optimization (RPSO), Quantum Simulated Annealing (QSA) and the Hybrid method that combines Genetic Algorithm with Simulated Annealing (HGASA). The results obtained show that the CGA, QSA and HGASA are able to solve the Sudoku puzzle with CGA finding a solution in 28 seconds, while QSA finding a solution in 65 seconds and HGASA in 1.447 seconds. This is mainly because HGASA combines the parallel searching of GA with the flexibility of SA. The RPSO was found to be unable to solve the puzzle.Comment: 13 page

Genetic Transfer or Population Diversification? Deciphering the Secret Ingredients of Evolutionary Multitask Optimization

Evolutionary multitasking has recently emerged as a novel paradigm that enables the similarities and/or latent complementarities (if present) between distinct optimization tasks to be exploited in an autonomous manner simply by solving them together with a unified solution representation scheme. An important matter underpinning future algorithmic advancements is to develop a better understanding of the driving force behind successful multitask problem-solving. In this regard, two (seemingly disparate) ideas have been put forward, namely, (a) implicit genetic transfer as the key ingredient facilitating the exchange of high-quality genetic material across tasks, and (b) population diversification resulting in effective global search of the unified search space encompassing all tasks. In this paper, we present some empirical results that provide a clearer picture of the relationship between the two aforementioned propositions. For the numerical experiments we make use of Sudoku puzzles as case studies, mainly because of their feature that outwardly unlike puzzle statements can often have nearly identical final solutions. The experiments reveal that while on many occasions genetic transfer and population diversity may be viewed as two sides of the same coin, the wider implication of genetic transfer, as shall be shown herein, captures the true essence of evolutionary multitasking to the fullest.Comment: 7 pages, 6 figure

Diversification and Intensification in Hybrid Metaheuristics for Constraint Satisfaction Problems

Metaheuristics are used to find feasible solutions to hard Combinatorial Optimization Problems (COPs). Constraint Satisfaction Problems (CSPs) may be formulated as COPs, where the objective is to reduce the number of violated constraints to zero. The popular puzzle Sudoku is an NP-complete problem that has been used to study the effectiveness of metaheuristics in solving CSPs. Applying the Simulated Annealing (SA) metaheuristic to Sudoku has been shown to be a successful method to solve CSPs. However, the ‘easy-hard-easy’ phase-transition behavior frequently attributed to a certain class of CSPs makes finding a solution extremely difficult in the hard phase because of the vast search space, the small number of solutions and a fitness landscape marked by many plateaus and local minima. Two key mechanisms that metaheuristics employ for searching are diversification and intensification. Diversification is the method of identifying diverse promising regions of the search space and is achieved through the process of heating/reheating. Intensification is the method of finding a solution in one of these promising regions and is achieved through the process of cooling. The hard phase area of the search terrain makes traversal without becoming trapped very challenging. Running the best available method - a Constraint Propagation/Depth-First Search algorithm - against 30,000 benchmark problem-instances, 20,240 remain unsolved after ten runs at one minute per run which we classify as very hard. This dissertation studies the delicate balance between diversification and intensification in the search process and offers a hybrid SA algorithm to solve very hard instances. The algorithm presents (a) a heating/reheating strategy that incorporates the lowest solution cost for diversification; (b) a more complex two-stage cooling schedule for faster intensification; (c) Constraint Programming (CP) hybridization to reduce the search space and to escape a local minimum; (d) a three-way swap, secondary neighborhood operator for a low expense method of diversification. These techniques are tested individually and in hybrid combinations for a total of 11 strategies, and the effectiveness of each is evaluated by percentage solved and average best run-time to solution. In the final analysis, all strategies are an improvement on current methods, but the most remarkable results come from the application of the “Quick Reset” technique between cooling stages

A sublinear Sudoku solution in cP Systems and its formal verification

Sudoku is known as a NP-complete combinatorial number-placement puzzle. In this study, we propose the first cP system solution to generalised Sudoku puzzles with m×m cells grouped in m blocks. By using a fixed constant number of rules, our cP system can solve all Sudoku puzzles in sublinear steps. We evaluate the cP system and discuss its formal verification

Solving Sudoku with Ant Colony Optimization

In this paper we present a new algorithm for the well-known and computationally-challenging Sudoku puzzle game. Our Ant Colony Optimization-based method significantly out-performs the state-of-the-art algorithm on the hardest, large instances of Sudoku. We provide evidence that – compared to traditional backtracking methods – our algorithm offers a much more efficient search of the solution space, and demonstrate the utility of a novel anti-stagnation operator. This work lays the foundation for future work on a general-purpose puzzle solver, and establishes Japanese pencil puzzles as a suitable platform for benchmarking a wide range of algorithms

Hyper Sudoku Solver dengan Menggunakan Harris Hawks Optimization Algorithm

Sudoku merupakan salah satu permainan klasik yang digemari banyak orang. Sebagai salah satu permainan papan, Sudoku mempunyai banyak varian, salah satunya Hyper Sudoku. Hyper Sudoku mempunyai tingkat kesulitas yang lebih tinggi daripada Sudoku biasa. Tingkat kompleksitas yang tinggi membuat pemainan ini menjadi brain teaser yang baik dan sangat cocok diambil sebagai media untuk menguji algoritma metaheuristik. Algoritma yang populer pada dekade terakhir ini adalah algoritma metaheuristik berbasis populasi, yang mengadaptasi perilaku binatang dalam memecahkan permasalahan optimasi, salah satunya adalah Harris Hawks Optimization (HHO). Seperti kebanyakan metode swarm intelligence (SI) lainnya, algoritma ini mengandalkan proses diversification dan intensification. Selain itu, HHO mempunyai empat strategi khusus untuk mencari solusi dengan kondisi yang berbeda. HHO mampu mencakup solusi multi dimensi, sehingga sangat cocok diimplementasikan pada persoalan Hyper Sudoku. Untuk uji coba, peneliti menggunakan bantuan aplikasi Visual Studio 2017 dan MATLAB R2018a. Pada proses pengujian, digunakan dua setting parameter yang berbeda, tiga macam persoalan Hyper Sudoku, dan tiga puluh independent run untuk mencapai hasil yang diinginkan. Berdasarkan hasil pengujian, dapat disimpulkan bahwa tingkat keberhasilan untuk mencari solusi pada persoalan Hyper Sudoku dengan menggunakan HHO berkisar antara 86 hingga 88%, dilihat dari fitness value-nya

Geometric Particle Swarm Optimization for Multi-objective Optimization Using Decomposition

Multi-objective evolutionary algorithms (MOEAs) based on decomposition are aggregation-based algorithms which transform a multi-objective optimization problem (MOP) into several single-objective subproblems. Being effective, efficient, and easy to implement, Particle Swarm Optimization (PSO) has become one of the most popular single-objective optimizers for continuous problems, and recently it has been successfully extended to the multi-objective domain. However, no investigation on the application of PSO within a multi-objective decomposition framework exists in the context of combinatorial optimization. This is precisely the focus of the paper. More specifically, we study the incorporation of Geometric Particle Swarm Optimization (GPSO), a discrete generalization of PSO that has proven successful on a number of single-objective combinatorial problems, into a decomposition approach. We conduct experiments on manyobjective 1/0 knapsack problems i.e. problems with more than three objectives functions, substantially harder than multi-objective problems with fewer objectives. The results indicate that the proposed multi-objective GPSO based on decomposition is able to outperform two version of the wellknow MOEA based on decomposition (MOEA/D) and the most recent version of the non-dominated sorting genetic algorithm (NSGA-III), which are state-of-the-art multi-objective evolutionary approaches based on decomposition