Entropy guided evolutionary search for solving Sudoku

Abstract

The Sudoku puzzle solving, a Constraint Satisfaction Problem (CSP), is challenging due to various complexity levels. Although, deterministic as well as meta-heuristics techniques are present to solve the Sudoku puzzle. Still, such techniques are insufficient for solving ‘hard’ frames and suffer from local minima in terms of increased complexity levels of Sudoku. Moreover, the entropy concept is used in designing the solution framework and rating of Sudoku frames. However, it provides superior results mostly up to medium levels. Therefore, we extend the concept of entropy by hybridization with an Evolutionary Algorithm (EA) for solving ‘harder’ instances. To improve results further, we attempt to reduce the uncertainty of Sudoku cells toward zero. With this hybridized EA framework, we embed problem-specific knowledge mapped in terms of a cell’s position uncertainty to construct a fitness function. Moreover, a performance metric, symmetric count ($S_\mathrm{c}$) is devised for assessing the obtained solutions. The empirical results show that the proposed hybrid EA framework is capable of efficiently solving a wide range of benchmark Sudoku instances; moreover, this strategy outperforms the existing solution strategies. The convergence rate of the proposed technique is faster in most cases, and this approach works for most instances of ‘hard’ and ‘very hard’ levels of the Sudoku grid. The results indicate that approx. 75% of the frames reach closer to an optimal solution, i.e., the saturation point ($S_\mathrm{c}=1$).