Tri-Paradigm analysis of computational complexity in NP-Hard Sudoku solvers using explainable AI

Sudoku is a demanding test of computational complexity and constraint satisfaction problems. This work develops a comprehensive and replicable benchmark that reformulates 9×9 Sudoku puzzles under three solver paradigms of constraint programming (CP), Boolean satisfiability (SAT), and integer programming (IP) in order to assess efficacy and explainability. For each paradigm, a parametrically equivalent model is implemented and solved with common datasets and experiment control conditions. We measure wall-clock running time, peak memory usage, and paradigm-data related measures of effort in backtracks or fails in CP, decisions or conflicts with conflict-driven clause learning(CDCL) in SAT, and branch & bound nodes or linear program relaxations in IP. Aggregate trends provide empirical evidence consistent with NP-hard/NP-complete behavior: runtime grows approximately exponentially with constraint tightness, while memory remains nearly flat for standard grids. An explicit Explainable Artificial Intelligence (XAI) trade-off emerges. SAT delivers the lowest runtime via clause learning and pruning; CP yields the most interpretable reasoning through transparent propagation and traceable search; IP offers algebraic auditability and flexibility for optimization variants but incurs higher search effort on pure feasibility. The benchmark, figures, and tables establish a verifiable claim–evidence link and supply a practical baseline for integrating explainability metrics into analytic decision pipelines. The results inform the design of intelligent and explainable solver architectures, aligning with the aim of Intelligent systems with applications. We conclude with a roadmap toward hybrid CP–SAT approaches that retain CP-level interpretability while leveraging SAT-level efficiency.