Graph Coloring Approach to Solving Sudoku with Oscillatory Neural Networks

2026-07-17

AI Summary

This work formulates the Sudoku puzzle as a constrained graph coloring problem and develops an efficient solver based on an oscillatory neural network (ONN). By explicitly embedding Sudoku-specific constraints—pertaining to rows, columns, and subgrids—into the energy function and refining the existing ONN architecture for graph coloring to reduce computational overhead, the proposed approach achieves substantially improved solution accuracy. The method attains near-perfect accuracy on 4×4 Sudoku instances and significantly outperforms existing Hopfield neural network (HNN) and ONN-based solvers on standard 9×9 puzzles. These results demonstrate the effectiveness of the proposed constraint-embedding strategy and the lightweight ONN design in solving combinatorial constraint satisfaction problems such as Sudoku.

Abstract

Oscillatory Neural Networks (ONNs) present an attractive physics-based computing paradigm rooted in the dynamics of a network of typically fully coupled oscillators aiming to minimize an underlying energy function. In this paper, we propose an ONN-based solver for one well-known constrained combinatorial optimization problem, namely a Sudoku, by formulating the problem as a Graph Coloring problem. By modifying the already existing Graph Coloring solver to a computationally cheaper version and introducing an additional term ensuring the fulfillment of the Sudoku constraints, our solver is shown to significantly outperform the existing HNN- and ONN solvers in terms of accuracy. In particular, we are able to achieve nearly flawless accuracies on $4 \times 4$ as well as rather high accuracies on $9 \times 9$ Sudoku puzzles for different numbers of unknown digits.

Problem

• Sudoku
• Graph Coloring
• Oscillatory Neural Networks
• Combinatorial Optimization
• Constraint Satisfaction

Innovation

• Oscillatory Neural Networks
• Graph Coloring
• Sudoku Solver
• Combinatorial Optimization
• Constraint Satisfaction