## Abstract

The generalised Sudoku problem with N symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where N is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has O(N
^{3}
) vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion to undirected Hamiltonian cycle problem is also provided so that it can be submitted to the best heuristics. A simple algorithm for obtaining the valid Sudoku solution from the Hamiltonian cycle is provided. Techniques to reduce the size of the resultant graph are also discussed.

Original language | English |
---|---|

Pages (from-to) | 272-282 |

Number of pages | 11 |

Journal | AKCE International Journal of Graphs and Combinatorics |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Dec 2016 |

## Keywords

- Hamiltonian cycle problem
- NP-complete
- Reduction
- Sudoku