Reducing the generalised Sudoku problem to the Hamiltonian cycle problem

TL;DR

This paper presents a constructive reduction of the generalized Sudoku problem to the Hamiltonian cycle problem, enabling the use of existing heuristics for solving Sudoku via graph algorithms.

Contribution

It introduces a novel, explicit reduction method from generalized Sudoku to Hamiltonian cycle, producing sparse, directed graphs suitable for heuristic solutions.

Findings

Reduction produces sparse, directed graphs with O(N^3) vertices.

Conversion to undirected graphs allows use of existing Hamiltonian cycle heuristics.

Algorithm enables solving Sudoku by solving a Hamiltonian cycle problem.

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.