Exact Method for Generating Strategy-Solvable Sudoku Clues

TL;DR

This paper introduces an exact, strategy-based framework for generating Sudoku clues, enabling precise solvability control and minimal clue determination, surpassing brute-force approaches in efficiency.

Contribution

We propose the first exact method for generating Sudoku clues based on strategy solvability, extending beyond trivial brute-force techniques.

Findings

Method terminates within 1 minute for many grids

Running time increases rapidly as clues approach 20

Efficiently evaluates minimal clues from known minimum Sudokus

Abstract

A Sudoku puzzle often has a regular pattern in the arrangement of initial digits and it is typically made solvable with known solving techniques, called strategies. In this paper, we consider the problem of generating such Sudoku instances. We introduce a rigorous framework to discuss solvability for Sudoku instances with respect to strategies. This allows us to handle not only known strategies but also general strategies under a few reasonable assumptions. We propose an exact method for determining Sudoku clues for a given set of clue positions that is solvable with a given set of strategies. This is the first exact method except for a trivial brute-force search. Besides the clue generation, we present an application of our method to the problem of determining the minimum number of strategy-solvable Sudoku clues. We conduct experiments to evaluate our method, varying the position and the number of clues at random. Our method terminates within $1$ minutes for many grids. However, as the number of clues gets closer to $20$, the running time rapidly increases and exceeds the time limit set to $600$ seconds. We also evaluate our method for several instances with $17$ clue positions taken from known minimum Sudokus to see the efficiency for deciding unsolvability.