It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.

翻译：已证明，任何9×9数独谜题必须包含至少17个线索才能保证唯一解。本文研究一个更具针对性的问题：对于任意已完成的数独棋盘，使得该棋盘成为唯一解的谜题最少需要多少个线索？我们将该问题定义为最小数独线索问题（MSCP）。我们构建了MSCP的二元双层线性规划模型，提出了一类全局有效不等式，并在50个9×9数独棋盘的MSCP实例上开展了计算研究。采用通用双层求解器，我们求解了95%的实例至最优解，并证明了适量增加不等式有助于优化求解过程。最后，我们将所提模型扩展至其他关注解唯一性的组合优化问题。