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It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17clues to have a unique solution. This paper investigates the more specificquestion: given a particular completed Sudoku grid, what is the minimum numberof clues in any puzzle whose unique solution is the given grid? We call thisproblem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binarybilevel linear program, present a class of globally valid inequalities, andprovide 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, andshow that the solution process benefits from the addition of a moderate amountof inequalities. Finally, we extend the proposed model to other combinatorialproblems in which uniqueness of the solution is of interest.

How Many Clues To Give? A Bilevel Formulation For The Minimum Sudoku Clue Problem