Using Constraint Processing to Model, Solve, and Support Interactive Solving of Sudoku Puzzles

1. Using Constraint Processing to Model, Solve, and Support Interactive Solving of Sudoku Puzzles Christopher G. Reeson Constraint Systems Laboratory Department of Computer Science & Engineering University of Nebraska-Lincoln Research supported by a UCARE grant Reeson–Undergraduate Thesis

2. Goals • How to use CP techniques on Sudoku puzzles? • Study & compare the effectiveness of various constraint propagation algorithms • How can we use CP to interactively support and guide human players? • Is Sudoku useful to illustrate and teach CP? • … How to deconstruct Sudoku solving to • Demystify the puzzle • Prevent students from losing time playing it Reeson–Undergraduate Thesis

3. Outline • Background information: the game, CSPs • Our contributions • Algorithms • Conjectures • Interface: interactive solver • Conclusions and future research • Live demonstration Reeson–Undergraduate Thesis

4. The game • Number place • Main properties • NP-complete [Yato 03] • Well-formed Sudoku: has 1 solution [Simonis 05] • Minimal Sudoku • In a 9x9 Sudoku: smallest known number of givens is 17 [Royle] • Symmetrical puzzles • Many axes of symmetry • Position of the givens, not their values • Often makes the puzzle non-minimal • Level of difficulties • Varied ranking systems exist • Mimimality and difficulty not related #15 on Royle’s web site Reeson–Undergraduate Thesis

5. V1 V2 {c, d, e, f} {d} V4 V3 {a, b, d} {a, b, c} Constraint Satisfaction Problem (CSP) • GivenP = (V, D, C) • V: set of variables • D: set of their domains • C: set of constraints (relations) restricting the acceptable combination of values for variables • Solution is a consistent assignment of values to variables • Query: find 1 solution, all solutions, etc. Reeson–Undergraduate Thesis

6. Sudoku as a CSP (9x9 puzzles) • Variables are the cells • Domains are sets {1,…,9} • Two models • Binary constraints: 810 all-different binary constraint between variables • Non-binary constraints: 27 all-different 9-ary constraints Reeson–Undergraduate Thesis

7. Solving Sudoku as a CSP • Search • Builds solutions by enumerating consistent combinations • Constraint propagation • Removes values that do not appear in a solution • Example, arc-consistency: Reeson–Undergraduate Thesis

8. Search • Backtrack search • Systematically enumerate solutions by instantiating one variable after another • Backtracks when a constraint is broken • Is sound and complete (guaranteed to find solution if one exists) • Look-ahead • Cleans up domain of ‘future’ variables, during search, given current instantiations • Forward Checking (FC): weak form of arc-consistency • Maintaining Arc-Consistency (MAC): arc-consistency Reeson–Undergraduate Thesis

9. Forward Checking (FC) • Forward Checking on the binary model Reeson–Undergraduate Thesis

10. Maintaining Arc-Consistency (MAC) • Arc Consistency on the binary model Reeson–Undergraduate Thesis

11. Generalized Arc-Consistency (GAC) • Operates on the non-binary model • Subsumes AC • Can be done in polynomial time for all-different constraints [Régin 94] Reeson–Undergraduate Thesis

12. Consistency Algorithms - SAC • Shaving Arc-Consistency • Binary • Stronger than Arc-Consistency Michael Mepham Extreme #26 Reeson–Undergraduate Thesis

13. Consistency Algorithms - SGAC • Shaving Generalized Arc-Consistency • Non-binary • Subsumes SAC Michael Mepham Extreme #26 Reeson–Undergraduate Thesis

14. Consistency relationships Sets denote the values removed by propagation Reeson–Undergraduate Thesis

15. R(1,2)C • Takes two constraints and • Removes any value that does not appear in a solution to both constraints Reeson–Undergraduate Thesis

16. Consistency R(1,2)C Reeson–Undergraduate Thesis

17. Conjecture 1: R(1,2)CSGAC • R(1,2)C is at least as strong as SGAC • Is SGAC at least as strong as R(1,2)C? • We proved R(1,2)CSGAC when the 2 relations overlap over one variable Reeson–Undergraduate Thesis

18. Conjecture 2: SGAC solves.. … every: • Well-formed • 9x9 Sudoku? • Proof still open, but empirical evidence is strong • Warning • Sudoku is NP-Complete • SGAC is polynomial • When a 9x9 Sudoku has multiple solutions, SGAC does not solve it • Proof may exploit the fact that all non-binary constraints are 1-tight Reeson–Undergraduate Thesis

19. Interactive interface • Loading instances • Buttons for propagation • On individual constraints • For consistency algorithms: AC, GAC, SAC, SGAC • For interactive look-ahead • Finding solutions • Hint panel • Highlighting error • Manual domain filtering • Manual assignments • Do/Undo buttons supporting every functionality Reeson–Undergraduate Thesis

20. Buttons for constraint propagation • Buttons allow users to run • AC • GAC • Over each • Column • Row • Block Reeson–Undergraduate Thesis

21. Finding solutions • Find the number of solutions at any point in the game • Creates a copy of the CSP displayed on the board • Runs consistency algorithms to speed up search • Runs exhaustive search to find all solutions • Found Sudokus that are not well-formed on the web and in… the LA Times Reeson–Undergraduate Thesis

22. Hint panel • Two types of hints: Singleton & Vital • Eight Levels (6 implemented) • FC, AC, Single GAC, GAC, Single SAC, SAC, Single SGAC, SGAC • Highlight a cell with a hint • Option to move to the next hint • Display the number of hints • Preliminary tests show that this new strategy can teach people to play Sudoku effectively and… get them quickly bored with the game Reeson–Undergraduate Thesis

23. Web-based implementation • XML files • Solver • Constructor Reeson–Undergraduate Thesis

24. The way humans play • ‘Cross-hash,’ sweep through lines, columns, and blocks • Pencil in possible positions of values • Generally, look for patterns, some intricate, and give them funny names: • Naked single, locked pairs, swordfish, medusa, etc. • ‘Identified’ dozens of strategies • Many fall under a unique constraint propagation technique • But humans do not seem to be able to carry simple inference (i.e., propagation) in a systematic way for more than a few steps Reeson–Undergraduate Thesis

25. Impact • Inspired a PhD student @ USC • to use propagation to infer the applicable constraints for automatically modeling various types of Sudoku puzzles • Collaborating with an MS student @ USC on Constructor • Inputting, storing instances • Generalization to other Sudoku variations Reeson–Undergraduate Thesis

26. Conclusions: CP solving • Propagation is useful for • effectively guide humans & train them in Sudoku solving • SAC (often), SGAC (always) can solve any well-formed Sudoku, also noticed by Simonis • Sudoku good illustration of the power of SAC and SGAC • otherwise thought to be more expensive than effective • No need for extra CP modeling as Simonis did: • Use the simple constraint model on row, columns, blocks • Stick to general purpose propagation algorithms • But… Who wants to play Sudoku any more when they know that SGAC can do it? Reeson–Undergraduate Thesis

27. Finally • Future work: • Use the interface to generate Sudoku puzzles • Investigate theoretically the relationship between • Sudoku size • And the consistency level necessary for solving it • Live demonstration Reeson–Undergraduate Thesis

28. Thank you for your attention I would be glad to answer your questions Reeson–Undergraduate Thesis