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  1. Coloration des graphes de reines michel.vasquez@mines-ales.fr LGI2P Ecole des Mines d’Alès

  2. Outline • About the Queen Graph Coloring Problem • Definition • Conjecture ? • A Complete Algorithm • Reformulation of the coloring problem • Efficient filtering • A Geometric Based Heuristic • Geometric Operators • Results synthesis • Coloring Extension

  3. Rule for moving the queen on the chessboard • Each queen controls: • 1 column • 1 row • 2 diagonals

  4. Graph definition • 1 square of the chessboard  vertex • 2 squares controlled by the same queenedge

  5. Graph definition: from chessboard to queen graph a queen graph instance  G(V,E) with :V n2 vertices and E  n3 edges

  6. The QueenGraph Coloring Problem: definition Given a chessboard, what is the minimum number of colors required to cover it without clash between two queens of the same color ?

  7. The Queen Graph Coloring Problem: what we know The chromatic number of Queen-72is 7 :  (7)  7 (and (n)  n if n is prime with 2 and 3)

  8. Conjecture ? The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3 • M. Gardner,1969 : The Unexpected Hanging and Other Mathematical Diversions, Simon and Schuster, New York.

  9. Conjecture ? The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3 The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3 • E. Y. Gik,1983 : Shakhmaty i matematika, Bibliotechka Kvant, vol. 24, Nauka, Moscow.

  10. Intox…

  11. Intox…

  12. Until 2003 no result are available for the queen graph chromatic number when n is greater than 9 and n is multiple of 2 or 3

  13. Outline • About the Queen Graph Coloring Problem • A Complete Algorithm • Reformulation of the coloring problem • Efficient filtering • A Geometric Based Heuristic • Geometric Operators • Results synthesis • Coloring Extension

  14. Property (1) • The n rows, the n columns and the 2 main diagonals are cliques with n verticesof the Queen-n2 graph •  (n)  n

  15. Question (1) For a given n, is  (n) equal to n ? saying it differently Is there a partition of the Queen-n2 graph in n independent sets ?

  16. Property (2) • A stable set cannot contain more than n vertices To answer yes to question (1) and cover nn squares : each independent set must contain at least n vertices

  17. Question (2) • Are there n independent sets with exactly n vertices which do not cover themselves ?

  18. General Algorithm Step 1) Enumerate the independent sets with n vertices (n queens that do not attack themselves) Step 2) Findn among them which do not intersect (solve the CSP)

  19. Avoiding many equivalent coloring permutations n squares belonging to a same clique are colored once for all:

  20. Computing IS by backtracking • Enumeration : backtracking

  21. A CSP with n variables (corresponding to a n squares) • Spreading of the independent sets for Queen-102

  22. Branching on the smallest domain variable • Non overlapping constraints propagation • The search space size is decreasing geometrically

  23. First result n = 10 : no solution 7000 seconds   (10) = 11

  24. Filtering (principle) • Consider the cliques of the graph constituted by the uncolored vertices • If such a clique contains k vertices then you need at least k colors (i.e. k independent sets) to complete the process

  25. Efficient Filtering (computationally) • Diagonals constitute cliques (and are easy to handle): • for a given diagonal there is at most one vertex that can come from a specific stable set, • at level k of the search tree, diagonals must contain less than n-k empty squares Delete all the independent sets that do not verify this condition

  26. Efficient Filtering (experimentally) • At the root of the search tree this independent set is excluded from the search space

  27. Efficient Filtering (experimentally) • Search space reduction

  28. Efficient Filtering (experimentally) • At each level : 4 more constraints

  29. First Results : complete method •  (10) no solution 1second (maximum depth of backtrack in the search tree : 5 rather than 10) •  (12) 12 454 solutions 6963seconds (exhaustive search) •  (14) 14 1 solution en 142 hours (search aborted after one week)

  30. Interest of filtering • Comparative results on n=12

  31. Outline • About the Queen Graph Coloring Problem • Definition • Intox/Conjecture ? • A Complete Algorithm • Reformulation of the coloring problem • Efficient filtering • A Geometric Based Heuristic • Geometric Operators • Results synthesis • Coloring Extension

  32. Certificate for n = 12

  33. Certificate for n = 12

  34. Certificate for n = 12

  35. Certificate for n = 12

  36. Exact but incomplete method • Assumption on the distribution of the colors on the chessboard • Enumerate several independent sets at the same time

  37. Geometric operator (1) n = 2  p symmetry H Search tree depth: n/2  (22)  22

  38. Geometric operator (1) n = 2  p symmetry H Search tree depth: n/2  (22)  22

  39. Geometric operator (1) n = 2  p symmetry H Search tree depth: n/2  (22)  22

  40. Geometric operator (2) n = 3  p  central symmetry Search tree depth: (n/2) - 1  (15)  15

  41. Geometric operator (2) n = 3  p  central symmetry Search tree depth: (n/2) - 1  (15)  15

  42. Geometric operator (2) n = 3  p  central symmetry Search tree depth: (n/2) - 1  (15)  15

  43. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  44. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  45. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  46. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  47. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  48. Geometric operator (3) n = ( 4  p ) + 1 /2 rotations: R, R2 et R3 Search tree depth: (n/4) - 1  (21)  21

  49. Geometric operator (4) n = ( 4  p )  symmetries H & V Search tree depth: (n/4)  (32)  32

  50. Geometric operator (4) n = ( 4  p )  symmetries H & V Search tree depth: (n/4)  (32)  32