lgi2p research center coloration des graphes de reines [email protected] lgi2p ecole des...
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LGI2PResearch Center
Coloration des graphes de reines
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
A
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 Queen Graph 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-72 is 7 : (7) 7
(and (n) n if n is prime with 2 and 3)
1 2 3 4 5 6 7
4 5 6 7 1 2 3
7 1 2 3 4 5 6
3 4 5 6 7 1 2
6 7 1 2 3 4 5
2 3 4 5 6 7 1
5 6 7 1 2 3 4
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
E. Y. Gik, 1983 : Shakhmaty i matematika, Bibliotechka Kvant, vol. 24, Nauka, Moscow.
The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3
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 vertices of the Queen-n2 graphÞ (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 nn squares : each independent set must contain at least n vertices
AA
AA
AA
A
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)
Find n 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:
1 2 3 4 5 6 7
20
Computing IS by backtracking
Enumeration : backtracking
n |V| |E| |I.S.| sec.10 100 1470 724 011 121 1980 2680 012 144 2596 14200 013 169 3328 73712 214 196 4186 365596 1515 225 5180 2279184 73
21
A CSP with n variables (corresponding to a n squares)
Spreading of the independent sets for Queen-102
56 36 48 69 63 63 69 48 36 56 176 946 839 785 046 00036 52 67 51 66 66 51 67 52 3648 67 66 43 48 48 43 66 67 4869 51 43 56 53 53 56 43 51 6963 66 48 53 42 42 53 48 66 6363 66 48 53 42 42 53 48 66 6369 51 43 56 53 53 56 43 51 6948 67 66 43 48 48 43 66 67 4836 52 67 51 66 66 51 67 52 3656 36 48 69 63 63 69 48 36 56
22
21 24 31 31 25 27 21 14 27 2 595 187 803 60013 21 32 34 31 27 28 21 1423 33 33 21 19 28 18 27 1929 24 25 18 24 24 21 25 3130 25 14 29 15 21 29 23 3533 25 13 28 22 15 16 34 3533 27 19 17 25 17 29 23 31
19 39 19 24 20 19 30 32 1917 28 25 21 33 28 31 22 1622 19 22 30 27 33 28 15 25
Branching on the smallest domain variable
Non overlapping constraints propagation
12 15 17 14 11 9 6 18 458 045 280 15 13 19 17 19 9 11 5
12 19 15 11 13 13 11 1016 12 14 8 8 15 12 2017 12 4 16 11 11 17 2117 16 11 13 7 8 11 20 12 16 10 8 17 9 14 19 5 18 15 11 11 16 19 86 18 10 13 17 20 11 916 9 13 16 17 11 8 12
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):
Þ for a given diagonal there is at most one vertex that can come from a specific stable set,
Þ 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
AA
AA
AA
A
this independent set is excluded from the search space
27
Efficient Filtering (experimentally)
Search space reduction
n |V| |E| |I.S.| |I.S.| f iltered10 100 1470 724 54411 121 1980 2680 174412 144 2596 14200 944013 169 3328 73712 5200814 196 4186 365596 23808815 225 5180 2279184 1484400
28
Efficient Filtering (experimentally)
At each level : 4 more constraints
29
First Results : complete method
(10) no solution 1 second(maximum depth of backtrack in the search tree : 5 rather than 10)
(12) 12 454 solutions 6963 seconds
(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
1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1
10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3
33
Certificate for n = 12
1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1
10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3
34
Certificate for n = 12
1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1
10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3
35
Certificate for n = 12
1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1
10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3
36
Exact but incomplete method
Assumption on the distribution of the colors on the chessboard
Þ Enumerate several independent sets at the same time
37
Geometric operator (1) n = 2 p symmetry H
1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13
Search tree depth: n/2
(22) 22
38
Geometric operator (1) n = 2 p symmetry H
1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13
Search tree depth: n/2
(22) 22
39
Geometric operator (1) n = 2 p symmetry H
1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13
Search tree depth: n/2
(22) 22
40
Geometric operator (2) n = 3 p central symmetry
Search tree depth: (n/2) - 1
(15) 15
1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13
11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3
10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12
14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2
41
Geometric operator (2) n = 3 p central symmetry
Search tree depth: (n/2) - 1
(15) 15
1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13
11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3
10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12
14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2
42
Geometric operator (2) n = 3 p central symmetry
Search tree depth: (n/2) - 1
(15) 15
1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13
11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3
10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12
14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2
43
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
44
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
45
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
46
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
47
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
48
Geometric operator (3) n = ( 4 p ) + 1 /2 rotations:
R, R2 et R3
Search tree depth: (n/4) - 1
(21) 21
4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11
21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16
15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19
10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12
17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13
14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2
49
Geometric operator (4) n = ( 4 p ) symmetries H & V
Search tree depth: (n/4)
(32) 32
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w x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i j
f # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &
x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s i
k f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & v
n + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s
$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c
+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a
@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d
* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b
# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d e
o u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l r
t z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g m
r p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q o
h g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z y
l q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p u
j s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n w
i l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u x
y v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h
& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w f
s i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x n
v & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k
50
Geometric operator (4) n = ( 4 p ) symmetries H & V
Search tree depth: (n/4)
(32) 32
a b c d e f g h i j k l m n o p q r s t u v w x y z & # @ $ * +
c d a b g r e x t u h w k f q s n p & v j y l m i # o z * + @ $
b e g c a t d q r k n f i y j u l w h x & s v o p @ m + $ z # *
g a b e d o c t n h l p w v f x i & k j q u y s m $ r @ # * + z
d m w a b e v p c r y u f i n z g s x & l h o $ q k # * + j t @
e c q l m y z d a b s k o w x f & i j r v n * + @ g h t u p $ #
m r e y c d p & l x a v b j z n s g w * k + i u f q @ $ h # o t
z y d q o s u c j i b & a t # k v e m + f * x w $ l n r p @ h g
q o n u t b & v # g j @ y c a i x + $ h d w z e k f * m l s r p
u h x o q a @ g & v # s $ m b j w * t c n e k f z d + p r i y l
p t l $ w z r # y s * i @ k + & f a v d x b n h e o g j c u m q
w x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i j
f # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &
x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s i
k f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & v
n + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s
$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c
+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a
@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d
* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b
# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d e
o u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l r
t z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g m
r p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q o
h g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z y
l q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p u
j s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n w
i l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u x
y v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h
& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w f
s i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x n
v & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k
51
Geometric operator (4) n = ( 4 p ) symmetries H & V
Search tree depth: (n/4)
(32) 32
a b c d e f g h i j k l m n o p q r s t u v w x y z & # @ $ * +
c d a b g r e x t u h w k f q s n p & v j y l m i # o z * + @ $
b e g c a t d q r k n f i y j u l w h x & s v o p @ m + $ z # *
g a b e d o c t n h l p w v f x i & k j q u y s m $ r @ # * + z
d m w a b e v p c r y u f i n z g s x & l h o $ q k # * + j t @
e c q l m y z d a b s k o w x f & i j r v n * + @ g h t u p $ #
m r e y c d p & l x a v b j z n s g w * k + i u f q @ $ h # o t
z y d q o s u c j i b & a t # k v e m + f * x w $ l n r p @ h g
q o n u t b & v # g j @ y c a i x + $ h d w z e k f * m l s r p
u h x o q a @ g & v # s $ m b j w * t c n e k f z d + p r i y l
p t l $ w z r # y s * i @ k + & f a v d x b n h e o g j c u m q
w x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i j
f # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &
x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s i
k f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & v
n + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s
$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c
+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a
@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d
* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b
# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d e
o u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l r
t z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g m
r p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q o
h g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z y
l q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p u
j s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n w
i l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u x
y v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h
& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w f
s i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x n
v & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k
52
Results synthesis
New results for the graphs counting more than ………………… 81 vertices (10) 11 1 sec. (20) 20 1 sec.
(12) 12 1 sec. (21) 21 30844 sec.
(14) 14 5 sec. (22) 22 233404 sec.
(15) 15 4897 sec. (24) 24 10 sec.
(16) 16 1 sec. (28) 28 1316 sec.
(18) 18 2171 sec. (32) 32 73000 sec. … 1024 vertices
53
Results synthesis
Some improvements : fixing the first stable set (and its symmetric set) according to other certificates, branching heuristic , …
(26) 26 1,400,000 seconds
54
The 26 letters of the alphabet are enough for coloring the 26 X 26 chessboard
55
26 = ( 3 x 2 ) + …
56
26 = ( 5 x 4 ) + ( 3 x 2 )
57
(26) = 26 2216 sec. (better than 1,400,000)
58
30 = ( 6 x 4 ) + ( 3 x 2 )
“Too long” still more heuristic
Evaluate the nodes by counting the number of edges in the no colored sub graph
Partial branching : at each level of the search tree take only the best node among 10
59
X(30)=30cpu 2965 sec.
60
Results synthesis
Nous avons 13 contre exemples qui prouvent que n n’a pas besoin d’être
premier avec 6 pour que (n) n
61
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
62
Remark
An independent set with 12 vertices for n=12
63
… remains an independent set for n = 24 ...
Remark
64
Idea
… replacing 1 square that uses 1 color by 52 squares that use only 5 colors ...
65
Idea
… (5 12) = (60) = 60 ...
66
Complementary Diagonals
67
Polynomial formula for n no multiple of 2 or 3
r(i,j) ( 2.i + j ) modulo p
complementary diagonals
68
Characteristic of Complementary Diagonals
69
Characteristic of Complementary Diagonals
70
Coloring Extension Formula
If (n) n and if p is prime with 2 and 3 then (np) np
given a coloring C(i,j) for the chessboard n n
r(i,j) ( 2i + j ) modulo p
R(i,j) = r(i,j) + p C(i/p,j/p)
(elementary algebra in the ring /p)
71
Coloring Extension Formula R(i,j) = r(i,j) + p.C(i/p,j/p)
If R(i,j) = R(i’,j)
r(i,j) + p.C(i/p,j/p) = r(i,j) + p.C(i’/p,j/p)
r(i,j)=r(i’,j) and C(i/p,j/p)= C(i’/p,j/p) because r < p
i=i’ modulo p because 2 is prime with p ( recall : r(i,j) 2i+j )and
i/p=i’/p because C is a coloration and we are on the same column
then i = i’
72
Coloring Extension Formula R(i,j) = r(i,j) + p.C(i/p,j/p)
Exercise
Prove the same kind of results for lines and diagonals
73
(60) (5 12) 5 12 60,
(70) (5 14) 5 14 70,
(75) (5 15) 5 15 75,
(84) (7 12) 7 12 84,
…
(2010) (67 30) 67 30 2010
…
74
Conclusion
We know that (n) n for n 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 30 and 32
We know that there is an infinity of integers n that are multiples of 2 or 3 such that (n) n
Is (n) n n 11 ?
But we don’t know, today, the value of (27)
75
Publications
Comptes Rendus Mathématiques de l’Académie des Sciences, Paris, Elsevier, Ser. I 342 (2006) p 157-160. “Coloration des graphes de reines”.
Journal of Heuristics (2004), 10 : p 407-413 : “New Results on Queen Graph Coloring Problem”.
ECAI'04, 16th European Conference on Artificial Intelligence, Valencia, Spain, August 22-27, 2004 : “Complete and Incomplete Algorithms for the Queen Graph Coloring Problem”.
Programmation en logique avec contraintes, JFPLC 2004, Hermes Science, ISBN
2-7462-0937-3, p 239-252 : “Algorithmes complet et incomplet pour la coloration des graphes de reines”.