1. Constraints and Search
Logic & AR Summer School, 2002
Constraints and Search
Toby Walsh Cork Constraint Computation Centre (4C)

2. Constraint satisfaction
Constraint satisfaction problem (CSP) is a triple <V,D,C> where:
V is set of variables
Each X in V has set of values, D_X
Usually assume finite domain
{true,false}, {red,blue,green}, [0,10], …
C is set of constraints
Goal: find assignment of values to variables to satisfy all the constraints

3. Constraint solver Tree search Number of choices
Assign value to variable
Deduce values that must be removed from future/unassigned variables
Constraint propagation
If any future variable has no values, backtrack else repeat
Number of choices
Variable to assign next, value to assign
Some important refinements like nogood learning, non-chronological backtracking, …

4. Constraint propagation
Enfrocing arc-consistency (AC)
A binary constraint r(X1,X2) is AC iff
for every value for X1, there is a consistent value (often called support) for X2 and vice versa
E.g. With 0/1 domains and the constraint X1 =/= X2
Value 0 for X1 is supported by value 1 for X2
Value 1 for X1 is supported by value 0 for X2 …
A problem is AC iff every constraint is AC

5. Tree search Backtracking (BT) Forward checking (FC)
Maintaining arc-consistency (MAC)
Limited discrepany search (LDS)
Non-chronological backtracking & learning

6. Example: n-queens
Place n-queens on an n  n board so that no pair of queens attacks each other

7. Example: 4-queens Variables: x1, x2 , x3 , x4 Domains: {1, 2, 3, 4}
Constraints:
xi  xj and
| xi - xj |  | i - j | x1 x2 x3 x4 1 2 3 4

8. Example: 4-queens One solution: x1  2 x2  4 x3  1 x4  3 x1 x2 x3

9. BT on 4-queens x1 x2 x3 x4
Try:
x1  1 1 Q 2 3 4

10. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  1 1 Q Q 2 3 4

11. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  1
Backtrack 1 Q Q 2 3 4

12. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  2 1 Q 2 Q 3 4

13. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  2
Backtrack 1 Q 2 Q 3 4

14. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3 1 Q 2 3 Q 4

15. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3
x3  1 1 Q Q 2 3 Q 4

16. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 x3  1 Backtrack 1 2 3 4

17. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3
x3  2 1 Q 2 Q 3 Q 4

18. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 x3  2 Backtrack 1 2 3 4

19. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3
x3  3 1 Q 2 3 Q Q 4

20. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 x3  3 Backtrack 1 2 3 4

21. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3
x3  4 1 Q 2 3 Q 4 Q

22. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 x3  4 Backtrack 1 2 3 4

23. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  4 1 Q 2 3 4 Q

24. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  4
x3  1 1 Q Q 2 3 4 Q

25. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  4 x3  1 Backtrack 1 2 3 4

26. BT on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  4
x3  2 1 Q 2 Q 3 4 Q

27. BT on 4-queens x1 x2 x3 x4 Try: x1  1 x2  4 x3  2 And so on! 1 2 3

28. BT search tree for 4-queensx1 1 2 3 4 … x2 … x3 x4
(2,4,1,3)
(3,1,4,2)

29. Forward checking (FC) Maintains limited form of arc-consistency
Makes constraints involving current var and any future var arc-consistent
Only looks at each constraint once!
No need to re-visit constraints to ensure support remains when a domain value is pruned
Very cheap, little extra cost over BT

30. FC on 4-queens x1 x2 x3 x4
Try:
x1  1 1 Q 2 3 4

31. FC on 4-queens x1 x2 x3 x4 Try: x1  1 Forward check 1 2 3 4 Q X X X X

32. FC on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  3 1 Q X X X 2 X 3 Q X 4 X

33. FC on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 Forward check 1 2 3 4 Q

34. FC on 4-queens Try: x1  1 x2  3 Domian wipeout for x3 x1 x2 x3 x4 1

35. FC on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 Backtrack 1 2 3 4 Q X X

36. FC on 4-queens x1 x2 x3 x4 Try: x1  1 x2  3 Backtrack 1 2 3 4 Q X X

37. FC on 4-queens x1 x2 x3 x4
Try:
x1  1
x2  4 1 Q X X X 2 X 3 X 4 Q X

38. FC on 4-queens x1 x2 x3 x4 Try: x1  1 x2  4 Forward check 1 2 3 4 Q

39. FC on 4-queens Try: x1  1 x2  4 x3  2 x1 x2 x3 x4 1 2 3 4 Q X X X X

40. FC on 4-queens Try: x1  1 x2  4 x3  2 Forward check x1 x2 x3 x4 1 2

41. FC on 4-queens Try: x1  1 x2  4 x3  2 Domain wipeout for x4 x1 x2

42. FC on 4-queens x1 x2 x3 x4 Try: x1  1 x2  4 Backtrack 1 2 3 4 Q X X

43. FC on 4-queens x1 x2 x3 x4
Try:
x1  1
Backtrack 1 Q X X X 2 X 3 X 4 X

44. FC on 4-queens x1 x2 x3 x4
Try:
x1  2 1 2 Q 3 4

45. FC on 4-queens x1 x2 x3 x4 Try: x1  2 Forward check 1 2 3 4 X Q X X X

46. FC on 4-queens x1 x2 x3 x4
Try:
x1  2
x2  4 1 X 2 Q X X X 3 X 4 Q X

47. FC on 4-queens x1 x2 x3 x4 Try: x1  2 x2  4 Forward check 1 2 3 4 X

48. FC on 4-queens x1 x2 x3 x4 Try: x1  2 x2  4 x3  1 1 2 3 4 X Q Q X X

49. FC on 4-queens x1 x2 x3 x4 Try: x1  2 x2  4 x3  1 Forward check 1 2

50. FC on 4-queens x1 x2 x3 x4 Try: x1  2 x2  4 x3  1 x4  3 1 2 3 4 X

51. FC on 4-queens x1 x2 x3 x4 Try: x1  2 x2  4 x3  1 x4  3 1 2 3 4 Q

52. Maintaining arc-consistency (MAC)
Enforces arc-consistency at each node in the search tree
Much more expensive than FC
But can save exploring many fruitless subtrees

53. MAC on 4-queens x1 x2 x3 x4
Try:
x1  1 1 Q 2 3 4

54. MAC on 4-queens x1 x2 x3 x4
Try:
x1  1
Enforce AC 1 Q 2 3 4

55. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC 1 2 3 4 Q X X X X X

56. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC 1 2 3 4 Q X X X X X

57. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC 1 2 3 4 Q X X X X X

58. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC 1 2 3 4 Q X X X X X

59. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC
Domain wipeout for x3 1 Q X X X 2 X X 3 X X 4 X X

60. MAC on 4-queens x1 x2 x3 x4 Try: x1  1 Enforce AC Backtrack 1 2 3 4 Q

61. MAC on 4-queens x1 x2 x3 x4
Try:
x1  1
Enforce AC
Backtrack 1 Q 2 3 4

62. MAC on 4-queens x1 x2 x3 x4
Try:
x1  2 1 2 Q 3 4

63. MAC on 4-queens x1 x2 x3 x4
Try:
x1  2
Enforce AC 1 2 Q 3 4

64. MAC on 4-queens x1 x2 x3 x4 Try: x1  2 Enforce AC 1 2 3 4 X Q X X X X

65. MAC on 4-queens x1 x2 x3 x4 Try: x1  2 Enforce AC 1 2 3 4 X Q X X X X

66. MAC on 4-queens x1 x2 x3 x4 Try: x1  2 Enforce AC 1 2 3 4 X Q X X X X

67. MAC on 4-queens x1 x2 x3 x4 Try: x1  2 Enforce AC 1 2 3 4 X X Q X X X

68. Each var has single value
MAC on 4-queens x1 x2 x3 x4
Try:
x1  2
Each var has single value
Gives solution 1 X X 2 Q X X X 3 X X 4 X X

69. Each var has single value
MAC on 4-queens x1 x2 x3 x4
Try:
x1  2
Each var has single value
Gives solution 1 X Q X 2 Q X X X 3 X X Q 4 Q X X

70. Each var has single value
MAC on 4-queens x1 x2 x3 x4
Try:
x1  2
Each var has single value
Gives solution 1 Q 2 Q 3 Q 4 Q

71. Heuristics for backtracking algorithms
Variable ordering (important)
what variable to branch on next
Value ordering (not yet so important)
given a choice of variable, what order to try values
Constraint ordering (not so important)
what order to propagate constraints
most likely to fail or cheapest propagated first

72. Variable ordering Domain dependent heuristics
Domain independent heuristics
Static variable ordering
fixed before search starts
Dynamic variable ordering
chosen during search

73. Variable ordering: Possible goals
Minimize expected depth of any branch
Encourage constraint propagation
Minimize expected number of branches
Minimize size of search space explored by backtracking algorithm
intractable to find “best” variable

74. Basic idea
Assign a heuristic value to a variable that estimates how difficult it is to find a satisfying value for that variable
Principle: most likely to fail first
or don’t postpone the hard part

75. Some variable ordering heuristics
minimum domain size (dom)
maximum degree (deg)
most constraining
plus: combining — tie break strategies
e.g., dom + deg, dom / deg

76. Value ordering All solutions One solution
value ordering not so important
One solution
if a solution exists, there exists a perfect value ordering

77. Value ordering: Intuition
Goal: minimize size of search space explored
Principle:
given that we have already chosen the next variable to instantiate, choose first the values that are most likely to succeed

78. Comparing algorithms How do BT, FC and MAC compare?
Theoretical comparison based on size of their search tree
MAC beats FC beats BT
Experimental comparison
MAC is more expensive at each node!

79. Theoretical comparison
Alg1 dominates Alg2 (written Alg1 >= Alg2) iff
given same fixed variable and value ordering
Alg1 visits search node implies Alg2 also visits this node
Alg1 strictly dominates Alg2 (written Alg1 > Alg2) iff
Alg1 domiantes Alg2
it visits fewer nodes on at least one problem

80. Theoretical comparison
MAC > FC > BT
Homework exercise!
Exist problems on which search trees are exponentially smaller
But do such problems turn up in practice?
Only experiment will tell you if MAC is better than FC for your class of problems
Usually it is!

81. Limited discrepancy search
Often first decision is wrong
Depth-first search (used in BT/FC/MAC) can take long time to undo this decision
E.g. in 4-queens, first decision was x1=1
But first solution has x1=2

82. Limited discrepancy search
Explore search tree in order of number of discrepancies
“discrepancy” = branch against heuristic
LDS search goes in waves:
0th iteration, explore heuristic branch
1st iteration, explore all branches with 1 branch against heuristic
2nd iteration, explore all branches with 2 branches agaisnt heuristic …

83. LDS
0th iteration
Follow heuristic to root (or backtrack point)

84. LDS
1st iteration
Follow all banches that go against heuristic once

85. LDS
2nd iteration
Follow all banches that go against heuristic twice

86. Non-chronological backtracking
Backjumping (BJ)
Conflict directed backjumping (CBJ)
Key idea in each of these is to backtrack to a “relevant” variable, decision that led to failure
Ignore decisions that didn’t contribute to the failure
Can combine with FC or MAC
E.g. FC-CBJ and MAC-CBJ

87. Backjumping
For each var xi, keep index of deepest var which xi performed consistency check
If xi has no consistent values, jump back to that var

88. Backjumping
For each var xi, keep index of deepest var which xi performed consistency check
If xi has no consistent values, jump back to that var
Only last backtrack down branch is non-chronological

89. Conflict-directed backjumping
For each var xi, keep conflict set
Set of vars which ruled out values from xi
If xi has no consistent values, jump back to deepest var
Not only first backtrack can be non-chronological
Multiple “backjumps” across conflicts possible

90. Theoretical comparison
We can apply the same dominance ordering
MAC-CBJ > FC-CBJ > CBJ > BJ > BT
FC > BJ

91. Learning Another use for conflict sets
At end of branch, identify cause for failure
This is a conflict set
Heuristically add to set of constraints
Heuristics include size bounded and relevance bounded learning

92. Local search Initial assignment 1 2 Pick queen in conflict Q x1 x2 x3
Move to min. conflicts Q x1 x2 x3 x4 2
Pick queen in conflict Q x1 x2 x3 x4

93. Summary Search important part of constraint solving
Backtracking algorithms often perform inference/constraint propagation at each node
Exist various refinements of chronological backtracking like CBJ, LDS, …