W10

1. Path consistency
Arc consistence can help reduce the domains of variables.
However, sometimes it tasks no effect.
Path consistency:
A two-variable set {Xi, Xj} is path-consistent wrt variable Xm if, for every
assignment {Xi = a, Xj = b} consistent with the constraints on {Xi, Xj}, there is an
assignment to Xm that satisfies the constraints on {Xi, Xm} and {Xm, Xj}.

2. Example
In the map coloring problem: if we use only
2 colors {red, blue}
Constraint: X ≠ Y
Arc-consistent?
Yes, since with each color of X,
Path-consistent?
Consider {HaNoi = red, BacNinh = blue},

3. PC-2 algorithm can enforce path consistency.
It is very similar to AC-3 algorithm.

4. K-consistency
A general form of local consistency.
A CSP is k-consistent if, for any set of k − 1 variables and for any consistent assignment
to those variables, a consistent value can always be assigned to any kth variable.
Some special cases:
 1-consistency:
 2-consistency:
 3-consistency:

5. Global constraints
A global constraint is one involving an arbitrary number of variables
For example, the Alldiff constraint

6. Global constraints
Example 2: Atmost constraint: in the car assembly scheduling problem,
 We can enforce consistency by deleting the maximum value of any domain if i

7. Sodoku problem
No digit appears twice in any row, column, or 3 ×
3 box.
Image:
(Russell,
2016)

8. Sudoku puzzle as a CSP
Variables:
Domains:
Constraints:
Image:
(Russell,
2016)

9. Alldiff constraints of a sudoku problem:
Source: (Russell, 2016)

10. Solving a Sudoku puzzle
Approach 1: AC-3 algorithm
Directly reducing domains of variables using the
constraints.
Image:
(Russell,
2016)

11. Solving a Sudoku puzzle
Approach 2: Triplets
1. In an unit, find 3 squares
2. Remove these 3 numbers from domains of
Image:
(Russell,
2016)

12. Notes
The mentioned approaches can apply to any CSPs, not just Sodoku.

13. Solve CSPs using
Backtracking Search

14. Motivation
Sudoku problems can be solved by inference over constraints (constraints propagation).
But many other CSPs cannot be solved by inference alone;
We could apply depth-first search:


However,
NOTE: CSPs have a property
 Backtracking search

15. Ideas of Backtracking search
A depth first search that chooses values for one variable at a time and backtracks when
a variable has no legal values left to assign.
Main steps:
Backtracking search uses CSP components.


16. Backtracking search algorithm
Source:
(Russell,
2016)

17. Demo

18. Comments on backtracking search algorithm
Source:
(Russell,
2016)

19. 1. Select next variable and value
For the selection of next variable:
 Random selection:
For example,
 Minimum remaining-values heuristic:

For example,
 Degree heuristic:

For example,

20. 1. Select next variable and value
For the selection of next value:
 Random selection:
 Least-constraining-value heuristic:
For example:

21. 2. Inference
Propagation constraint to reduce domains of other
variables when having a new assignment (var = value).
 Forward checking:
For example,
Forward checking doesn’t detect all inconsistencies, for example
 Maintaining arc consistency:
PhuTho VinhPhuc BacNinh BacGiang
HaNoi = blue
HaiDuong =

22. 3. Backtracking strategies
 Chronological backtracking:

 Backjumping: store a conflict set – set of assignments conflicted with the value of
current variable.
For example,