1. CSC5240 - Modeling with FD Constraints
Controlling Search
Where we examine how modeling and controlling search interact with finite domain constraints
CSC Modeling with FD Constraints

2. CSC5240 - Modeling with FD Constraints
Domains and Labeling
New requirement: need to define the domains of variables
arithmetic X :: [1,2,3,4] or X :: [1..4]
non-arithmetic X :: [red, blue, yellow]
multiple variables [X,Y,Z] :: [0..10]
If no domain is given a default is used (say [ ]
CSC Modeling with FD Constraints

3. CSC5240 - Modeling with FD Constraints
Domains Example - 1
Goal for the smugglers knapsack problem:
[W,P,C] :: [0..9], 4*W + 3*P + 2*C <= 9,
15*W + 10*P + 7*C >= 30.
The constraint solver returns unknown.
But how do we find a solution?
Invoke a complete (backtracking) solver
CSC Modeling with FD Constraints

4. CSC5240 - Modeling with FD Constraints
Domains Example - 2
Complete solvers for FD are expensive and should not be invoked every time a new constraint is added to the constraint store
Thus, most constraint solver provide only an incomplete solver
When necessary, however, the complete backtracking solver for FD can be invoked explicitly by the programmer, usually in the form of built-in labeling
CSC Modeling with FD Constraints

5. CSC5240 - Modeling with FD Constraints
Labeling - 1
Built-in predicate labeling invokes the complete constraint solver
labeling(Vs) takes a list of finite domain variables Vs and finds a solution
[W,P,C] :: [0..9], 4*W + 3*P + 2*C <= 9,
15*W + 10*P + 7*C >= 30, labeling([W,P,C]).
has solutions:
CSC Modeling with FD Constraints

6. Constrain-and-Generate
Typical form of a finite domain program
variables and domains are defined
constraints modeling problem are given
labeling is added to invoke complete solver
Minimization can be applied on labeling
[W,P,C] :: [0..9], 4*W + 3*P + 2*C <= 9,
15*W + 10*P + 7*C >= 30,
minimize(labeling([W,P,C]), -15*W-10*P-7*C).
CSC Modeling with FD Constraints

7. Send+More=Money Example - 1
Cryptarithmetic problem: each digit is different and the equation holds
[S,E,N,D,M,O,R,Y] :: [0..9],
S != 0, M != 0,
alldifferent_neq([S,E,N,D,M,O,R,Y]),
1000*S + 100*E + 10*N + D
*M + 100*O + 10*R + E
= 10000*M *O + 100*R + 10*E + Y,
labeling([S,E,N,D,M,O,R,Y]).
CSC Modeling with FD Constraints

8. Send+More=Money Example - 2
After domain declarations:
After then
alldifferent_neq adds disequations (no change)
final equation: (one propagation rule)
With the current domain:
CSC Modeling with FD Constraints

9. Send+More=Money Example - 3
Hence D(M) = [1..1]
Propagation continues arriving at
Note: 3 variables fixed and all domains reduced
Making use of the current domains of variables, labeling tries S=9. Then E=4 (this fails eventually), then E=5 which gives
CSC Modeling with FD Constraints

10. CSC5240 - Modeling with FD Constraints
Generate-and-Test
Methodology without constraints: first generate a valuation and then test if it is a solution
[S,E,N,D,M,O,R,Y] :: [0..9],
labeling([S,E,N,D,M,O,R,Y]),
S != 0, M != 0,
alldifferent_neq([S,E,N,D,M,O,R,Y]),
1000*S + 100*E + 10*N + D
*M + 100*O + 10*R + E
= 10000*M *O + 100*R + 10*E + Y.
This program requires choices before finding the solution, and the previous required 2
CSC Modeling with FD Constraints

11. CSC5240 - Modeling with FD Constraints
Labeling - 3
There are two choices made in labeling
choice of which variable to label first
choice of which value to try first
Default labeling
try variables in order of the given list
try value in order min to max (returned by dom)
We can program different strategies
CSC Modeling with FD Constraints

12. CSC5240 - Modeling with FD Constraints
Choice of Variable
Variable choice effects the size of derivation tree
Choose variable with smallest current domain (hence smallest possible answers)
Example
Labeling first tries S, M, O (need do nothing), then either E or N. Much better than Y first
CSC Modeling with FD Constraints

13. First-Fail Labeling - 2 Application of the first-fail principle
“To succeed, try first where you are most likely to fail”
CSC Modeling with FD Constraints

14. CSC5240 - Modeling with FD Constraints
First-Fail Labeling - 3
It is reasonable to assume that a variable involved in many constraints is more likely to cause failure when it is set to a value
Thus another application of the first-fail principle is to label first a variable that is involved in the most number of constraints
CSC Modeling with FD Constraints

15. CSC5240 - Modeling with FD Constraints
Choice of Domain Value
Value choice only effects order in which branches are explored
Problem specific knowledge may lead to finding a solution faster
Also important for optimization
Good solutions found earlier reduce search later
CSC Modeling with FD Constraints

16. CSC5240 - Modeling with FD Constraints
N-Queens Example Q1 Q2 Q3 Q4 1 2 3 4
Problem of placing N queens on an NN chessboard so that none can take another.
For large N solutions are more likely with values in the middle of variable domains
CSC Modeling with FD Constraints

17. CSC5240 - Modeling with FD Constraints
Labeling Efficiency
We can use both variable and value ordering together
Different number of choices for N-queens in different ranges
CSC Modeling with FD Constraints

18. CSC5240 - Modeling with FD Constraints
Domain Splitting - 1
Labeling doesn’t have to set a variable to a value---principle of least commitment
Simply has to reduce the variable domains
Domain splitting splits the current domain of chosen variable in half
Advantage: more search space is pruned when failure is detected
Disadvantage: less propagation
CSC Modeling with FD Constraints

19. CSC5240 - Modeling with FD Constraints
Domain Splitting - 3
The idea is to split recursively the domains of each variable in half until all domains are singletons.
Domain splitting is better than regular labeling when
the variable domains are large, and
the constraints involving the variables to be labeled can gain substantial consistency information from the split domain
This way, half of the variable’s domain may be eliminated
CSC Modeling with FD Constraints

20. CSC5240 - Modeling with FD Constraints
Search in MiniZinc
Search order defined using the variables that appear in the output statement
Search annotations allow you to specify search order
There are basic search annotations and advanced ones can be defined too
Read Chapter 6 of the MiniZinc tutorial
CSC Modeling with FD Constraints