An Introduction to Artificial Intelligence
Lecture 5: Constraint Satisfaction Problems
Ramin Halavati (halavati@ce.sharif.edu)
In which we see how treating states as more than just black boxes
leads to the invention of a range of powerful new search methods
and a deeper understanding of problem structure and complexity.
Overview
• States/Goal(s)
– Black Box vs. Structured
– General Purpose Heuristics
What is Constraint Satisfaction?
• A set of variables: X1,X2,…,Xn.
• A set of constraints: C1,C2,…,Cm.
• Each variable, a domain: D1,D2,…,Dn.
• State: An assignment of values to Xis.
• Consistent: Obeying all constraints.
• Complete: All variables having values.
• Solution: A complete, consistent state.
• Objective Function: Extra!
Real World Samples of CSP
• Assignment problems
– e.g., who teaches what class
–
• Timetabling problems
•
– e.g., which class is offered when and where?
–
• Transportation scheduling
•
• Factory scheduling
CSP, Sample
• To assign one of {red, green, blue} to each of 7
variables: WA, BT, SA, Q, NSW, V, T.
• Constraints:
– WA ≠ NT,WA ≠ SA ,
NT ≠ Q, SA ≠ Q , …
• Solution:
– {WA = red , NT = green,
Q = red , NSW = green, …
Formulation
• Initial State: { }
• Successor Function: Value assignment to one
variable with no conflict.
• Goal Test: Consistent, Complete, assignment.
• Path Cost: 1 for each step.
Constraint Types
• Constraint Types:
– Unary / Binary / More…
• Auxiliary Variables
• Variables: F T U W, R O X1 X2 X3
• Constraints: All Different (F, T, U, W, R, O)
•
–
–
–
–
–
–
O + O = R + 10 · X1
X1 + W + W = U + 10 · X2
X2 + T + T = O + 10 · X3
X3 = F, T ≠ 0, F ≠ 0
TWO
+TWO
FOUR
Constraint Graph
Recursive Back Track Algorithm
• Recursive Back Track( assignment , CSP )
If assignment is complete, return assignment.
Var  Select an unassigned variable.
For each Value in Domain (Var),
if Consistent( assignment + Var  Value ) Then
result  RBT (assignment + Var  Value )
if ( result ≠ failure ) return result.
Return failure.
Major Questions…
1. Which variable to assign.
2. What to assign.
3. How to prevent further repetitions of a failure.
Minimum Remaining Values (MRV)
• To choose one which is most likely to fail
• Immediate recognition
of failure
Benchmark
Problem
Backtracking
Backtracking + MRV
USA 4-Coloring
> 1000 K
> 1000 K
N-Queens (2-50)
> 40000 K
13500 K
3859 K
1K
Zebra
Most Constraining Variable (MCS)
• To choose one who interferes the others most!
• Major reduction in
3
3
branching factor.
2
5
3
2
0
Least Constraining Value (LCS)
• To choose a value which puts minimum constraint on
other variables.
• To leave maximum
flexibility.
Forward Checking
• Keep track of remaining legal values for unassigned variables
• Terminate search when any variable has no legal values
Benchmark
Problem
BT +
Forward
FC +
MRV
Checking
MRV
>1M
>1M
2K
60
N-Queens
> 40 M
13.5 K
>40 M
817 K
Zebra
3859 K
1K
35 K
500
USA
BT
Constraint Propagation
• How to find fast that implications of one variable on
other variables.
Arc Consistency
• AB is consistent if for each remaining value in
domain of A, there may be a consistent value in
domain of B.
– Consistent:
• SANSW, NSWV,…
– Not C.:
• NSWSA, NTSA,…
Arc Consistency Checking Algorithm (AC-3!)
• AC3( csp )
Fill queue with all available arcs of csp.
While queue not empty,
(Xi,Xj) RemoveHead( queue )
If RemoveInconsistentValues(Xi,Xj) Then
for each Xk in Neighbor (Xi) do
Add (Xk,Xi) to queue.
• Remove Inconsistent Values (Xi,Xj)
removed  false.
For each x in Domain (Xi)
if no allowed value in Domain (Xj) after assignment of x to Xi,
delete x from Domain (Xi); removed  true.
Return removed.
Special Constraints…
• All Different
– Sort remaining variables based on their number of choices…
• Resource Constraints
– Checking the sums, …
• Flight271 [0,165] , Flight272 [0,385]
• Flight271+Flight272 = 420
Flight271 [35,165] , Flight272 [255,385]
• Bounds Propagation
Intelligent Backtracking
1. Q  Red
1
2. NSW  Green
3. V  Blue
4. T  Red
2
?
5. SA  ?
•
Chronological Backtracking
3
4
Back Jumping
1. Q  Red
2. NSW  Green
1
3. V  Blue
4. T  Red
5. SA  ?
•
2
?
3
Conflict Set: {Q, V, NSW}
4
NOTE: Back Jumping doesn’t help Forward Checking.
Local Search for Constraint Satisfaction Problems
• Min Conflict Algorithm
– Create a random complete state.
– For a fixed number of times
• If current state is consistent, return.
• Choose a random variable v, and change assignment of v to a
value that causes minimum conflict.
• Scheduling Hubble’s weekly observation:
– From 3 weeks to 10 minutes!
• Extra advantage: Online Updating
Benchmark
Problem
BT +
Forward
FC +
Min
MRV
Checking
MRV
Conflicts
>1M
>1M
2K
60
64
N-Queens
> 40 M
13.5 K
>40 M
817 K
4K
Zebra
3859 K
1K
35 K
500
2K
USA
BT
Heuristics based on Structure
• Sub Problems
– Finding Connected Components
• Constraint Trees
A
E
B
C
D
F
Heuristics based on Structure
A
•
E
B
Constraint Trees:
D
C
F
– Order nodes.
A
– From last to first, remove all
1
values from domain of parent
2
B
which may violate arc-consistency.
– From first to last, assign a
remaining value.
3
C
4
D
5
F
6
E
How to convert Constraint Graph to Constraint Tree
• Cut-Set Conditioning
– To select a set of nodes that subtracting them results in a tree.
– To check the rest for all valid assignments of this set.
How to convert Constraint Graph to Constraint Tree
•
Tree Decomposition
Tree Decomposition
•
Include all variables
•
Each constraint must be in at least one sub problem.
•
If a variable is in two sub-probs, it must be in all subprobs along the path.
Summery…