An Introduction to Artificial Intelligence

Lecture 5: Constraint Satisfaction ProblemsRamin 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+TWOFOUR

TWO+TWOFOUR

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

Zebra 3859 K 1 K

Most Constraining Variable (MCS)

• To choose one who interferes the others most!

• Major reduction in

branching factor.

55

33

33

33

22

22

00

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 BT +

MRV

Forward

Checking

FC +

MRV

USA > 1 M > 1 M 2 K 60

N-Queens > 40 M 13.5 K >40 M 817 K

Zebra 3859 K 1 K 35 K 500

Constraint Propagation

• How to find fast that implications of one variable on

other variables.

Arc Consistency

• AB is consistent if for each remaining value in

domain of A, there may be a consistent value in

domain of B.

– Consistent:

• SANSW, NSWV,…

– Not C.:

• NSWSA, NTSA,…

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

2. NSW Green

3. V Blue

4. T Red

5. SA ?

• Chronological Backtracking

1111

2222

3333

4444

????

Back Jumping

1. Q Red

2. NSW Green

3. V Blue

4. T Red

5. SA ?

• Conflict Set: {Q, V, NSW}

NOTE: Back Jumping doesn’t help Forward Checking.

1111

2222

3333

4444

????

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 BT +

MRV

Forward

Checking

FC +

MRV

Min

Conflicts

USA > 1 M > 1 M 2 K 60 64

N-Queens > 40 M 13.5 K >40 M 817 K 4 K

Zebra 3859 K 1 K 35 K 500 2 K

Heuristics based on Structure

• Sub Problems

– Finding Connected Components

• Constraint Trees

AA

BB

CC

DD

EE

FF

Heuristics based on Structure

• Constraint Trees:

– Order nodes.

– From last to first, remove all

values from domain of parent

which may violate arc-consistency.

– From first to last, assign a

remaining value.

AA

BB

CC

DD

EE

FF

AA

BB

CC DD

EEFF

11

33

22

55

44

66

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 sub-

probs along the path.

Summery…