Constraint Satisfaction Constraint Satisfaction ProblemsProblems

Russell and Norvig: Chapter 5.1-3

Intro Example: 8-QueensIntro Example: 8-Queens

Generate-and-test, 88 combinations

Intro Example: 8-QueensIntro Example: 8-Queens

What is Needed?What is Needed?

Not just a successor function and goal testBut also a means to propagate the constraints imposed by one queen on the others and an early failure test Explicit representation of constraints and constraint manipulation algorithms

Constraint Satisfaction Constraint Satisfaction ProblemProblem

Set of variables {X1, X2, …, Xn}Each variable Xi has a domain Di of possible values Usually Di is discrete and finite

Set of constraints {C1, C2, …, Cp} Each constraint Ck involves a subset of

variables and specifies the allowable combinations of values of these variables

Assign a value to every variable such that all constraints are satisfied

Example: 8-Queens Example: 8-Queens ProblemProblem

8 variables Xi, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms: Xi = k Xj k for all j = 1 to 8, ji Xi = ki, Xj = kj |i-j| | ki - kj|

for all j = 1 to 8, ji

Example: Map ColoringExample: Map Coloring

• 7 variables {WA,NT,SA,Q,NSW,V,T}• Each variable has the same domain {red, green, blue}• No two adjacent variables have the same value: WANT, WASA, NTSA, NTQ, SAQ, SANSW, SAV,QNSW, NSWV

WA

NT

SA

Q

NSWV

T

WA

NT

SA

Q

NSWV

T

Example: Task SchedulingExample: Task Scheduling

T1 must be done during T3T2 must be achieved before T1 startsT2 must overlap with T3T4 must start after T1 is complete

T1

T2

T3

T4

Constraint GraphConstraint Graph

Binary constraints

T

WA

NT

SA

Q

NSW

V

Two variables are adjacent or neighbors if theyare connected by an edge or an arc

T1

T2

T3

T4

CSP as a Search ProblemCSP as a Search Problem

Initial state: empty assignmentSuccessor function: a value is assigned to any unassigned variable, which does not conflict with the currently assigned variablesGoal test: the assignment is completePath cost: irrelevant

RemarkRemark

Finite CSP include 3SAT as a special case 3SAT is known to be NP-complete So, in the worst-case, we cannot expect to solve a finite CSP in less than exponential time

Backtracking example

Backtracking example

Backtracking example

Backtracking example

Backtracking AlgorithmBacktracking Algorithm

CSP-BACKTRACKING(PartialAssignment a) If a is complete then return a X select an unassigned variable D select an ordering for the domain of X For each value v in D do

If v is consistent with a then Add (X= v) to a result CSP-BACKTRACKING(a) If result failure then return result Remove (X= v) from a

Return failure

Start with CSP-BACKTRACKING({})

Improving backtracking efficiency

Which variable should be assigned next?

In what order should its values be tried?

Can we detect inevitable failure early?

Most constrained variable

Most constrained variable:choose the variable with the fewest

legal values

a.k.a. minimum remaining values (MRV) heuristic

Most constraining variable

Tie-breaker among most constrained variablesMost constraining variable: choose the variable involved in

largest # of constraints on remaining variables

Least constraining value

Given a variable, choose the least constraining value: the one that rules out the fewest values

in the remaining variables

Combining these heuristics makes 1000 queens feasible

Forward CheckingForward Checking After a variable X is assigned a value v,

look at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain any value that is inconsistent with v

Forward checking

Forward checking

Forward checking

Forward checking

Constraint propagation

Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures:

NT and SA cannot both be blue!

Definition (Arc consistency)

A constraint C_xy is said to be arc consistent w.r.t. x if for each value v of x

there is an allowed value of y.

Similarly, we define that C_xy is arc consistent w.r.t. y.

A binary CSP is arc consistent iff every constraint C_xy is arc consistent wrt x

as well as wrt y.

When a CSP is not arc consistent, we can make it arc consistent, e.g. by using AC3.

This is also called “enforcing arc consistency”.

ExampleLet domains be D_x = {1,2,3}, D_y = {3,4,5,6}A constaint C_xy = {(1,3),(1,5),(3,3),(3,6)}

C_xy is not arc consistent w.r.t. x, neither w.r.t. y. By enforcing arc consistency, we get reduced domains

D’_x = {1,3}, D’_y={3,5,6}

Arc consistency

Simplest form of propagation makes each arc consistentX Y is consistent ifffor every value x of X there is some allowed y

Arc consistency

Simplest form of propagation makes each arc consistentX Y is consistent ifffor every value x of X there is some allowed y

Arc consistency

If X loses a value, neighbors of X need to be rechecked

Arc consistency

Arc consistency detects failure earlier than forward checkingCan be run as a preprocessor or after each assignment

Example: domain reduction

Consider constraints: X > Y, Y > Z Domains: Dx = Dy = Dz = {1,2,3}

- 1 in Dx is removed by maintaining arc consistency, w.r.t. the constraint X < Y.

- You work out the rest. The resulting domains are

D’x = {1}, D’y = {2}, D’z = {3} No search is needed

General CP for Binary General CP for Binary ConstraintsConstraints

Algorithm AC3

contradiction false Q stack of all variables while Q is not empty and not contradiction do X UNSTACK(Q) For every variable Y adjacent to X do

If REMOVE-ARC-INCONSISTENCIES(X,Y) then

If Y’s domain is non-empty then STACK(Y,Q)

Else return false

Complexity Analysis of Complexity Analysis of AC3AC3

e = number of constraints (edges) (or n2 where n is the # of variables)

d = number of values per variable Each variable is inserted in Q up to d times REMOVE-ARC-INCONSISTENCY takes O(d2) time AC3 takes O(ed3) time to run

Solving a CSPSolving a CSP

Search: can find solutions, but must examine

non-solutions along the way

Constraint Propagation: can rule out non-solutions, but this is

not the same as finding solutions:

Interweave constraint propagation and search Perform constraint propagation at

each search step.

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

Local search for CSPs

Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned

To apply to CSPs: allow states with unsatisfied constraints operators reassign variable values

Variable selection: randomly select any conflicted variable

Value selection by min-conflicts heuristic: choose value that violates the fewest constraints

SummarySummary

Constraint Satisfaction Problems (CSP)CSP as a search problem Backtracking algorithm General heuristics

Forward checkingConstraint propagation (Arc consistency)Interweaving CP and backtracking