Constraint Constraint Satisfaction Problems Satisfaction Problems Russell and Norvig: Chapter 5.1-3
Mar 31, 2015
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