Constraint Satisfaction
CPSC 386 Artificial Intelligence
Ellen Walker
Hiram College
Constraint Satisfaction Problem
• Variables: X1, X2, … XN
• Constraints: C1, C2, … CN
– Often mathematical e.g. X1+X2=0
• Domains: D1, D2, … DN (possible values for
variables)
• A solution assigns values to variable (X1=V1, X2=V2
etc.) so all constraints are satisfied
• A partial solution (or consistent assignment) doesn’t
violate any constraints, but may leave some variables
unassigned
• If there are multiple solutions, choose the one that
maximizes an objective function (e.g. contiinuous
problems)
Domain Types
• Finite domain (fixed number of variables, possible
assignments)
– Depth is #variables, breadth is #assignments/variable
• Discrete variables with infinite domains (e.g. integers,
strings)
– Cannot search by enumeration, need a language to
represent constraints
• Continuous variables (e.g. real numbers)
– Linear programming and other continuous optimization
techniques
Example Problems
• Map Coloring (see text)
• Course / room scheduling
– Each course has a room
– Each room has no more than one course at a
given time
– Additional constraints (e.g. size, projector)
• Cryptarithmetic (see text)
• Line labeling
• Sudoku, Kakuro, KenKen, etc.
A Cryptarithmetic Problem
• SEND + MORE = MONEY
• Constraints
–
–
–
–
S≠E≠N≠D≠M≠O≠R≠Y (rules)
M=1
(It is a carry out of the high-order digit)
D+E = Y or D+E=10+Y
N+R+ (Y-(D+E))/10 = E (or 10+E) (etc)
• Solution
– One technique: trial and error (with backtracking…)
– Set M=1, that constrains S to be 8 or 9 and 0 can’t be 1, so it
must be 0…
Consistency
• Node Consistency
– Every value in the variable’s domain meets all
unary constraints
• Arc Consistency
– Every value in the variable’s domain meets all
binary constraints (with at least one value in the
relevant neighbor)
• If not consistent, remove value(s) from
domain(s) until it is.
– Book calls this “applying {Node, Arc} consistency
Arc Consistency Algorithm AC-3
• Initialize a queue (really a set) of all arcs
(pairs of constrained nodes) in the problem
• While the queue isn’t empty:
–
–
–
–
Remove an arc from the queue
Remove values that do not satisfy the constraint
If domain is empty return false
If domain changed, then add pairs of this node
and all neighbors (except the one just checked) to
the queue
• Return true (domains reduced as much as
possible)
Arc Consistency Isn’t Enough
• Consider 2-coloring problem
– Color Texas Flag with 3 colors, no two that touch
and are the same.
– Arc consistency does not reveal the problem
– Need Path Consistency: for every pair of
consistent elements, there is a third element so
that first and third are consistent, and second and
third are also consistent. (I.e. there is a path of
consistency across two constraint pairs)
Line Labeling
1
Label each line in the
picture (left)
2
5
3
4
>
–
>
+
>
–
+
<
Both ends of the
segment must have the
same label!
+
–
+
>
Constraints based on
vertices (“L” and “arrow”
vertices shown)
<
+
<
<
+
–
>
>
–
Constraint Problem as Search
• Formulation
– Initial state: no assignments
– Successor function: assign a value to an unassigned
variable without violating constraints
– Goal test: all variables are assigned
– Path cost: constant for each step
• Order of assignments doesn’t matter
• Known solution depth (# of variables)
• Appropriate search is depth-limited DFS
Backtracking Search for CSP’s
Pick an unassigned variable
For each possible value that meets constraints
Assign the value
If all variables are assigned, return the solution
Else
Recursively solve the rest of the problem
If the recursion succeeded, return the solution
Return failure
Searching More Efficiently
• Choose the most constrained variable (fewest legal
values) to assign next
– # of iterations of loop is minimized
– “MRV” Minimum Remaining Values heuristic
• In case of tie, choose the variable that is involved in
the most constraints (highest degree)
• Once a variable is chosen, choose the least
constraining value, i.e. rule out the fewest future
choices
Constraint Propagation
• Forward checking
– Whenever a variable is assigned, delete inconsistent values
from other domains
– Works well with MRV heuristic
• Maintaining Arc Consistency (MAC)
– Consider results of constraints between variables affected by
forward checking
– Needs to be recursively applied, because removal could
make a future arc inconsistent; after each change use AC3
starting with only the neighbors of changed nodes.
– Tradeoff between consistency checking (takes time) and
overall search time, but polynomial time is likely impossible
Chronological vs. Intelligent
Backtracking
• Chronological Backtracking
– Undoes most recent decision
• Intelligent Backtracking
– Undoes the decision that caused the problem in
the first place!
– Set of variables that caused the failure is “conflict
set”
– Backjumping: undo most recent variable in conflict
set (redundant with forward-checking; see text)
Local Search for CSP
• Hill Climbing with min conflicts as heuristic
• Every state is a complete assignment, count how
many variables are conflicted (violate constraints)
• At each step: choose a conflicted variable at
random, change its assignment to the value that
causes the minimum number of conflicts
• Surprisingly effective for many CSP’s e.g. N-queens
problem!
• Does depend (like all hill climbing) on initial state
though.