Constraint Satisfaction

1
Constraint Satisfaction

• CPSC 386 Artificial Intelligence
• Ellen Walker
• Hiram College

2
Constraint Satisfaction Problem

• Variables X1, X2, XN
• Constraints C1, C2, CN
• Often mathematical e.g. X1X20
• Domains D1, D2, DN (possible values for
  variables)
• A solution assigns values to variable (X1V1,
  X2V2 etc.) so all constraints are satisfied
• A partial solution (or consistent assignment)
  doesnt 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)

3
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

4
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.

5
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)
• DE Y or DE10Y
• NR (Y-(DE))/10 E (or 10E) (etc)
• Solution
• One technique trial and error (with
  backtracking)
• Set M1, that constrains S to be 8 or 9 and 0
  cant be 1, so it must be 0

6
Consistency

• Node Consistency
• Every value in the variables domain meets all
  unary constraints
• Arc Consistency
• Every value in the variables 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

7
Arc Consistency Algorithm AC-3

• Initialize a queue (really a set) of all arcs
  (pairs of constrained nodes) in the problem
• While the queue isnt 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)

8
Arc Consistency Isnt 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)

9
Line Labeling
Label each line in the picture (left) Constraints
based on vertices (L and arrow vertices
shown) Both ends of the segment must have the
same label!
10
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 doesnt matter
• Known solution depth ( of variables)
• Appropriate search is depth-limited DFS

11
Backtracking Search for CSPs

• 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

12
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

13
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

14
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)

15
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 CSPs e.g.
  N-queens problem!
• Does depend (like all hill climbing) on initial
  state though.