Constraint Satisfaction Problems

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Constraint Satisfaction Problems

• Russell and Norvig Chapter 5.1-3

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Intro Example 8-Queens
Generate-and-test, 88 combinations
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Intro Example 8-Queens
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What is Needed?

• Not just a successor function and goal test
• But 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

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Constraint Satisfaction Problem

• 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

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Example 8-Queens Problem

• 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, j?i
• Xi ki, Xj kj ?i-j ? ki - kj
• for all j 1 to 8, j?i

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Example 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
• WA?NT, WA?SA, NT?SA, NT?Q, SA?Q, SA?NSW,
  SA?V,Q?NSW, NSW?V

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Example Task Scheduling
T1 must be done during T3 T2 must be achieved
before T1 starts T2 must overlap with T3 T4 must
start after T1 is complete
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Constraint Graph

• Binary constraints

Two variables are adjacent or neighbors if
they are connected by an edge or an arc
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CSP as a Search Problem

• Initial state empty assignment
• Successor function a value is assigned to any
  unassigned variable, which does not conflict with
  the currently assigned variables
• Goal test the assignment is complete
• Path cost irrelevant

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Remark

• 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

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Backtracking example
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Backtracking example
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Backtracking example
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Backtracking example
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Backtracking 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()

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Improving backtracking efficiency

• Which variable should be assigned next?
• In what order should its values be tried?
• Can we detect inevitable failure early?

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Most constrained variable

• Most constrained variable
• choose the variable with the fewest legal values
• a.k.a. minimum remaining values (MRV) heuristic

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Most constraining variable

• Tie-breaker among most constrained variables
• Most constraining variable
• choose the variable involved in largest of
  constraints on remaining variables

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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

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Forward 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 Ys domain
  any value that is inconsistent with v

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Forward checking
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Forward checking
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Forward checking
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Forward checking
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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!

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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.

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• 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.

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Example

• Let 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_y3,5,6

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Arc consistency

• Simplest form of propagation makes each arc
  consistent
• X ?Y is consistent iff
• for every value x of X there is some allowed y

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Arc consistency

• Simplest form of propagation makes each arc
  consistent
• X ?Y is consistent iff
• for every value x of X there is some allowed y

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Arc consistency

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

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Arc consistency

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

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Example domain reduction

• Consider constraints
• X gt Y, Y gt Z
• Domains
• Dx Dy Dz 1,2,3
• - 1 in Dx is removed by maintaining arc
  consistency, w.r.t. the constraint X lt Y.
• - You work out the rest. The resulting domains
  are
• Dx 1, Dy 2, Dz 3
• No search is needed

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General CP for Binary Constraints

• 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 Ys domain is non-empty then STACK(Y,Q)
• Else return false

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Complexity Analysis of AC3

• 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

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Solving 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.

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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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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

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Summary

• Constraint Satisfaction Problems (CSP)
• CSP as a search problem
• Backtracking algorithm
• General heuristics
• Forward checking
• Constraint propagation (Arc consistency)
• Interweaving CP and backtracking