Constraint Satisfaction

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Constraint Satisfaction
R N Chapter 5
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Constraint Satisfaction Early Use
From http//www-2.cs.cmu.edu/awm/731/constraint07
-2x2.pdf
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The Waltz Algorithm
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More on Waltz Algorithm
The constraints come from the fact that a line
must have the same label at both ends.
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Crossword Puzzles
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Latin Squares
Using only the numbers 1, 2, 3, and 4, arrange
four sets of these numbers into a four-by-four
array so that no column or row contains the same
two numbers. The result is known as a Latin
square. Here are two examples of Latin squares of
order 4
1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1
1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3
http//www.sciencenews.org/20000506/mathtrek.asp
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Satisfiability
(P ? Q ? ?S) ? (A ? ? Q ? B) ? (P ? ? B ? ?C)
? (? A ? B ? C) ? (Q ? ?A ? ?C) Example
applications
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Real Examples
Scheduling What else?
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Algorithms for CSPs

• Branching
• State space usually incremental formulation
• Straight line
• State space usually complete-state formulation

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Cryptarithmetic
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PEAS description

• State space when formulated as a CSP variables,
  domain, constraints
• Incremental state space formulation
• Complete state formulation

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Cryptarithmetic
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Representing the constraints As formulas As a
graph Using special constraints Alldiff
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Cryptarithmetic Best First Search
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M 2
M 2
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Cryptarithmetic Best First Search
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M 2
M 2
M 1 S 2 O 3
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Cryptarithmetic Branching Algorithms

• Backtracking
• Minimum remaining values
• Degree heuristic
• Least-constraining value

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M 2
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Cryptarithmetic Constraint Propagation
Forward checking After assigning a value to a
variable X, look at all the variables connected
to X and prune inconsistent values. Constraint
propagation using arc consistency The
tradeoff how much time to spend propagating vs.
searching.
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Backtracking Chronological vs. Dependency
Directed
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Straight-Line Algorithms Min-Conflicts
function Min-Conflicts(csp, max_steps) returns a
solution or failure inputs csp, a constraint
satisfaction problem max_steps, the number of
steps allowed before giving up current ? an
initial assignment for csp for i 1 to
max_steps do if current is a solution for csp
then return current var ? a randomly chosen,
conflicted variable from Variablescsp value
? the value v for var that minimizes
Conflicts(var, v, current, csp) set var
value in current return failure
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Do Straight-Line Algorithms Work?
Does Min-Conflicts work?
From http//www.cs.mu.oz.au/303/slides/week03ah.p
df
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Do Straight-Line Algorithms Work?
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Another Win of Straight Line Algorithms
They can be used to perturb a solution that
ceases to be correct Most scheduling problems