CSP: Definition, Creation, and Algorithms

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CSP Definition, Creation, and Algorithms

• Mr. Tianbing Lin t.lin_at_ieee.org
• Dr. Goodwin Scott
• University of Windsor

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Outline

• Introduction of Constraint Satisfaction Problem
• Algorithms Backtracking, Forward Checking, Back
  Jump, Dynamic Backtracking
• Evaluation Random CSP, N-Queen
• Experimental Result
• Question?

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Introduction of CSP

• Given a set of variables Xx1, x2xn and a
  finite set Di of possible values (its domain) for
  every variable.
• Given some constraints restricting the values
• How to assign values for all the variables so
  that all the constraints are satisfied?

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Introduction of CSP N-Queen

• Variables Q1, Q2, Q3, Q4
• Domain 1, 2, 3, 4
• Constraints
• QiltgtQj Not in the same row
• Qi-Qjltgti-j Not in diagonal
• Extentional Constraints
• (Q1, Q2) (1,3), (1,4), (2, 4), (3,1), (4, 1),
  (4, 2)
• (Q1, Q3)

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Introduction of CSP Graph Coloring

• Paint the graph with different colors
• 3 Color?

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Algorithms Overview

• Systematic Algorithms Backtracking, Forward
  Checking
• Search the whole solution space systematically
• In most cases, slow
• Heuristical Algorithms Hill-climbing, Simulated
  Annealing, Genetic Algorithm
• In some cases, fast
• Might not get solution at all.

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Algorithms Backtracking
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Algorithms Forward Checking
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Algorithms Back Jumping

• if we find all the values of Variable C are
  invalid because they are conflict with the value
  of Variable A, then we don't need to change the
  value of B to check C again. We can directly jump
  to another value of Variable A, skip other values
  of B (under A1 value).
• N-Queen problem can not demonstrate this
  algorithm.

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Algorithms Dynamic Backtracking

• When we back jump to A from C, we reset value of
  B.
• Do we really have to reset B, if its not
  conflict with A or C?
• We dynamically change the order of variables
  Before, A-gtB-gtC. Now B-gtA-gtC

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Algorithms Dynamic Backtracking

• Example of Graph Coloring
• Weve finished painting the 3 slides in the north
  and 2 slides in the south.
• We dont need to change the value of south slides.

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Evaluation N-Queen

• N-Queen is a nature test scheme.
• Normally a PC can solve 30-Queen problem in
  reasonable time using above algorithms.
• Drawback Every assignment create (almost) same
  number of no-good for other variables.

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Evaluation Random CSP

• Randomized constraints can be created according
  number of variables, average number of
  constraints, average tightness.
• Use average solving time to compare different
  algorithms.

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Experimental Results
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Experimental Results Sort by Variable Number
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Experimental Results Sort by Tightness
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Considering Tightness

• Constraint density cant assure the tightness of
  the CSP. For example, we have CSP A and CSP B
  (domain size 3, 3 variables)
• Density of Constraint 1 and Density of Constraint
  2 are 33.3. Solution is empty, so the tightness
  is 0/27

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Considering Tightness

• Density of Constraint 1 and Density of Constraint
  2 are 33.3, Can have 9 solutions(1, 1, 1) (1,
  1, 2) (1, 1, 3) (2, 1, 1) (2, 1, 2) (2, 1, 3) (3,
  1, 1) (3, 1, 2) (3, 1, 3), which means the
  tightness is 9/2733.

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Considering Tightness

• The smallest constraint density of all the
  constraints is the maximum value of tightness the
  CSP can have. The exact value of the tightness
  depends on the relation of the constraint tables.

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Reference

• 1 Barták, R., Constraint Programming In
  Pursuit of the Holy Grail, in Proceedings of the
  Week of Doctoral Students (WDS99), Part IV,
  MatFyzPress, Prague, June 1999, pp. 555-564.
• 2 Malek Mouhoub, class notes of Artificial
  Intelligence.
• 3 Prosser, P., Binary constraint satisfaction
  problems Some are harder than others,
  Proceedings ECAI-94 (11th European Conference on
  Artificial Intelligence)
• 4 White, S, Enhancing Knowledge Acquisition
  with Constraint Technology, PhD Thesis,
  University of Aberdeen
• 5 Pedro Meseguer, CSP Constraint Programming
• 6 Joe Culberson and Toby Walsh , Tightness of
  Constraint Satisfaction Problems
• 7 Peter van Beek and Rina Dechter, Constraint
  Tightness and Looseness versus Local and Global
  Consistency

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Thank you

• Question?