Constraint Propagation

1
Constraint Propagation

• Artificial Intelligence
• CMSC 25000
• January 22, 2002

2
Agenda

• Constraint Propagation Motivation
• Constraint Propagation Example
• Waltz line labeling
• Constraint Propagation Mechanisms
• Arc consistency
• CSP as search
• Forward-checking
• Back-jumping
• Summary

3
Constraint Satisfaction Problems

• Very general Model wide range of tasks
• Key components
• Variables Take on a value
• Domains Values that can be assigned to vars
• Constraints Restrictions on assignments
• Constraints are HARD
• Not preferences Must be observed
• E.g. Cant schedule two classes same room, same
  time

4
Constraint Satisfaction Problem

• Graph/Map Coloring Label a graph such that no
  two adjacent vertexes same color
• Variables Vertexes
• Domain Colors
• Constraints If E(a,b), then C(a) ! C(b)

5
Constraint Satisfaction Problems

• Resource Allocation
• Scheduling N classes over M terms, 4
  classes/term
• Aircraft at airport gates
• Satisfiability e.g. 3-SAT
• Assignments of truth values to variables such
  that 1) consistent and 2) clauses true

6
Constraint Satisfaction Problem

• N-Queens
• Place N queens on an NxN chessboard such that
  none attacks another
• Variables Queens (1/column)
• Domain Rows
• Constraints Not same row, column, or diagonal

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N-queens
Q3
Q1
Q4
Q2
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Constraint Satisfaction Problem

• Range of tasks
• Coloring, Resource Allocation, Satisfiability
• Varying complexity E.g. 3-SAT NP-complete
• Complexity Property of problem NOT CSP
• Basic Structure
• Variables Graph nodes, Classes, Boolean vars
• Domains Colors, Time slots, Truth values
• Constraints No two adjacent nodes with same
  color,
• No two classes in same time, consistent,
  satisfying asst

9
Constraint Propagation Method

• For each variable,
• Get all values for that variable
• Remove all values that conflict with ALL adjacent
• AFor each neighbor, remove all values that
  conflict with ALL adjacent
• Repeat A as long as some label set is reduced

10
Constraint Propagation Example

• Image interpretation
• From a 2-D line drawing, automatically determine
  for each line, if it forms a concave, convex or
  boundary surface
• Segment image into objects
• Simplifying assumptions
• World of polyhedra
• No shadows, no cracks

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CSP Example Line Labeling

• 3 Line Labels
• Boundary line regions do not abut gt
• Arrow direction right hand side walk
• Interior line regions do abut
• Concave edge line _
• Convex edge line
• Junction labels
• Where line labels meet

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CSP Example Line Labeling

• Simplifying (initial) restrictions
• No shadows, cracks
• Three-faced vertexes
• General position
• small changes in view-gt no change in junction
  type
• 42 labels for 2 line junction L
• 43 3-line junction Fork, Arrow, T
• Total 208 junction labelings

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CSP Example Line Labeling

• Key observation Not all 208 realizable
• How many? 18 physically realizable

All Junctions
L junctions Fork Junctions T junctions
Arrow junctions


_

_

_

_



_
_
_
_

_
_
_
_

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CSP Example Line Labeling

• Label boundaries clockwise
• Label arrow junctions

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CSP Example Line Labeling

• Label boundaries clockwise

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CSP Example Line Labeling

• Label fork junctions with s

17
CSP Example Line Labeling

• Label arrow junctions with -s

_


_


_
_

_
_
_



_



_



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Waltzs Line Labeling

• For each junction in image,
• Get all labels for that junction type
• Remove all labels that conflict with ALL adjacent
• AFor each neighbor, remove all labels that
  conflict with ALL
• Repeat A as long as some label set is reduced

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Waltz Propagation Example
C
D
X
B
X
A
X
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Waltzs Line Labeling

• Full version
• Removes constraints on images
• Adds special shadow and crack labels
• Includes all possible junctions
• Not just 3 face
• Physically realizable junctions
• Still small percentage of all junction labels
• O(n) n lines in drawing

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Constraint Propagation Mechanism

• Arc consistency
• Arc V1-gtV2 is arc consistent if for all x in D1,
  there exists y in D2 such that (x,y) is allowed
• Delete disallowed xs to achieve arc consistency

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Arc Consistency Example

• Graph Coloring
• Variables Nodes
• Domain Colors (R,G,B)
• Constraint If E(a,b), then C(a) ! C(b)
• Initial assignments

X
R,G,B
B
R,B
Z
Y
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Arc Consistency Example
Arc Rm Value
Assignments X G Y R Z B
X -gt Y None
X -gt Z XB
gt
Y-gt Z YB
X -gt Y XR
X -gt Z None
Y -gt Z None
X
R,G,B
B
R,B
Z
Y
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Limitations of Arc Consistency
X
G,B

• Arc consistent
• No legal assignment
• Arc consistent
• gt1 legal assignment

G,B
G,B
Z
Y
X
R,G
G,B
G,B
Z
Y
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CSP as Search

• Constraint Satisfaction Problem (CSP) is a
  restricted class of search problem
• State Set of variables assignment info
• Initial State No variables assigned
• Goal state Set of assignments consistent with
  constraints
• Operators Assignment of value to variable

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CSP as Search

• Depth Number of Variables
• Branching Factor Domain
• Leaves Complete Assignments
• Blind search strategy
• Bounded depth -gt Depth-first strategy
• Backtracking
• Recover from dead ends
• Find satisfying assignment

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CSP as Search
No!
No!
No!
No!
No!
Yes!
No!
Yes!
No!
No!
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Backtracking Issues

• CSP as Search
• Depth-first search
• Maximum depth of variables
• Continue until (partial/complete) assignment
• Consistent return result
• Violates constraint -gt backtrack to previous
  assignment
• Wasted effort
• Explore branches with no possible legal
  assignment

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Backtracking with Forward Checking

• Augment DFS with backtracking
• Add some constraint propagation
• Propagate constraints
• Remove assignments which violate constraint
• Only propagate when domain 1
• Forward checking
• Limit constraints to test

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BacktrackingForward Checking
No!
No!
Yes!
Yes!
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Heuristic CSP

• Improving Backtracking
• Standard backtracking
• Back up to last assignment
• Back-jumping
• Back up to last assignment that reduced current
  domain
• Change assignment that led to dead end

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Heuristic CSP

• Backtracking
• Pick first/leftmost node to expand
• Heuristic CSP
• Most constrained variable
• Minimize branching factor
• Least constraining value
• Pick value that removes fewest values from other
  vars
• Improve chance of finding SOME assignment
• Can increase feasible size of CSP problem

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Iterative Improvement

• Alternate formulation of CSP
• Rather than DFS through partial assignments
• Start with some complete, valid assignment
• Search for optimal assignment wrt some criterion
• Example Traveling Salesman Problem
• Minimum length tour through cities, visiting each
  one once

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Iterative Improvement Example

• TSP
• Start with some valid tour
• E.g. find greedy solution
• Make incremental change to tour
• E.g. hill-climbing - take change that produces
  greatest improvement
• Problem Local minima
• Solution Randomize to search other parts of
  space
• Other methods Simulated annealing, Genetic algs

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CSP Summary

• General problem structure
• Variables Domain values to assign
• Constraints Restrictions on valid assignments
• N-Queens Graph-coloring 3-Sat Scheduling
• Constraint propagation mechanisms
• Arc consistency
• Search model BacktrackForward-checking
• Heuristics Back-jump Variablevalue selection
• Iterative improvement reformulation