Constraint Networks

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Constraint Networks Overview
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Suggested reading
Russell and Norvig. Artificial Intelligence
Modern Approach. Chapter 5.
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Good source of advanced information
Rina Dechter, Constraint Processing, Morgan
Kaufmann
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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Example The N-queens problem
The network has four variables, all with domains
Di 1, 2, 3, 4. (a) The labeled chess board.
(b) The constraints between variables.
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Example configuration and design
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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Operations with relations

• Intersection
• Union
• Difference
• Selection
• Projection
• Join
• Composition

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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Backtracking search
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Search space
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Backtracking
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The search space

• A tree of all partial solutions
• A partial solution (a1,,aj) satisfying all
  relevant constraints
• The size of the underlying search space depends
  on
• Variable ordering
• Level of consistency possessed by the problem

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Search space and the effect of ordering
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The effect of variable ordering
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Dependency on consistency level

• After arc-consistency z5 and l5 are removed
• After path-consistency
• R_zx
• R_zy
• R_zl
• R_xy
• R_xl
• R_yl

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Backtrack-free network
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Backtracking

• Complexity of extending a partial solution
• Complexity of consistent O(e log t), t bounds
  tuples, e constraints
• Complexity of selectvalue O(e k log t)

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A coloring problem example
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Backtracking Search for a Solution
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Backtracking search for a solution
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Backtracking Search for a Solution
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Backtracking Search for All Solutions
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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Improving Backtracking O(exp(n))

• Before search (reducing the search space)
• Arc-consistency, path-consistency, i-consistency
• Variable ordering (fixed)
• During search
• Look-ahead schemes
• Value ordering/pruning (choose a least
  restricting value),
• Variable ordering (Choose the most constraining
  variable)
• Look-back schemes
• Backjumping
• Constraint recording
• Dependency-directed backtracking

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Consistency methods

• Constraint propagation inferring new
  constraints
• Can get such an explicit network that the search
  will find the solution without dead-ends.
• Approximation of inference
• Arc, path and i-consistency
• Methods that transform the original network into
  a tighter and tighter representations

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Arc-consistency
- infer constraints based on pairs of variables
X
Y
?
3
2,
1,
3
2,
1,
1 ? X, Y, Z, T ? 3 X ? Y Y Z T ? Z X ??T

?
3
2,
1,
3
2,
1,
?
T
Z
Insures that every legal value in the domain of a
single variable has a legal match In the domain
of any other selected variable
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Arc-consistency algorithm

• domain of x domain of y

Arc is arc-consistent if for any
value of there exist a matching value of
Algorithm Revise makes an arc
consistent Begin 1. For each a in Di if there
is no value b in Dj that matches a then delete a
from the Dj. End. Revise is , k is the
number of value in each domain.
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Algorithm AC-3

• Begin
• 1. Q lt--- put all arcs in the queue in both
  directions
• 2. While Q is not empty do,
• 3. Select and delete an arc from the
  queue Q
• 4. Revise
• 5. If Revise cause a change then add to the queue
  all arcs that touch Xi (namely (Xi,Xm) and
  (Xl,Xi)).
• 6. end-while
• End
• Complexity
• Processing an arc requires O(k2) steps
• The number of times each arc can be processed is
  2k
• Total complexity is

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Sudoku Constraint Satisfaction

• Variables empty slots
• Domains 1,2,3,4,5,6,7,8,9
• Constraints
• 27 all-different

• Constraint
• Propagation
• Inference

2 34 6
2
Each row, column and major block must be
alldifferent Well posed if it has unique
solution 27 constraints
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Path-consistency
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Example before and after path-consistency

• PC-1 requires 2 processings of each arc while
  PC-2 may not
• Can we do path-consistency distributedly?

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The Effect of Consistency Level

• After arc-consistency z5 and l5 are removed
• After path-consistency
• R_zx
• R_zy
• R_zl
• R_xy
• R_xl
• R_yl

Tighter networks yield smaller search spaces
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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Forward-checking on Graph-coloring
FW overhead MAC overhead
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Propositional Satisfiability
Example party problem

• If Alex goes, then Becky goes
• If Chris goes, then Alex goes
• Query
• Is it possible that Chris goes to the party
  but Becky does not?

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Unit Propagation

• Arc-consistency for cnfs.
• Involve a single clause and a single literal
• Example (A, not B, C) (B)
  (A,C)

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Look-ahead for SAT(Davis-Putnam, Logeman and
Laveland, 1962)
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Look-ahead for SAT DPLLexample
(AVB)(CVA)(AVBVD)(C)
(Davis-Putnam, Logeman and Laveland, 1962)
Backtracking look-ahead with Unit propagation
Generalized arc-consistency
Only enclosed area will be explored with
unit-propagation
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Outline

• CSP Definition, and simple modeling examples
• Representing constraints
• Basic search strategy
• Improving search
• Consistency algorithms
• Look-ahead methods
• Look-back methods

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Look-back Backjumping / Learning

• Backjumping
• In deadends, go back to the most recent culprit.
• Learning
• constraint-recording, no-good recording.
• good-recording

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Backjumping

• (X1r,x2b,x3b,x4b,x5g,x6r,x7r,b)
• (r,b,b,b,g,r) conflict set of x7
• (r,-,b,b,g,-) c.s. of x7
• (r,-,b,-,-,-,-) minimal conflict-set
• Leaf deadend (r,b,b,b,g,r)
• Every conflict-set is a no-good

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A coloring problem
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Example of Gaschnigs backjump
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The cycle-cutset method

• An instantiation can be viewed as blocking cycles
  in the graph
• Given an instantiation to a set of variables that
  cut all cycles (a cycle-cutset) the rest of the
  problem can be solved in linear time by a tree
  algorithm.
• Complexity (n number of variables, k the domain
  size and C the cycle-cutset size)

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Tree Decomposition
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GSAT local search for SAT(Selman, Levesque and
Mitchell, 1992)

• For i1 to MaxTries
• Select a random assignment A
• For j1 to MaxFlips
• if A satisfies all constraint,
  return A
• else flip a variable to maximize the
  score
• (number of satisfied constraints
  if no variable
• assignment increases the score,
  flip at random)
• end
• end

Greatly improves hill-climbing by adding
restarts and sideway moves
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WalkSAT (Selman, Kautz and Cohen, 1994)
Adds random walk to GSAT

• With probability p
• random walk flip a variable in some
  unsatisfied constraint
• With probability 1-p
• perform a hill-climbing step

Randomized hill-climbing often solves large and
hard satisfiable problems
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More Stochastic Search Simulated Annealing,
reweighting

• Simulated annealing
• A method for overcoming local minimas
• Allows bad moves with some probability
• With some probability related to a temperature
  parameter T the next move is picked randomly.
• Theoretically, with a slow enough cooling
  schedule, this algorithm will find the optimal
  solution. But so will searching randomly.
• Breakout method (Morris, 1990) adjust the
  weights of the violated constraints

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