Constraint Networks

( slides courtesy of Natalia Flerova,based on slides courtesy of Rina Dechter)

Reading

Russell and Norvig

Chapter 6 (3rd ed.); Chapter 5 (2nd ed.)

For next week:

Chapter 7 (for Tuesday)

Chapter 8 (for Thursday)

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Good source of advanced information

Rina Dechter,

Constraint Processing,Morgan Kaufmann

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Outline

Constraint Satisfaction Problems (CSPs): Definition, and simple modeling examples

Representing constraints Consistency algorithms (arc-, path- and i-consistency) General search strategy

Look-ahead methods

A Bred greenred blackgreen redgreen blackblack greenblack red

Constraint Satisfaction

Example: map coloring Variables - countries (A,B,C,etc.)

Values - colors (e.g., red, green, black)

Constraints: A≠B, A≠D, D≠E , etc .

C

A

B

D

E

F

G

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Constraint Network; DefinitionA constraint network is: R=(X,D,C)X variables

D domains

C constraints

R expresses allowed tuples over scopes

A solution is an assignment to all variables that satisfies all constraints (join of all relations).

Tasks: consistency?, one or all solutions, counting, optimization

X={X1 , .. . ,X n}

D={D1 ,. . . , Dn}, Di={v1 , . .. vk }

C={C1 ,. .. C t },,, C i= S i , Ri

Example The 4-queen problem

Q

QQ

Q QQ

QQ

Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.

Standard CSP formulation of the problem:• Variables: each row is a variable.

QQ

QQ

X1

X 4

X 3

X 2

1 2 3 4

• Domains: Di={1,2,3,4}.

• Constraints: There are = 6 constraints involved:42( )

R12={1,31,4 2,4 3,1 4,1 4,2 }R13={1,2 1,4 2,12,3 3,2 3,4 4,1 4,3}R14={1,2 1,3 2,12,3 2,4 3,1 3,2 3,4 4,2 4,3 }R23={1,31,4 2,4 3,1 4,1 4,2 }R24={1,2 1,4 2,1 2,3 3,2 3,4 4,1 4,3 }R34={1,3 1,4 2,4 3,1 4,1 4,2 }

• Constraint Graph :

X1

X 2 X 4

X 3

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Example: configuration and design

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Configuration and design Want to build: recreation area, apartments, houses, cemetery,

dump

Recreation area near lakeSteep slopes avoided except for recreation areaPoor soil avoided for developmentsHighway far from apartments, houses and recreationDump not visible from apartments, houses and lakeLots 3 and 4 have poor soilLots 3, 4, 7, 8 are on steep slopesLots 2, 3, 4 are near lakeLots 1, 2 are near highway

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Outline

CSP: Definition, and simple modeling examples Representing constraints Consistency algorithms (arc-, path- and i-consistency) General search strategy

Look-ahead methods

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Constraint’s representations

Relation: allowed tuples

Algebraic expression:

Propositional formula:X Y 2≤ 10 , X ≠Y

a∨b ¬c

X Y Z1 3 22 1 3

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Operations with relations Intersection Union Difference Selection Projection Join Composition

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Figure 1.8: Example of set operations intersection, union, and difference applied to relations.

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selection, projection, and join operations on relations.

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Constraint Graphs: Primal, Dual and Hypergraphs

A (primal) constraint graph: a node per variable arcs connect constrained variables. A dual constraint graph: a node per constraint’s scope, an

arc connect nodes sharing variables =hypergraph

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Outline

CSP: Definition, and simple modeling examples Representing constraints Consistency algorithms (arc-, path- and i-consistency) General search strategy

Look-ahead methods

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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

32,1,

32,1, 32,1,

1 X, Y, Z, T 3X YY = ZT ZX T

X Y

T Z

32,1,

=

- infer constraints based on pairs of variables

Insures that every legal value in the domain of a single variable hasa legal match In the domain of any other selected variable

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1 X, Y, Z, T 3X YY = ZT ZX T

X Y

T Z

=

1 3

2 3

Arc-consistency

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Arc-consistency

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Revise for arc-consistency

Di Di∩π i Rij⊗ D j

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AC-1

O enk 3 Complexity (Mackworth and Freuder, 1986): e = number of arcs, n variables, k values (ek^2, each loop, nk number of loops), best-case = ek, Arc-consistency is:

ek 2

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AC-3

O ek 3 Complexity: Best case O(ek), since each arc may be processed in O(2k) e = number of arcs, n variables, k values

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Sudoku –Constraint Satisfaction

Each row, column and major block must be alldifferent

“Well posed” if it has unique solution: 27 constraints

2 34 62

•Variables: empty slots

•Domains = {1,2,3,4,5,6,7,8,9}

•Constraints: 27 all-different

•Constraint •Propagation

•Inference

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Path-consistency

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Path-consistency

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Revise-3

Complexity: O(k^3) Best-case: O(t) Worst-case O(tk) e = number of arcs, n variables, k values, t = tightness: number of tuples in the

constraint relation

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PC-1

Complexity: e = number of arcs, n variables, k values, t = tightness: number of tuples in the

constraint relation

O n 5 k 5

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PC-2

O n 3 k 5 Complexity: Optimal PC-4: (each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2) size of Q is

O(n^3 k^2), processing is O(k^3)) e = number of arcs, n variables, k values, t = tightness: number of tuples in the

constraint relation

O n 3 k 3

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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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I-consistency

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Outline

CSP: Definition, and simple modeling examples Representing constraints Consistency algorithms (arc-, path- and i-consistency) General search strategy

Look-ahead methods

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Backtracking algorithm

No matter how much we reason about the problem – we are still left with choices → need to search the space of possible choices.

Backtracking search – assign values to each variable in turn, making sure each assigned value is consistent with values assigned thus far.

Encounter variable with no domain value consistent with previous assignment – dead-end → backtrack – change the value of previous variables.

Backtracking

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

Search space and the effect of ordering

20 legal states1 dead-end leaf

48 legal states18 dead-end leaves

Z evenly divides x,y,l

Dependency on consistency level

After arc-consistency z=5 and l=5 are removed

After path-consistency R’_zx R’_zy R’_zl R’_xy R’_xl R’_yl

Backtrack-free network

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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Outline

CSP: Definition, and simple modeling examples Representing constraints Consistency algorithms (arc-, path- and i-consistency) General search strategy

Look-ahead methods

Look-ahead: value orderings

Intuition: Choose value least likely to yield a dead-end Approach: apply propagation at each node in the search tree

Forward-checking (check each unassigned variable separately)

Maintaining arc-consistency (MAC) (apply full arc-consistency)

Full look-ahead One pass of arc-consistency (AC-1)

Partial look-ahead directional-arc-consistency

Forward-checking

The most limited form of constraint propagation during search

Propagates the effect of a tentative value selection to each future variable separately

If domain of one of the future variables becomes empty – value is not selected → try next value.

Forward-checking example

Forward-checking

O ek 2 Complexity of selectValue-forward-checking at each node:

Arc-consistency look-ahead(Gashnig, 1977)

Applies full arc-consistency on all un-instantiated variables following each value assignment to the current variable.

Complexity: If optimal arc-consistency is used:

SelectValue-Arc-Consistency: The repeat...until loop – essentially AC-1 with some variables instantiated.

Maintaining-Arc-Consistency (MAC): Performs full arc-consistency after each domain value is rejected.

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Arc-consistency look-ahead:(maintaining arc-consistency MAC)

Full and partial look-ahead

Full looking ahead: Make one full pass through future variables

(remove repeat-until from selectValue-arc-consistency)

Partial look-ahead: Applies (similar-to) directional arc-consistency to

future variables. - future variables are only compared with those variables following them.

Complexity: also More efficient than MAC

O ek 3

Example of partial look-ahead