Solving Constraint Satisfaction Problems: Arc Consistency and - - PDF document

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Solving Constraint Satisfaction Problems: Arc Consistency and Constraint Propagation

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Brian Williams 16.410 - 13 September 29th, 2003

Slides adapted from: 6.034 Tomas Lozano Perez and AIMA Stuart Russell & Peter Norvig

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Outline

• Recap: constraint satisfaction problems (CSP)
• Solving CSPs
• Arc-consistency and propagation
• Analysis of constraint propagation
• Search

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Solving CSPs

Solving CSPs involves some combination of:

• 1. Constraint propagation
• eliminates values that can’t be part of any

solution 2. Search

• explores valid assignments

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Arc Consistency

• Directed arc (Vi, Vj) is arc consistent if
• For every x in Di, there exists some y in Dj such that

assignment (x,y) is allowed by constraint Cij

• Or ∀x∈Di ∃y∈Dj such that (x,y) is allowed by constraint Cij

where

• ∀ denotes “for all”
• ∃ denotes “there exists”
• ∈ denotes “in”

Arc consistency eliminates values of each variable domain that can never satisfy a particular constraint (an arc). Vi Vj {1,2,3} {1,2} =

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Arc Consistency

• Directed arc (Vi, Vj) is arc consistent if
• ∀x∈Di ∃y∈Dj such that (x,y) is allowed by constraint Cij

Arc consistency eliminates values of each variable domain that can never satisfy a particular constraint (an arc). Vi Vj {1,2,3} {1,2} = Example: Given: Variables V1 and V2 with Domain {1,2,3,4} Constraint: {(1, 3) (1, 4) (2, 1)} What is the result of arc consistency?

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Achieving Arc Consistency via Constraint Propagation

• Directed arc (Vi, Vj) is arc consistent if

∀x∈Di ∃y∈Dj such that (x,y) is allowed by constraint Cij Arc consistency eliminates values of each variable domain that can never satisfy a particular constraint (an arc). Constraint propagation: To achieve arc consistency:

• Delete every value from each tail domain Di of each arc

that fails this condition,

• Repeat until quiescence:
• If element deleted from Di then
• check directed arc consistency for each arc with head Di
• Maintain arcs to be checked on FIFO queue (no duplicates).

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Constraint Propagation Example

R,G,B G R, G

Graph Coloring

Initial Domains Different-color constraint

V1 V2 V3 Each undirected constraint arc denotes two directed constraint arcs.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 Value deleted Arc examined

R,G,B G R, G

V2 V3 V1

Graph Coloring

Initial Domains

Arcs to examine V1-V2, V1-V3, V2-V3

• Introduce queue of arcs to be examined.
• Start by adding all arcs to the queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1 > V2 Value deleted Arc examined

R,G,B G R, G

V2 V3 V1

Graph Coloring

Initial Domains

Arcs to examine V1<V2, V1-V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V1 > V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1<V2, V1-V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V2 > V1 none V1 > V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1-V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V2 > V1 none V1 > V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1-V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1-V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1>V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1<V3, V2-V3

• Vi – Vj denotes two arcs between Vi and Vj.
• Vi < Vj denotes an arc from Vj and Vi.
• Delete unmentioned tail values

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1(G) V1>V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1<V3, V2-V3 IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1(G) V1>V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R,G,B G R, G

V2 V3 V1 Arcs to examine V1<V3, V2-V3, V2>V1, V1<V3, IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1<V3 V1(G) V1>V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R, G

V2 V3 V1 Arcs to examine V2-V3, V2>V1

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V1<V3 V1(G) V1>V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R, G

V2 V3 V1 Arcs to examine V2-V3, V2>V1

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V2 >V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R, G

V2 V3 V1 Arcs to examine V2<V3, V2>V1

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V2(G) V2 >V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R, G

V2 V3 V1 Arcs to examine V2<V3, V2>V1

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V2(G) V2 >V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R, G

V2 V3 V1 Arcs to examine V2<V3, V2>V1 , V1>V2

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V3 > V2 V2(G) V2 >V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine V2>V1 , V1>V2

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V3 > V2 V2(G) V2 >V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine V2>V1 , V1>V2

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V2>V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine V1>V2

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V2>V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine V1>V2

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1>V2 none V2>V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1(R) V1>V2 none V2>V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V1(R) V1>V2 none V2>V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains R, B G R

V2 V3 V1 Arcs to examine V2>V1, V3>V1

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 V3>V1 V2>V1 V1(R) V2-V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains B G R

V2 V3 V1 Arcs to examine

• Delete unmentioned tail values

IF An element of a variable’s domain is removed, THEN add all arcs to that variable to the examination queue.

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Constraint Propagation Example

R,G,B G R, G Different-color constraint

V1 V2 V3 none V3>V1 none V2>V1 V1(R) V2-V1 V2(G) V2 -V3 V1(G) V1-V3 none V1 – V2 Value deleted Arc examined

Graph Coloring

Initial Domains B G R

V2 V3 V1 Arcs to examine IF examination queue is empty THEN arc (pairwise) consistent.

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Outline

• What is a constraint satisfaction problem (CSPS)?
• Solving CSPs
• Arc-consistency and propagation
• Analysis of constraint propagation
• Search

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What is the Complexity of Constraint Propagation?

Assume:

• domains are of size at most d
• there are e binary constraints.

Which is the correct complexity? 1. O(d2) 2. O(ed2 ) 3. O(ed3) 4. O(ed)

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Complexity of Constraint Propagation

Assume:

• domains are of size at most d
• there are e binary constraints.

Complexity:

• Straight forward arc consistency is O(ed3)
• There are 2 * e arcs to check
• Verifying arc consistency takes O(d2) for each arc.
• An arc is checked at most O(d) times.

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Is arc consistency sound and complete?

R, G R, G R, G

Each arc consistent solution selects a value for every variable from the arc consistent domains. Completeness: Does arc consistency rule out any valid solutions?

• Yes
• No

Soundness: Is every arc-consistent solution a solution to the CSP?

• Yes
• No

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Arc consistency doesn’t rule out all infeasible solutions

R, G R, G R, G R, G R, G B, G

Graph Coloring

arc consistent but no solutions arc consistent but 2 solutions, not 8. B,R,G B,G,R

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Next: To Solve CSPs we combine arc consistency and search

1. Arc consistency (Constraint propagation),

• eliminates values that are shown locally to not be a

part of any solution. 2. Search

• explores consequences of committing to particular

assignments. Methods Incorporating Search:

• Standard Search
• BackTrack search (BT)
• BT with Forward Checking (FC)
• Dynamic Variable Ordering (DV)
• Iterative Repair
• Backjumping (BJ)