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Foundations of Constraint Processing CSCE421/821, Spring 2009

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More on Constraint Consistency

Properties & Algorithms

Foundations of Constraint Processing

CSCE421/821, Spring 2009

www.cse.unl.edu/~choueiry/S09-421-821/

All questions: cse421@cse.unl.edu

Berthe Y. Choueiry (Shu-we-ri)

Avery Hall, Room 360

choueiry@cse.unl.edu

Tel: +1(402)472-5444

Outline

- Global properties
- Local properties
- Binary CSPs Dechter Sections 3.1, 3.2, 3.3, 3.4
- Non-Binary CSPsDechter Sections 3.5.1, 8.1
- Effects of Consistency Algorithms
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domainsDechter Sections 3.5.3
- Weighted CSPs

Global Consistency Properties

- Minimality & Decomposability
- Originally defined for binary CSPs
- Easily extendable to non-binary CSPs
- Minimality Dechter Definition 2.6
- Every constraint is as tight as it can be
- Minimality n-consistency
- In DB, the relations are said to “join completely”
- Decomposability
- Every consistent partial solution can be terminated backtrack free
- Decomposability ≡ strong n-consistency

Outline

- Global properties
- Local properties
- Binary CSPs
- Non-Binary CSPs
- Effects
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domains
- Weighted CSPs

Local Properties: Binary CSPs

- Classical ones
- Arc, path, i, strong i, (i,j)-consistency
- More recently
- Singleton Arc Consistency
- Inverse Consistency
- Neighborhood Inverse Consistency
- (Conservative) Dual consistency
- Special Constraints

Classical Local Consistency: Properties

- Arc consistency
- Every vvp can be extended to a partial solution of length 2
- Path consistency
- Every partial solution of length 2 can be extended to a partial solution of length 3
- i-consistency
- Every partial solution of length (i-1) can be extended to a partial solution of length i
- (i,j)-consistency
- Every partial solution of length i can be extended to a partial solution of length i+j

Classical Local Consistency: Algorithms

- Arc consistency:
- AC-1, 2, 3, …, 7, AC-2001, AC-*, …
- Effect: domain filtering
- Complexity: in n2
- Path consistency
- PC-1, 2, 3, …, 8, PC2001, PPC, …
- Effect: adds binary constraints, modifies the width of network
- Complexity: in n3
- i-consistency
- Dechter Figure 3.14 & 3.15
- Effect: adds constraints of arity i-1, modifies the arity of network
- Complexity: in ni

Local Properties: Binary CSPs

- Classical ones
- Arc, path, i, strong i, (i,j)-consistency
- More recently
- Singleton Arc Consistency
- Inverse Consistency
- Neighborhood Inverse Consistency
- (Conservative) Dual consistency
- Special Constraints

Singleton Arc Consistency (SAC)

- Property: The CSP is AC for every vvp
- (Sketchy) Algorithm

Repeat until no change occurs

Repeat for each variable

Repeat for each value in domain

Assign this value to this variable.

If the CSP is AC, keep the value.

Otherwise, remove it.

- Effect: domain filtering
- Note
- Proposed by Debruyne &Bessière, IJCAI 97
- Quite expensive, but can be quite effective

Inverse Consistency

- Path Inverse Consistency (PIC)
- Equivalent to (1,2)-consistency
- Inverse m-consistency
- Equivalent to(1,m)-consistency
- Neighborhood Inverse Consistency (NIC)
- Every vvp participates in a solution in the CSP induced by its neighborhood

Neighborhood Inverse Consistency (NIC): Algorithm

Repeat until no change occurs

Repeat for each variable

Consider only the neighborhood of the variable

Repeat for each value for the variable

If the value appears in a complete solution for the neighborhood, then keep it.

Otherwise, remove it.

- Effect: domain filtering
- Note
- Proposed by Freuder & Elfe, AAAI 96
- Very effective, very expensive

Summary: Binary CSPs

- Arc, path, i-consistency
- (i,j)-consistency
- SAC
- PIC
- (1,m)-consistency
- NIC

Local Properties: Binary CSPs

- Classical ones
- Arc, path, i, strong i, (i,j)-consistency
- More recently
- Singleton Arc Consistency
- Inverse Consistency
- Neighborhood Inverse Consistency
- (Conservative) Dual consistency
- Special Constraints

AC for Special Constraints [Van Hentenryck et al. AIJ 92]

- Specialized AC algorithms exist for special constraints
- Functional

A constraint C is functional with respect to a domain D iff for all vD (respectively wD) there exists at most one wD (respectivelyvD) such that C(v,w)

- Anti-functional

A constraint C is anti-functional with respect to a domain D iff C is functional with respect to D

- Monotonic

A constraint C is monotonic with respect to a domain D iff there exists a total ordering on D such that, for all vand wD,C(v,w) holds implies C(v’,w)’ holds for all values all v’and w’D such that v’ v and w’ w

Outline

- Global properties
- Local properties
- Binary CSPs
- Non-Binary CSPs
- Effects of Consistency Algorithms
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domains
- Weighted CSPs

How about Non-binary CSPs?

- (Almost) all properties (& algorithms) discussed so far were restricted to binary CSPs
- Consistency properties for non-binary CSPs are the topic of current research
- Mainly, properties and algorithms for:
- Domain filtering techniques (a.k.a. domain reduction, domain propagation)
- Do not change ‘topology’ of network (width/arity)
- Do not modify constraints definitions
- Relational m-consistency [Dechter, Chap 8]
- Add constraints/change constraint definitions

Non-Binary CSPs

- Domain filtering
- Generalized Arc Consistency (GAC) Dechter 3.5.1
- Singleton Generalized Arc Consistency (SGAC)
- maxRPWC, rPIC, RPWC, etc. [Bessiere et al., 08]
- Relational consistency
- (strong) Relational m-consistency
- Relational Arc-Consistency (R1C)
- Relational Path-Consistency (R2C)
- Relational (i,m)-consistency
- i = 1, Relational (1,m)-consistency is a domain filtering technique
- i=1 and m=2, Relational (1,m)-consistency is known as rPIC
- Relational (*,m)-consistency (m-wise consistency)

Generalized Arc-Consistency : Property

- First introduced by [Mohr & Masini, ECAI 88]
- Every value in the domain of every variable has a support in every constraint in the problem
- In every constraint, every vvp participates in a consistent tuple (can be extended to all other variables in the scope of the constraint)

Generalized Arc-Consistency: Algorithm1

- (Sketchy) Algorithm
- Project the constraint on each of the variables in its scope to tighten the domain of the variable.
- As domains are filtered, filter the constraint
- Repeat the above until quiescence

- When constraint is not defined in extension, GAC may be problematic (e.g., NP-hard in TCSP)

Generalized Arc-Consistency: Algorithm2

- Another (Sketchy) Algorithm
- Iterate over every combination of a variable and a constraint where it appears (Vx, Ci)
- For every value for Vx, identify a support for this value in Ci, where a support is a tuple where all vvps in the tuple are alive
- Repeat the above until quiescence
- Does not filter the constraints
- Check GAC2001 [Bessière et al., AIJ05]

- When constraint is not defined in extension, GAC may be problematic (e.g., NP-hard in TCSP)

SGAC

- Idea: Similar to SAC
- (Sketchy) Algorithm

Repeat until quiescence

For each vvp

Assign the vvp; Enforce GAC on the CSP;

If CSP is GAC, keep the vvp, else remove it

- Note
- Costly in practice, but polynomial as long as GAC is polynomial
- SGAC has been empirically shown to solve every known 9x9 Sudoku puzzle

Relational Consistency

- Dechter generalizes consistency Dechter 8.1.1

properties to non-binary constraints

- Relational m-consistency
- Relational 1-consistency relational arc-consistency
- Relational 2-consistency relational path-consistency
- Relational (i,m)-consistency
- Relational (1,1)consistency GAC
- m-wise consistency (Databases)
- Relational (*,m)-consistency

Relational 1-Consistency Dechter Def 8.1

- Property
- For every constraint C
- Let k be the arity of C
- Every consistent partial solution of length k-1
- Can be extended to a consistent partial solution of length k
- (Sketchy) Algorithm Dechter Equation (8.2), (8.3)
- For each constraint C, generate all constraints of arity k-1 by
- Joining C with the domain of each variable x in scope of C and
- Projecting result on remaining variables (possibly intersecting with existing constraints)
- Effect: Adds a huge number of new constraints
- Complexity: polynomial in the largest scope

Relational 2-Consistency Dechter Def 8.2

- Property
- For every two constraint C1and C2
- Let s = scope(C1) scope(C2)
- Every consistent partial solution of length |s|-1
- Can be extended to a consistent partial solution of length |s|
- (Sketchy) Algorithm Dechter Equation (8.4)
- For each constraints C1and C2, generate all constraints of arity|s|-1 by
- Joining C1, C2, and the domain of a variable (in C1 and C2 ) and
- Projecting the result on remaining variables
- Effect: Adds a huge number of new constraints
- Complexity: polynomial in the largest |s|

Relational m-Consistency Dechter Def 8.3

- Property
- For every m constraints C1, C2 , .., Cm
- Let s = imscope(Ci)
- Every consistent partial solution of length |s|-1
- Can be extended to a consistent partial solution of length |s|
- (Sketchy) Algorithm Dechter Equation (8.5)
- For each m constraints, generate all constraints of arity|s| -1 by
- Joining the m constraints and the domain of a variable (at the intersection of their scopes) and
- Projecting the result on remaining variables
- Effect: Adds a huge number of new constraints
- Complexity: polynomial in the sum of largest 2 scopes

Relational (i,m)-Consistency Dechter Def 8.4

- Property
- For every m constraints C1, C2 , .., Cm
- Let s = imscope(Ci)
- Every consistent partial solution of length i
- Can be extended to a consistent partial solution of length |s|
- Algorithm Dechter Fig 8.1
- For each m constraints, generate all constraints of arity iby
- Joining the m constraints and the domain of a variable (at the intersection of their scopes) and
- Projecting the result on every combination of i variables
- Effect: Adds a huge number of new constraints, except for i=1
- Complexity: exponential in s (largest union of scope of m constraints) in time and space

m-wise consistency

- Property
- For every set of m constraints,
- Every tuple in each constraint appears in a consistent solution to the m constraints
- That is, each constraint is as tight as it can be for the set of m constraints
- (Sketchy) Algorithm

Repeat until quiescence

Join each set of m constraints

Project it on each existing constraint to filter the constraint

- Effect: Filters the constraints, w/o introducing new constraints
- Note:
- Defined in DB: pairwise consistency, relations join completely
- Woodward defined R(*,m)C + new algorithms that are linear space, currently under evaluation

Summary: Non-Binary CSPs

- Domain filtering
- Generalized Arc Consistency (GAC)
- Singleton Generalized Arc Consistency (SGAC)
- maxRPWC, rPIC, RPWC, etc. [Bessiere et al., 08]
- Relational consistency
- (strong) Relational m-consistency
- Relational Arc-Consistency (R1C)
- Relational Path-Consistency (R2C)
- Relational (i,m)-consistency
- i = 1, Relational (1,m)-consistency is a domain filtering technique
- i=1 and m=2, Relational (1,m)-consistency is known as rPIC
- Relational (*,m)-consistency (m-wise consistency)

Outline

- Global properties
- Local properties
- Binary CSPs
- Non-Binary CSPs
- Effects of Consistency Algorithms
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domains
- Weighted CSPs

Effects of Consistency Algorithms

- Filter the domains
- Old algorithms: AC-*, GAC-*, etc.
- New algorithms: maxRPWC, R(1,m)C, etc.
- Filter the constraints
- New algorithms: R(*,m)C
- Add new constraints to the problem
- Old algorithms: PC-2, etc.
- i-consistency (i>2), (i,j)-C, RmC, R(i,m)C
- Example: Solving the CSPs by Constraint Synthesis

Solving CSPs by Constraint synthesis[Freuder 78]

- From i=2 to i=n,
- achieve i-consistency by using (i-1)-arity constraints to synthesize i-arity constraints,
- then use the i-ary constraints to filter constraints of arity i-1, i-2, etc.
- Process ends
- with a unique n-ary constraint
- whose tuples are all the solutions to the CSP

Outline

- Global properties
- Local properties
- Binary CSPs
- Non-Binary CSPs
- Effects of Consistency Algorithms
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domains
- Weighted CSPs

Box Consistency (on interval constraints)

- Domains are (continuous) intervals
- Historically also called: continuous CSPs, continuous domains
- Domains are infinite:
- We cannot enumerate consistent values/tuples
- [Davis, AIJ 87] (see recommended reading) showed that even AC may be incomplete or not terminate
- We apply consistency (usually, arc-consistency) on the boundaries of the interval
- Sometimes, domains are split, so that boundaries can be further filtered

Weighted CSPs

- Weighted CSPs
- Tuples have weights in [0,m], m: intolerable cost
- Costs are added ab=min{m,a+b}
- Soft Arc Consistency (Cooper, de Givry, Schiex, etc.)
- VAC: Virtual Arc Consistency
- EDAC: Existential Directional Arc Consistency
- OSAC: Optimal Soft Arc Consistency

Summary

- Global properties
- Local properties
- Binary CSPs
- Non-Binary CSPs
- Effects of Consistency Algorithms
- Domain filtering
- Constraint filtering
- Constraint synthesis
- Beyond finite, crisp CSPs
- Continuous domains
- Weighted CSPs

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