1. More on Constraint Consistency
Properties & Algorithms
Problem Solving with Constraints
CSCE421/821, Fall 2014
All questions to Piazza
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)

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

3. 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

4. Graphical Illustration
Minimality
tuple
n-2 variables
Decomposability
any consistent assignment of length k
n-k variables
Arc consistency
(1,1)-consistency
vvp
Path consistency
(2,1)-consistency
tuple
(k-1,1)-consistency
k-consistency
any consistent assignment of length k-1
kth variables
(i,j)-consistency
any consistent assignment
of length i
j variables
(1,j)-consistency
inverse consistency
vvp
j variables
NIC
vvp
deg variables

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

6. Local Properties: Binary CSPs
Classical ones
Based on variations of (i,j)-consistency
More recently
Singleton Arc Consistency
Inverse Consistency
Neighborhood Inverse Consistency
(Conservative) Dual consistency
Special Constraints

7. 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

8. 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

9. Local Properties: Binary CSPs
Classical ones
Based on variations of (i,j)-consistency
More recently
Singleton Arc Consistency
Inverse Consistency
Neighborhood Inverse Consistency
(Conservative) Dual consistency
Special Constraints

10. 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

11. 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

12. 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

13. Summary: Binary CSPs (i,j)-consistency j=1 i=1, inverse consistency
Arc consistency: (1,1)-consistency
Path consistency: (2,1)-consistency
i-consistency: (i-1,1)-consistency
Strong i-consistency: k-consistency for all k ≤ i
i=1, inverse consistency
Path Inverse Consistency (PIC): (1,2)-consistency
Global Inverse Consisntecy (GIC): (1,n-1) consistency (minimal domains)
Neighborhood Inverse Consistency (NIC)

14. Local Properties: Binary CSPs
More recently
Singleton Arc Consistency
Partial Path Consistency (PPC)
(Conservative) Dual consistency
Conservative Path Consistency Debruyene ICTAI 99
Special Constraints

15. Special Constraints: AC [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 vD (respectively wD) there exists at most one wD (respectively vD) 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
Mon otonic
A constraint C is monotonic with respect to a domain D iff there exists a total ordering on D such that, for all v and wD, C(v,w) holds implies C(v’,w)’ holds for all values all v’ and w’D such that v’ v and w’ w

16. 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

17. 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

18. Non-Binary CSPs Domain filtering Relational consistency
Generalized Arc Consistency (GAC) Dechter
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)

19. 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)

20. 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)

21. 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)

22. SGAC Idea: Similar to SAC Note (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

23. 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

24. 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

25. Relational 2-Consistency Dechter Def 8.2
Property
For every two constraint C1 and 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 C1,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|

26. Relational m-Consistency Dechter Def 8.3
Property
For every m constraints C1 , C2 , .., Cm
Let s = im scope(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

27. Relational (i,m)-Consistency Dechter Def 8.4
Property
For every m constraints C1 , C2 , .., Cm
Let s = im scope(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 i by
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

28. Graphical Illustration1 s
s-1
sth
RmC s i
s-i
R(i,m)C s m
s-1
s-1 1
GAC
R1C: Relational Arc Consistency

29. m-wise consistency Property (Sketchy) Algorithm
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

30. 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)

31. 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

32. 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

33. 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

34. 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

35. 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

36. Weighted CSPs Weighted CSPs
Tuples have weights in [0,m], m: intolerable cost
Costs are added ab=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

37. 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