1. Path Consistency & Global Consistency Properties
Problem Solving with Constraints
CSCE421/821, Fall 2016
All questions: Piazza
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)

2. Lecture Sources Required reading Recommended
Algorithms for Constraint Satisfaction Problems, Mackworth and Freuder AIJ'85
Sections 3.1, 3.2, 3.3. Chapter 3. Constraint Processing. Dechter
Recommended
Sections 3.4—3.10. Chapter 3. Constraint Processing. Dechter
Networks of Constraints: Fundamental Properties and Application to Picture Processing, Montanari, Information Sciences 74
Bartak: Consistency Techniques (link)
Path Consistency on Triangulated Constraint Graphs, Bliek & Sam-Haroud IJCAI'99

3. Outline Motivation Path consistency and its complexity
Global consistency properties
Minimality
Decomposability
When PC guarantees global consistency

4. AC is not enough Example borrowed from Dechter
Arc-consistent?
Satisfiable?
 seek higher levels of consistency V V 1 1
b a
a b = = V V V V 2 2 3 3
a b
a b
b a
a b =

5. Outline Motivation Path consistency and its complexity
Global consistency properties
Minimality
Decomposability
When PC guarantees global consistency

6. Consistency of a path
A path (V0, V1, V2, …, Vm) of length m is consistent iff
for any value xDV0 and for any value yDVm that are consistent (i.e., PV0 Vm(x, y))
 a sequence of values z1, z2, … , zm-1 in the domains of variables V1, V2, …, Vm-1, such that all constraints between them (along the path, not across it) are satisfied
(i.e., PV0 V1(x, z1)  PV1 V2(z1, z2)  …  PVm-1 Vm(zm-1, zm) )
for all x  DV0
for all y  DVm V0 Vm
Vm-1 V2 V1

7. Note  The same variable can appear more than once in the path
 Every time, it may have a different value
 Constraints considered: PV0,Vm and those along the path
Universal constraints can be included in path
All other constraints are neglected
for all x  DV0
for all y  DVm V0 Vm
Vm-1 V2 V1

8. Example: consistency of a path
 Check path length = 2, 3, 4, 5, 6, ....
{a, b, c} V2 V3 V1 V4 V7 V5 V6
All mutex constraints            

9. Path consistency: definition
 A path of length m is path consistent
 A CSP is path consistent Property of a CSP
Definition: A CSP is path consistent (PC) iff every path is
consistent (i.e., any length of path)
Question: should we enumerate every path of any length?
Answer: No, only length 2, thanks to [Mackworth AIJ'77]

10. Tools for PC-1 Two operators Constraint composition: ( • )
R13 = R12 • R23
Constraint intersection: (  )
R13 R13, old  R13, induced

11. Path consistency (PC-1)
Achieved by composition and intersection (of binary relations expressed as matrices) over all paths of length two.
Procedure PC-1:
1 Begin
2 Yn  R
3 repeat
begin
Y0  Yn
For k  1 until n do
For i  1 until n do
For j  1 until n do
9 Ylij  Yl-1ij  Yl-1ik • Yl-1kk • Yl-1kj
end
11 until Yn = Y0
12 Y  Yn
10 end

12. Properties of PC-1 Discrete CSPs [Montanari'74]
PC-1 terminates
PC-1 results in a path consistent CSP
PC-1 terminates. It is complete, sound (for finding PC network)
PC-2: Improves PC-1 similar to how AC3 improves AC-1
Complexity of PC-1..

13. Basic Consistency Methods
Complexity of PC-1
Procedure PC-1:
1 Begin
2 Yn  R
3 repeat
4 begin
5 Y0  Yn
6 For k  1 until n do
7 For i  1 until n do
For j  1 until n do
Ylij  Yl-1ij  Yl-1ik • Yl-1kk • Yl-1kj
10 end
11 until Yn = Y0
12 Y  Yn
10 end
Line 9: a3
Lines 6–10: n3. a3
Line 3: at most n2 relations x a2 elements
PC-1 is O(a5n5)
PC-2 is O(a5n3) and (a3n3)
PC-1, PC-2 are specified using constraint composition
Basic Consistency Methods

14. Enforcing Path Consistency (PC)
General case: Complete graph
Theorem: In a complete graph, if every path of length 2 is consistent, the network is path consistent [Mackworth AIJ'77]
 PC-1: two operations, composition and intersection
 Proof by induction.

15. Some improvements Mohr & Henderson (AIJ 86) Han & Lee (AIJ 88)
PC-2 O(a5n3)  PC-3 O(a3n3)
Open question: PC-3 optimal?
Han & Lee (AIJ 88)
PC-3 is incorrect
PC-4 O(a3n3) space and time
Singh (ICTAI 95)
PC-5 uses ideas of AC-6 (support bookkeeping)
Also:
PC8: iterates over domains, not constraints [Chmeiss & Jégou 1998]
PC2001: an improvement over PC8, not tested [Bessière et al. 2005]
Note: PC is seldom used in practical applications unless in presence of special type of constraints (e.g., bounded difference)
Project!

16. Path consistency as inference of binary constraints
B < C
A < B
A < C
Path consistency corresponds to inferring a new constraint
(alternatively, tightening an existing constraint) between every two
variables given the constraints that link them to a third variable
 Considers all subgraphs of 3 variables
 3-consistency
B < C

17. Path consistency as inference of binary constraints
Another example: V4 V3
a b V2 V1  =

18. Question Adapted from Dechter
Given three variables Vi, Vk, and Vj and the constraints CVi,Vk, CVi,Vj, and CVk,Vj, write the effect of PC as a sequence of operations in relational algebra. B B B C A
A < B
A + 3 > C
A < C
B < C
A < B
A < B A A
B < C
B < C C C
-3 < A –C < 0
A + 3 > C
Solution: CVi,Vj  CVi,Vj  ij(CVi,Vk CVk,Vj)

19. Partial Path Consistency
Formal definition: Same as PC except that
Universal constraints cannot be included
Defined over cycles
Algorithm: Same as PC-i except that
We triangulate the graph
We run the closure loops over the triangles only O(n3)
(Correction: Careful for articulation points in graph)
Theorem: In a triangulated graph, if every path of length 2 is consistent, the network is partial path consistent [Bliek & Sam-Haroud ‘99]
 PPC (partially path consistent)

20. PPC versus PC Arbitrary binary constraints Algorithm Graph Filtering
Complete
Tight, not necessarily minimal
PPC
Triangulated
Weaker filtering than PC-2
PC property is strictly stronger than PPC property
Open question: Can PC detect insatisfiability when PPC does not?
Yes! Example found by Chris Reeson [TBP]

21. Constraint propagation courtesy of Dechter
After Arc-consistency:
After Path-consistency:
Are these CSPs the same?
Which one is more explicit?
Are they equivalent?
The more propagation,
the more explicit the constraints
the more search is directed towards a solution 1 2 3 1 2 3
( 0, 1 ) 
( 0, 1 ) 

22. PC can detect unsatisfiability
Arc-consistent?
Path-consistent? V1
a b    V2 V3
a b 
a b
a b  
a b V4

23. Warning: Does 3-consistency guarantee 2-consistency?B
{red, blue}
{red, blue}   A C
{ red }
{ red }
Question:
Is this CSP 3-consistent?
is it 2-consistent?
Lesson:
3-consistency does not guarantee 2-consistency

24. PC is not enough Arc-consistent? Path-consistent? Satisfiable?
{a, b, c} V2 V3 V1 V4 V7 V5 V6
All mutex constraints
Arc-consistent?
Path-consistent?
Satisfiable?
 we should seek (even) higher levels of consistency
k-consistency, k = 1, 2, 3, … …following lecture                 

25. Outline Motivation Path consistency and its complexity
Global consistency properties
Minimality
Decomposability
When PC guarantees global consistency

26. Minimality PC tightens the binary constraints
The tightest possible binary constraints yield the minimal network
Minimal network a.k.a. central problem
Given two values for two variables, if they are consistent, then they appear in at least one solution.
Note:
Minimal  path consistent
The definition of minimal CSP is concerned with binary CSPs, but it need not be

27. Minimal CSP Minimal network a.k.a. central problem
Given two values for two variables, if they are consistent, then they appear in at least one solution.
Informally
In a minimal CSP the remainder of the CSP does not add any further constraint to the direct constraint CVi, Vj between the two variables Vi and Vj [Mackworth AIJ'77]
A minimal CSP is perfectly explicit: as far as the pair Vi and Vj is concerned, the rest of the network does not add any further constraints to the direct constraint CVi, Vj [Montanari'74]
The binary constraints are explicit as possible [Montanari'74]

28. Decomposability Decomposable  Minimal  Path Consistent    
Any combination of values for k variables that satisfy the constraints between them can be extended to a solution.
Decomposability generalizes minimality
Minimality: any consistent combination of values for
any 2 variables is extendable to a solution
Decomposability: any consistent combination of values for
any k variables is extendable to a solution
Decomposable 
Minimal 
Path Consistent 
  
Strong n-consistent 
n-consistent 
Solvable

29. Terminology Minimal Globally consistent
Decomposable strongly n-consistent

30. Outline Motivation Path consistency and its complexity
Global consistency properties
Minimality
Decomposability
When PC guarantees global consistency

31. PC approximates.. In general:
Decomposability  minimality  path consistent
PC is used to approximate minimality (which is the central problem)
When is the approximation the real thing?
Special cases:
When composition distributes over intersection, [Montanari'74]
PC-1 on the completed graph guarantees minimality and decomposability
When constraints are convex [Bliek & Sam-Haroud 99]
PPC on the triangulated graph guarantees minimality and decomposability (and the existing edges are as tight as possible)

32. PPC versus PC Algorithm Graph Filtering/property Arbitrary Constraints
Complete
Tight, not necessarily minimal
PPC
Triangulated
Weaker filtering than PC-2
Composition distributes over intersection
Minimal & Decomposable

33. PC: Special Case Distributivity property
Outer loop in PC-1 (PC-3) can be ignored
Exploiting special conditions in temporal reasoning
Temporal constraints in the Simple Temporal Problem (STP): composition & intersection
Composition distributes over intersection
PC-1 is a generalization of the Floyd-Warshall algorithm (all pairs shortest path)
Convex constraints
PPC

34. Distributivity property
Intersection, 
Composition, •
In PC-1, two operations:
RAB • (RBC  R'BC) = (RAB • RBC)  (RAB • R’BC)
When ( • ) distributes over (  ), then [Montanari'74]
PC-1 guarantees that CSP is minimal and decomposable
The outer loop of PC-1 can be removed B
RAB
R’BC
RBC A C

35. Condition does not always hold
Constraint composition does not always distribute over constraint intersection
R12= R23= R’23=
⋅( ∩ ) = ⋅ =
( ⋅ ) ∩ ( ⋅ )= ∩ =
1 1
0 0
0 0
1 0
1 0
0 0
1 1
0 0
0 0
1 0
1 0
0 0
1 1
0 0
0 0
0 0
1 0
0 0
1 0
0 0
1 1
0 0
0 0
1 0
1 1
0 0
1 0
0 0
1 0
0 0

36. Temporal Reasoning constraints of bounded difference
Variables: X, Y, Z, etc.
Constraints: a  Y-X  b, i.e. Y-X = [a, b] = I
Composition: I 1 • I2 = [a1, b1] • [a2, b2] = [a1+ a2, b1+b2]
Interpretation:
intervals indicate distances
composition is triangle inequality.
Intersection: I1  I2 = [max(a1, a2), min(b1, b2)]
Distributivity: I1 • (I2  I3) = (I1 • I2)  (I1 • I3)
Proof: left as an exercise

37. Example: Temporal Reasoning
Composition of intervals + :
R’13 = R12 + R23 = [4, 12]
R01 + R13 = [2,5] + [3, 5] = [5, 10]
R01 + R'13 = [2,5] + [4, 12] = [6, 17]
Intersection of intervals: R13  R'13 = [4, 12]  [3, 5] = [4, 5]
R01 + (R13  R'13) = (R01 + R13)  (R01 + R'13)
R01 + (R13  R'13) = [2, 5] + [4, 5] = [6, 10]
(R01 + R13)  (R01 + R'13) = [5, 10]  [6,17] = [6, 10]
Here, path consistency guarantees minimality and decomposability
R’13
R13 =[3,5]
R23=[1,8]
R12=[3,4] V1 V3 V2 V0
R01=[2,5]

38. Composition Distributes over 
PC-1 generalizes Floyd-Warshall algorithm (all-pairs shortest path), where
composition is ‘scalar addition’ and
intersection is ‘scalar minimal’
PC-1 generalizes Warshall algorithm (transitive closure)
Composition is logical OR
Intersection is logical AND

39. Convex constraints: temporal reasoning (again!)
Thanks to Xu Lin (2002)
Constraints of bounded difference are convex
We triangulate the graph (good heuristics exist)
Apply PPC: restrict propagations in PC to triangles of the graph (and not in the complete graph)
According to [Bliek & Sam-Haroud 99] PPC becomes equivalent to PC, thus it guarantees minimality and decomposability

40. Summary
Alert: Do not confuse a consistency property with the algorithms for reinforcing it
Local consistency methods
Remove inconsistent values (node, arc consistency)
Remove Inconsistent tuples (path consistency)
Get us closer to the solution
Reduce the ‘size’ of the problem & thrashing during search
Are ‘cheap’ (i.e., polynomial time)
Global consistency properties are the goal we aim at
Sometimes (special constraints, graphs, etc) local consistency guarantees global consistency
E.g., Distributivity property in PC, row-convex constraints, special networks
Sometimes enforcing local consistency can be made cheaper than in the general case
E.g., functional constraints for AC, triangulated graphs for PC