Problem Solving with Constraints

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Path Consistency Global Consistency Properties

• Problem Solving with Constraints
• CSCE421/821, Fall2012
• www.cse.unl.edu/choueiry/F12-421-821
• All questions Piazza
• Berthe Y. Choueiry (Shu-we-ri)
• Avery Hall, Room 360
• Tel [phone]

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Lecture Sources

• Required reading
• Algorithms for Constraint Satisfaction Problems,
  Mackworth and Freuder AIJ'85
• Sections 3.1, 3.2, 3.3. Chapter 3. Constraint
  Processing. Dechter
• Recommended
• Sections 3.43.10. 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

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Outline

• Motivation
• Path consistency and its complexity
• Global consistency properties
• Minimality
• Decomposability
• When PC guarantees global consistency

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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
3
2
a b
3
b a
a b
a b
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Outline

• Motivation
• Path consistency and its complexity
• Global consistency properties
• Minimality
• Decomposability
• When PC guarantees global consistency

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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) )

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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
• ? All other constraints are neglected

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Example consistency of a path

• ? Check path length 2, 3, 4, 5, 6, ....

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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

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Tools for PC-1

• Two operators
• Constraint composition ( )
• R13 R12 R23
• Constraint intersection ( ? )
• R13? R13, old ? R13, induced

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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
• 4 begin
• 5 Y0 ? Yn
• 6 For k ? 1 until n do
• 7 For i ? 1 until n do
• 8 For j ? 1 until n do
• 9 Ylij ? Yl-1ij ? Yl-1ik Yl-1kk Yl-1kj
• 10 end
• 11 until Yn Y0
• 12 Y ? Yn
• 10 end

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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..

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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
• 8 For j ? 1 until n do
• 9 Ylij ? Yl-1ij ? Yl-1ik Yl-1kk
  Yl-1kj
• 10 end
• 11 until Yn Y0
• 12 Y ? Yn
• 10 end

• Line 9 a3
• Lines 610 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
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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.
• General case Triangulated graph
• Theorem In a triangulated graph, if every path
  of length 2 is consistent, the network is path
  consistent Bliek Sam-Haroud 99
• ? PPC (partially path consistent) ? PC

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PPC versus PC
Algorithm Graph PC-p? Filtering
Arbitrary Constraints PC-2 Complete ? Tight, not necessarily minimal
Arbitrary Constraints PPC Triangulated ? Weaker filtering than PC-2
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Some improvements

• Mohr Henderson (AIJ 86)
• 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!
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Path consistency as inference of binary
constraints

• 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 lt C
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Path consistency as inference of binary
constraints

• Another example

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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
A lt B
A lt B
A
A
B lt C
B lt C
C
C
-3 lt A C lt 0
A 3 gt C
Solution CVi,Vj ? CVi,Vj ? ??ij(CVi,Vk
CVk,Vj)
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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

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PC can detect unsatisfiability

• Arc-consistent?
• Path-consistent?

V1
a b
?
?
?
V2
V3
?
a b
a b
a b
?
?
a b
V4
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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

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PC is not enough

• Arc-consistent?
• Path-consistent?
• Satisfiable?
• ? we should seek (even) higher levels of
  consistency
• k-consistency, k 1, 2, 3, .
  following lecture

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Outline

• Motivation
• Path consistency and its complexity
• Global consistency properties
• Minimality
• Decomposability
• When PC guarantees global consistency

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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

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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

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Decomposability

• 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

Minimal ?
Decomposable ?
Path Consistent
? ? ?
?
n-consistent ?
Strong n-consistent ?
Solvable
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Relations to (theory of) DB
CSP Database
Minimal C-wise consistent The relations join completely
Decomposable ?
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Outline

• Motivation
• Path consistency and its complexity
• Global consistency properties
• Minimality
• Decomposability
• When PC guarantees global consistency

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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)

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PPC versus PC
Algorithm Graph PC-p? Filtering
Arbitrary Constraints PC-2 Complete ? Tight, not necessarily minimal
Arbitrary Constraints PPC Triangulated ? Weaker filtering than PC-2
Composition distributes over intersection PC-2 Complete ? Minimal Decomposable
Composition distributes over intersection PPC Triangulated ? Minimal Decomposable
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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

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Distributivity property
Intersection, ? Composition,

• In PC-1, two operations
• RAB (RBC ? R'BC) (RAB RBC) ? (RAB
  RBC)
• 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
RBC
RAB
RBC
A
C
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Condition does not always hold

• Constraint composition does not always distribute
  over constraint intersection
• R12 R23 R23
• ( n )
• ( ) n ( ) n
  

1 0 0 0
1 1 0 0
0 0 1 0
0 0 0 0
1 1 0 0
0 0 1 0
0 0 0 0
1 1 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 1 0 0
1 0 0 0
0 0 1 0
1 1 0 0
1 0 0 0
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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, b1b2
• 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

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Example Temporal Reasoning

• Composition of intervals
• R13 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

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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

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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

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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