Structural%20Decomposition%20Methods%20for%20Constraint%20Solving%20and%20Optimization

1
Structural Decomposition Methods for Constraint
Solving and Optimization
Please do not distribute beyond MERS group

• Martin Sachenbacher
• February 2003

2
Outline

• Decomposition-based Constraint Solving
• Tree Decompositions
• Hypertree Decompositions
• Solving Acyclic Constraint Networks
• Decomposition-based Optimization
• Dynamic Programming
• Generalized OCSPs involving State Variables
• Demo of Prototype
• Decomposition vs. other Solving Methods
• Conditioning-based Methods
• Conflict-directed Methods

3
Outline

• Decomposition-based Constraint Solving
• Tree Decompositions
• Hypertree Decompositions
• Solving Acyclic Constraint Networks
• Decomposition-based Optimization
• Dynamic Programming
• Generalized OCSPs involving State Variables
• Demo of Prototype
• Decomposition vs. other Solving Methods
• Conditioning-based Methods
• Conflict-directed Methods

4
Constraint Satisfaction Problems

• Domains dom(vi)
• Variables V v1, v2, , vn
• Constraints R r1, r2, , rm
• Tasks
• Find a solution
• Find all solutions

5
Methods for Solving CSPs
Guessing

• Generate-and-test
• Backtracking

• Truth Maintenance
• Kernels

• Analytic Reduction

Decomposition
Conflicts
6
Example
r(E,B)
E
r(E,D)
r(E,C)
1,2
1 22 1
1 22 1
1 21 32 12 3
?
?
1,2
1,2,3
?
D
?
C
r(B,C)
1 22 1
?
1,2
1,2
?
A
B
r(D,A)
r(A,B)
1 22 1
1 22 1
7
Example

• Eliminate Variable E

E
r(E,B)
r(E,D)
r(E,C)
1,2
1 22 1
1 22 1
1 21 32 12 3
?
?
1,2
1,2,3
?
D
?
C
r(B,C)
1 22 1
?
1,2
1,2
?
A
B
r(D,A)
r(A,B)
1 22 1
1 22 1
8
Example
r(B,C)
r(D,B,C)
1 22 1
2 2 22 2 31 1 11 1 3
1,2
1,2,3
D
C
?
1,2
1,2
?
A
B
r(D,A)
r(A,B)
1 22 1
1 22 1
9
Example (continued)

• Eliminate variable D

r(B,C)
r(D,B,C)
1 22 1
2 2 22 2 31 1 11 1 3
1,2
1,2,3
D
C
?
1,2
1,2
?
A
B
r(D,A)
r(A,B)
1 22 1
1 22 1
10
Example (continued)

• Eliminate variable C

r(A,B,C)
r(A,B)
1 2 21 2 32 1 12 1 3
1 22 1
1,2,3
C
1,2
1,2
A
B
11
Example (continued)

• Eliminate variable B

r(A,B)
1 22 1
1,2
1,2
A
B
12
Example (continued)

• Non-empty Satisfiable!

r(A)
Backtrack-free
12
Extend backwardsto find solutions
1,2
A
13
Idea of Decomposition

• Computational Steps Join, Project

Var A
r(A)
Var B
r(B,A)
r(A,B)
Var C
r(C,B)
r(A,B,C)
Var D
r(D,A)
r(B,C,D)
Var E
r(E,D) r(E,B) r(E,C)
14
Idea of Decomposition

• Computational Scheme (acyclic)

r(A)
r(B,A) ? r(A,B)
r(C,B) ? r(A,B,C)
r(D,A) ? r(B,C,D)
r(E,D) ? r(E,B) ? r(E,C)
15
Idea of Decomposition

• Alternative Scheme for the Example

r(E,B)
r(E,D) ? r(B,D)
r(E,C) ? r(B,C)
r(A,B) ? r(A,D)
16
Tree Decompositions

• A tree decomposition of a graph is a triple
  (T,?,?) where T(N,E) is a tree, ? and ? are
  labeling functions associating with each node n
  ? N two sets ?(n) ? V, ?(n) ? R such that
• For each rj ? R, there is at least one n ? N such
  that rj ? ?(n) and scope(rj) ? ?(n) (covering)
• For each variable vi ? V, the set n ? N vi ?
  ?(n) induces a connected subtree of T
  (connectedness)
• The tree-width of a tree decomposition is defined
  as max(?(n)), n ? N.

17
Tree Decompositions

• Comparing Tree Decompositions for the Example

R(E,B)
R(A)
Tree-Width 4
R(B,A) ? R(A,B)
R(E,C) ? R(B,C)
R(E,D) ? R(B,D)
R(C,B) ? R(A,B,C)
Tree-Width 3
R(A,B) ? R(A,D)
R(D,A) ? R(B,C,D)
R(E,D) ? R(E,B) ? R(E,C)
18
Structural Decomposition Methods

• Biconnected Components Freuder 85
• Treewidth Robertson and Seymour 86
• Tree Clustering Dechter Pearl 89
• Cycle Cutset Dechter 92
• Bucket Elimination Dechter 97
• Tree Clustering with Minimization Faltings 99
• Hinge Decompositions Gyssens and Paredaens 84
• Hypertree Decompositions Gottlob et al. 99

19
Tree Clustering

• Example

A
B
C
D
E
F
20
Tree Clustering

• Step 1 Select Variable Ordering

A
B
C
D
E
F
21
Tree Clustering

• Step 2 Make graph chordal (connect non-adjacent
  parents)

A
A
B
B
C
C
D
E
F
22
Tree Clustering

• Step 3 Identify maximal cliques

A
B
C
D
E
F
23
Tree Clustering

• Step 4 Form Dual Graph using Cliques as Nodes

A,B,C,E
B,C,E
E
D,E
B,C,D,E
D,E,F
24
Tree Clustering

• Step 5 Remove Redundant Arcs

Tree-Width 4
A,B,C,E
B,C,E
(E)
D,E
B,C,D,E
D,E,F
25
Tree Clustering

• Alternative variable order F,E,D,C,B,A produces

F,D
Tree-Width 3
D
B,C,D
B,C
A,B,C
(B)
B,A
A,B,E
26
Decomposing Hypergraphs

• Possible Approach Turn hypergraph into
  primal/dual graph, then apply graph decomposition
  method
• But sub-optimal, conversion loses information
• Idea Gottlob 99 Generalize decomposition to
  hypergraphs

27
Hypertree Decompositions

• A triple (T,?,?) such that
• For each rj ? R, there is at least one n ? N such
  that scope(rj) ? ?(n) (covering)
• For each variable vi ? V, the set n ? N vi ?
  ?(n) induces a connected subtree of T
  (connectedness)
• For each n ? N, ?(n) ? scope(?(n))
• For each n ? N, scope(?(n)) ? ?(Tn) ? ?(n), where
  Tn is the subtree of T rooted at n
• The hypertree-width of a hypertree decomposition
  is defined as max(?(n)), n?N.

28
Tree-Width vs. Hypertree-Width

• Class of CSPs with bounded HT-width subsumes
  class of CSPs with bounded tree-width (Gottlob
  00)
• Determining the HT-width of a CSP is NP-complete
• For each fixed k, it can be determined in
  polynomial time whether the HT-width of a CSP is
  ? k

29
Game Characterization Tree-Width

• Robber and k Cops
• Cops want to capture the Robber
• Each Cop controls a node of the graph
• At any time, Robber and Cops can move to
  neighboring nodes
• Robber tries to escape, but must avoid nodes
  controlled by Cops

30
Playing the Game
q
b
a
d
g
c
f
e
p
h
j
l
i
k
n
o
m
31
First Move of the Cops
q
b
a
d
g
c
f
e
p
h
j
l
i
k
n
o
m
32
Shrinking the Space
q
b
a
d
g
c
f
e
p
h
j
l
i
k
n
o
m
33
The Capture
q
b
a
d
g
c
f
e
p
h
j
l
i
k
n
o
m
34
Game Characterization HT-Width

• Cops are now on the edges
• Cop controls all the nodes of an edge
  simultaneously

35
Playing the Game
V
P
R
S
X
Y
Z
T
U
W
36
First Move of the Cops
V
P
R
S
X
Y
Z
T
U
W
37
Shrinking the Space
V
P
R
S
X
Y
T
Z
U
W
38
The Capture
V
P
R
S
X
Y
Z
T
U
W
39
Different Robbers Choice
V
P
R
S
X
Y
Z
T
U
W
40
The Capture
V
P
R
S
X
Y
Z
T
U
W
41
Strategies and Decompositions

• Theorem A hypergraph has hypertree-width ? k iff
  k Cops have a winning strategy
• Winning strategies correspond to decompositions
  and vice versa

42
First Move of the Cops
a(S,X,T,R) ? b(S,Y,U,P)
V
P
R
S
X
Y
Z
T
U
W
43
Possible Choice for the Robber
a(S,X,T,R) ? b(S,Y,U,P)
V
P
R
S
X
Y
Z
T
U
W
44
The Capture
a(S,X,T,R) ? b(S,Y,U,P)
c(R,P,V,R)
V
P
R
S
X
Y
Z
T
U
W
45
Alternative Choice for the Robber
a(S,X,T,R) ? b(S,Y,U,P)
c(R,P,V,R)
V
P
R
S
X
Y
Z
T
U
W
46
Shrinking the Space
a(S,X,T,R) ? b(S,Y,U,P)
c(R,P,V,R)
d(X,Y) ? e(T,Z,U)
V
P
R
S
X
Y
Z
T
U
W
47
The Capture
a(S,X,T,R) ? b(S,Y,U,P)
c(R,P,V,R)
d(X,Y) ? e(T,Z,U)
V
P
R
S
d(X,Y) ? f(W,X,Z)
X
Y
Z
T
U
W
48
Decomposition
a(S,X,T,R) ? b(S,Y,U,P)
HT-Width 2
c(R,P,V,R)
d(X,Y) ? e(T,Z,U)
d(X,Y) ? f(W,X,Z)
49
Decomposition-based CSP Solving

• Turn CSP into equivalent acyclic instance
• Solve equivalent acyclic instance (polynomial in
  width)

W(E,C)
R(A,B) ? S(D,A)
T(B,C)
U(E,D)
T(B,C) ? U(E,D)
V(E,B)
Compilation
S(D,A)
R(A,B)
V(E,B)
W(E,C)
50
Solving Acyclic CSPs

• Bottom-Up Phase
• Consistency check
• Top-Down Phase
• Solution extraction
• Polynomial complexity
• Highly parallelizable

51
Bottom-Up Phase

• Node Ordering
• Solve(node)
• For Each tuple ? node.relation
• For Each child ? node.children
• cons ? consistentTuples(child.relation,tup
  le)
• If cons ? Then
• node.relation ? node.relation \
  tuple
• Exit For
• End If
• Next child
• Next tuple

Semi-join,DAC
52
Example
9 9 98 8 87 7 76 6 6
non-emptysatisfiable
P(x,y,z)
9 9 28 8 37 8 5 7 5 87 7 58 9 2
9 7 29 6 77 5 5 5 4 84 3 5
Q(z,c,d)
R(x,a,b)
1 9 21 8 32 7 5 1 7 5
9 28 37 5
G(u,z,d)
H(c,d)
53
Top-Down Phase

• Node Ordering
• Search Queue
• Initialization Queue ? (True)
• Expand(entry)
• cons ? consistentTuples(entry.node.relation,entry.
  assignment)
• For Each tuple ? cons
• Queue ? (nextInOrdering(entry.node),
• tuple ? entry.assignment)
• Next tuple

54
Example
(True)
P(x,y,z)
9 9 97 7 7
9 7 29 6 77 5 5 5 4 84 3 5
9 9 28 8 37 7 5
Q(z,c,d)
R(x,a,b)
1 9 21 8 32 7 5 1 7 5
9 28 37 5
G(u,z,d)
H(c,d)
55
Example
(True)
(xyz 999)

P
(abxyz 72999)
R

Backtrack-free Search
(abcdxyz 7292999)
Q
G
(abcduxyz 72921999)
H
(abcduxyz 72921999)
56
Outline

• Decomposition-based Constraint Solving
• Tree Decompositions
• Hypertree Decompositions
• Solving Acyclic Constraint Networks
• Decomposition-based Optimization
• Dynamic Programming
• Generalized OCSPs involving State Variables
• Demo of Prototype
• Decomposition vs. other Solving Methods
• Conditioning-based Methods
• Conflict-directed Methods

57
Optimal CSPs

• Domains dom(vi)
• Variables V v1, v2, , vn
• Constraints R r1, r2, , rm
• Utility Functions u(vi) dom(vi) ? R
• mutual preferential independence
• Tasks
• Find best solution
• Find k best solutions
• Find all solutions up to utility u

58
Optimization for Acyclic CSPs

• Utility of tuple in n Utility of best
  instantiation for variables in subtree Tn that is
  compatible with tuple
• Dynamic Programming Best instantiation composed
  of tuple and best-utility consistentchild tuple
  for each child of n
• Proof Connectednessproperty of the
  treedecomposition

n
Tn
59
Example

• Dom(vi) 0,1,2

r(A,B,C) (A,B,C) A?B?C
B
A
r(A,E,F) (A,E,F) A?E?F
r(C,D,E) (C,D,E) C?D?E
C
r(A,C,E) (A,C,E) A?C?E
F
E
U 6A5B4C3D2EF
D
60
Example

• Tree Decomposition

001012002112
ACE
A,C
A,E
C,E
ABC
CDE
AEF
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
61
Example
Weight

• Child ABC

001012002112
U 8
5
ACE
A,C
A,E
C,E
CDE
AEF
ABC
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
U4
U9
62
Example

• Child AEF

001012002112
U 8
5
2
ACE
A,C
A,E
C,E
CDE
AEF
ABC
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
U6
63
Example

• Child CDE

001012002112
U 8
5
2
6 21
ACE
A,C
A,E
C,E
CDE
AEF
ABC
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
U11
U14
64
Example

• Best Solution utility 27

U 16 7
001012002112
U 813 21
ACE
U 49 12
U 1413 27
A,C
A,E
C,E
CDE
AEF
ABC
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
001,012,002,112,000,011,022,111,122,222
65
Solving Acyclic Optimal CSPs

• Bottom-Up Phase
• Consistency Check plus Utility Computation
• Top-Down Phase
• Solution Extraction Best-First
• Polynomial complexity
• Highly parallelizable

66
Bottom-Up Phase

• Solve(node)
• For Each tuple ? node.relation
• tuple.weight ? weight(tuple)
• For Each child ? node.children
• cons ? consistentTuples(child.relation,tup
  le)
• If cons ? Then
• node.relation ? node.relation \
  tuple
• Exit For
• Else tuple.weight ? tuple.weight
  bestUtilToGo(cons)
• End If
• Next child
• Next tuple

67
Top-Down Phase

• Initialization Queue ? (True, 0)
• Expand(entry)
• cons ? consistentTuples(entry.node.relation,entry.
  assignment)
• For Each tuple ? cons
• util ? entry.util tuple.util -
  bestUtil(cons)
• Queue ? (nextInOrdering(entry.node),
• tuple ? entry.assignment,
  util)
• Next tuple

68
Top-Down Phase Computing Utility

• Utility of extended assignment
• Util weight(tuple ? entry.assignment)
  utilToGo(tuple ? entry.assignment)
• weight(tuple ? entry.assignment)
  tuple.utilToGo entry.utilToGo -
  bestUtilToGo(cons)
• tuple.util entry.util -
  weightSharedVars(cons) -
  bestUtilToGo(cons)
• tuple.util entry.util - bestUtil(cons)

No need to call weight(), cancels out.
69
Example
(True, 0)
(ACE 112, 27)

ACE
(ABCE 1112, 27)

ABC
Backtrack-free A Search
(ABCEF 11122, 27)
AEF
CDE
(ABCDEF 111222, 27)
(ABCDEF 111122, 24)
70
OCSPs with State Variables

• Domains dom(vi)
• Variables V v1, v2, , vn
• Constraints R r1, r2, , rm
• Utility Functions u(vi) dom(vi) ? R for subset
  Dec ? V
• mutual preferential independence
• Tasks
• Find the best solution projected on Dec
• Find the k best solutions projected on Dec
• Find all solutions projected on Dec up to utility
  u

71
Example

• Boolean Polycell

X
A 1
Or1
And1
B 1
F 0
Y
Or2
C 1
And2
G 1
D 1
Z
Or3
E 0
72
Example

• State Variables A, B, C, D, E, F, G, X, Y, Z
• Domain 0,1
• Decision Variables O1, O2, O3, A1, A2
• Domain ok, faulty
• Utility function Mode Probabilities
• Or-gate u(ok) 0.99, u(faulty) 0.01
• And-gate u(ok) 0.995, u(faulty) 0.005

73
Example

• Hypertree Decomposition

Or3 ? And1
Y,Z
Y
C,X
Or2
And2
Or1
74
State Variables Challenges

• Infeasible to iterate over tuples with state
  variables
• Instead Must handle sets of tuples with same
  weight
• Problem Child assignments can now constrain
  themselfes mutually, hence they can no longer be
  considered independently.

75
Example
Decision Variables

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
76
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
77
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
78
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
79
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
Inconsistent!
80
Idea

• Best-First-Search for Consistent Child
  Assignments, until Parent Assignment is fully
  covered, or Child Assignments are exhausted.

81
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
82
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
U 9.7E-3
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
U 0.99
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
U 0.01
U 0.005
83
Example

• O3ok, A1ok

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1
0 1
O3A1CEFXYZ
U 9.7E-3
U 4.8E-5
Y,Z
Y
C,X
O2BDY
A2GYZ
O1ACX
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
U 0.01
U 0.01
U 0.005
84
Bottom-Up Phase

• Solve(node)
• For Each tuple ? projDec(node.relation)
• tuples ? consistentTuples(node.relation,
  tuple)
• tuple.weight ? weigth(tuple)
• Repeat
• childrenAssign ? nextBestChildrenAssignmen
  t(node.children)
• cons ? consistentTuples(childrenAssign,
  tuples)
• If cons ? ? Then
• cons.weight ? tuple.weight
  bestUtilToGo(cons)
• insertPartitionElement(node, cons)
• tuples ? tuples \ cons
• End If
• Until tuples ? Or childrenAssign ?
• node.relation ? node.relation \ tuples
• Next tuple

85
Example

• Root Node Partition

ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1ok ok 1 0 0 0
0 1
fty ok 1 0 0 0 1 1fty ok 1 0 0 1 0 1
U4.9E-9
U9.7E-3
U4.8E-5
U4.9E-7
U4.8E-5
fty fty 1 0 0 1 1 1fty fty 1 0 0 0 1 1fty fty 1
0 0 1 1 0fty fty 1 0 0 0 1 0fty fty 1 0 0 1 0 1
fty fty 1 0 0 1 0 0 fty fty 1 0 0 0 0 1 fty fty
1 0 0 0 0 0
ok fty 1 0 0 1 1 1ok fty 1 0 0 0 1 1ok fty 1 0
0 1 0 1ok fty 1 0 0 0 0 1
U4.8E-3
U4.9E-7
U4.8E-5
U2.4E-7
U2.4E-7
U2.4E-9
U2.4E-9
fty ok 1 0 0 0 1 1fty ok 1 0 0 0 1 0fty ok 1 0
0 1 0 0fty ok 1 0 0 1 0 1
U9.8E-5
U2.5E-11
U4.9E-7
86
Top-Down Phase

• Initialization Queue ? (True, 0)
• Ordering on Siblings
• Expand(entry)
• Repeat
• siblingsAssign ? nextBestAssignment(entry.sibl
  ings)
• cons ? consistentTuples(siblingsAssign,
  entry.assignment)
• If cons ? ? Then
• bestUtilSibAssign ? Max(bestUtilSibAssign,
  siblingsAssign.util)
• util ? entry.util siblingsAssign.util -
  bestUtilSibAssign
• Queue ? (nextInOrdering(entry.siblings),
• projSharedOrDec(cons ?
  entry.assignment), util)
• End If
• Until siblingsAssign 0

87
Example
(True, 0)

Or3 ? And1
(O3A1CXYZ ok ok 1011, 0.0097)
And2

Or2
Or1
(O1O2O3A1A2 fty ok ok ok ok, 0.0097)
88
Example Leading Four Solutions

• Search Queue

(True, 0)
89
Example Leading Four Solutions

• Search Queue

(O3A1CXYZ ok ok 1011, 9.7E-3)
(O3A1CXYZ ok fty 1111, 4.8E-3)
(O3A1CXYZ fty ok 1011, 9.8E-5)
(O3A1CXYZ ok ok 1101, 4.8E-5)
90
Example Leading Four Solutions

• Search Queue

(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
(O3A1CXYZ ok fty 1111, 4.8E-3)
(O3A1CXYZ fty ok 1011, 9.8E-5)
(O3A1CXYZ ok ok 1101, 4.8E-5)
91
Example Leading Four Solutions

• Search Queue

(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
(O3A1CXYZ ok fty 1111, 4.8E-3)
(O3A1CXYZ fty ok 1011, 9.8E-5)
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O3A1CXYZ fty ok 1010, 4.8E-5)
92
Example Leading Four Solutions

• Search Queue
• Solutions

(O3A1CXYZ ok fty 1111, 4.8E-3)
(O3A1CXYZ fty ok 1011, 9.8E-5)
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
93
Example Leading Four Solutions

• Search Queue
• Solutions

(O1O2O3A1A2 ok ok ok fty ok, 4.8E-3)
(O3A1CXYZ fty ok 1011, 9.8E-5)
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
94
Example Leading Four Solutions

• Search Queue
• Solutions

(O3A1CXYZ fty ok 1011, 9.8E-5)
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
(O1O2O3A1A2 ok ok ok fty ok, 4.8E-3)
Solution 2
95
Example Leading Four Solutions

• Search Queue
• Solutions

(O1O2O3A1A2 fty ok fty ok ok, 9.8E-5)
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
(O1O2O3A1A2 ok ok ok fty ok, 4.8E-3)
Solution 2
96
Example Leading Four Solutions

• Search Queue
• Solutions

(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
(O1O2O3A1A2 ok ok ok fty ok, 4.8E-3)
Solution 2
(O1O2O3A1A2 fty ok fty ok ok, 9.8E-5)
Solution 3
97
Example Leading Four Solutions

• Solutions

(O1O2O3A1A2 fty ok ok ok ok, 9.7E-3)
Solution 1
(O1O2O3A1A2 ok ok ok fty ok, 4.8E-3)
Solution 2
(O1O2O3A1A2 fty ok fty ok ok, 9.8E-5)
Solution 3
(O1O2O3A1A2 fty fty ok ok ok, 9.8E-5)
Solution 4
Search queue size bounded by k
98
Prototype for Decomposition-based Optimization

• Software Components

Decomp.-basedOptimization
OCSP.XML
Tree.XML
ConstraintSystem
OptkDecomp(HT-Decomp)
BDD
99
Outline

• Decomposition-based Constraint Solving
• Tree Decompositions
• Hypertree Decompositions
• Solving Acyclic Constraint Networks
• Decomposition-based Optimization
• Dynamic Programming
• Generalized OCSPs involving State Variables
• Demo of Prototype
• Decomposition vs. other Solving Methods
• Conditioning-based Methods
• Conflict-directed Methods

100
Methods for Solving CSPs
Guessing

• Generate-and-test
• Backtracking

• Truth Maintenance
• Kernels

• Analytic Reduction

Decomposition
Conflicts
101
Decomposition, Elimination, Resolution

• Basic Principle
• Analytic reduction to equivalent subproblems
• Advantages
• No search, no inconsistencies (unless no solution
  exists)
• Solutions obtained simultaneously (knowledge
  compilation)
• Time/Space Requirements
• Bound by structural properties (width)
• Worst case is average case
• Problems
• Space Requirements (large constraints, as
  variables are unassigned)

102
Conditioning, Search, Guessing

• Basic Principle
• Breaking up the problem into smaller subproblems
  by (heuristically) assigning values and testing
  candidates

103
Conditioning, Search, Guessing

• Basic Principle
• Breaking up the problem into smaller subproblems
  by (heuristically) assigning values and testing
  candidates
• Advantages
• Less space (small constraints, as variables are
  assigned)
• Works also for hard problems
• Time/Space Requirements
• Exponential (but average case much better than
  worst-case)
• Problems
• Backtracking necessary
• Solutions obtained only one-by-one

104
Conflicts, Truth Maintenance

• Basic Principle
• Find and generalize inconsistencies (conflicts)
  to construct descriptions of feasible regions
  (kernels)
• Advantages
• Re-use of information
• Avoids redundant exploration of the search space
• Time/Space Requirements
• Exponential (both in number of conflicts and size
  of kernels)
• Problems
• Complexity

105
Examples

• Elimination
• SAB (Structural Abduction) Dechter 95
• Conditioning
• CBA (Constraint-based A) Williams 0?
• Conflict Generation
• GDE (General Diagnosis Engine) de Kleer Williams
  87

106
Approximations

• Approximate Elimination
• Local constraint propagation (incomplete)
• Approximate Conditioning
• Hill Climbing, Particle Filtering (incomplete)
• Approximate Conflict Generation
• Focussed ATMS, Sherlock (incomplete)

107
Hybrids

• Elimination Conditioning
• DCDR(b) Rish Dechter 96
• Conditioning Conflict Generation
• CDA Williams 0?
• Elimination Conflict Generation
• XC1 Mauss Tatar 02

Challenge Elimination Conditioning Conflicts.
108
Resources

• Websites
• F. Scarcellos homepage http//ulisse.deis.unical
  .it/frank
• Software
• optkdecomp implements hypertree decomposition
  (Win32)
• decompOpSat implements tree-based optimization
  (Win32)
• Papers
• Gottlob, Leone, Scarcello A comparison of
  structural CSP decomposition methods. Artificial
  Intelligence 124(2), 2000
• Gottlob, Leone, Scarcello On Tractable Queries
  and Constraints, DEXA99, Florence, Italy, 1999
• Rina Dechter and Judea Pearl, Tree clustering for
  constraint networks, Artificial Intelligence
  38(3), 1989