Structural Decomposition Methods for Constraint Solving and Optimization

Structural Decomposition Methods for Structural Decomposition Methods for Constraint Solving and OptimizationConstraint Solving and Optimization

Martin SachenbacherFebruary 2003

Please do not distribute beyond MERS group

OutlineOutline

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




OutlineOutline

Constraint Satisfaction ProblemsConstraint Satisfaction Problems

Domains dom(vi)

Variables V = v1, v2, …, vn

Constraints R = r1, r2, …, rm

Tasks– Find a solution– Find all solutions

Methods for Solving CSPsMethods for Solving CSPs

Generate-and-test Backtracking …

“Guessing”

“Decomposition”“Conflicts”

Truth Maintenance Kernels …

Analytic Reduction

ExampleExample

r(E,D)1 22 1

r(E,B)1 22 1

r(E,C)1 21 32 12 3r(B,C)

1 22 1

r(D,A)1 22 1

r(A,B)1 22 1

E

A B

D C

{1,2}

{1,2}{1,2}

{1,2} {1,2,3}

ExampleExample

Eliminate Variable E

E

A B

D C

{1,2}

{1,2}{1,2}

{1,2} {1,2,3}

r(E,D)1 22 1

r(E,B)1 22 1

r(E,C)1 21 32 12 3r(B,C)

1 22 1

r(D,A)1 22 1

r(A,B)1 22 1

ExampleExample

A B

D C

{1,2}{1,2}

{1,2} {1,2,3}

r(D,B,C)2 2 22 2 31 1 11 1 3

r(D,A)1 22 1

r(A,B)1 22 1

r(B,C)1 22 1

Example (continued)Example (continued)

Eliminate variable D

A B

D C

{1,2}{1,2}

{1,2} {1,2,3}

r(D,B,C)2 2 22 2 31 1 11 1 3

r(D,A)1 22 1

r(A,B)1 22 1

r(B,C)1 22 1


Eliminate variable C

A B

C

{1,2}{1,2}

{1,2,3}

r(A,B,C)1 2 21 2 32 1 12 1 3

r(A,B)1 22 1


Eliminate variable B

A B{1,2}{1,2}

r(A,B)1 22 1


Non-empty: Satisfiable!

A{1,2}

r(A)12

Backtrack-free

Extend “backwards”to find solutions

Idea of DecompositionIdea of Decomposition

Computational Steps: Join, Project

Var E:

Var D:

Var C:

Var B:

r(E,D) r(E,B) r(E,C)

r(D,A) r(B,C,D)

r(C,B)

r(B,A)

Var A:

r(A,B,C)

r(A,B)

r(A)


Computational Scheme (acyclic)

r(E,D) ⋈ r(E,B) ⋈ r(E,C)

r(D,A) ⋈ r(B,C,D)

r(C,B) ⋈ r(A,B,C)

r(B,A) ⋈ r(A,B)

r(A)


Alternative Scheme for the Example

r(A,B) ⋈ r(A,D)

r(E,D) ⋈ r(B,D) r(E,C) ⋈ r(B,C)

r(E,B)

Tree DecompositionsTree 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:

1. For each rj R, there is at least one n N such that rj (n) and scope(rj) (n) (“covering”)

2. 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.

Tree DecompositionsTree Decompositions

Comparing Tree Decompositions for the Example

R(A,B) ⋈ R(A,D)

R(E,D) ⋈ R(B,D)

R(E,C) ⋈ R(B,C)

R(E,B)

R(E,D) ⋈ R(E,B) ⋈ R(E,C)

R(D,A) ⋈ R(B,C,D)

R(C,B) ⋈ R(A,B,C)

R(B,A) ⋈ R(A,B)

R(A) Tree-Width 4

Tree-Width 3

Structural Decomposition MethodsStructural 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]

Tree ClusteringTree Clustering

Example

A

E

D

C

B

F


Step 1: Select Variable Ordering

A

E

D

C

B

F


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

A

E

D

C

B

F

A

C

B


Step 3: Identify maximal cliques

A

E

D

C

B

F


Step 4: Form Dual Graph using Cliques as Nodes

A,B,C,E

B,C,D,ED,E,F

B,C,EE

D,E


Step 5: Remove Redundant Arcs

A,B,C,E

B,C,D,ED,E,F

B,C,E

D,E

(E)

Tree-Width 4


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

F,D

DB,C,D

A,B,C

A,B,E

B,C

B,A

(B)

Tree-Width 3

Decomposing HypergraphsDecomposing 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

Hypertree DecompositionsHypertree Decompositions

A triple (T,,) such that:

1. For each rj R, there is at least one n N such that scope(rj) (n) (“covering”)

2. For each variable vi V, the set {n N | vi (n)} induces a connected subtree of T (“connectedness”)

3. For each n N, (n) scope((n))

4. 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)|), nN.

Tree-Width vs. Hypertree-WidthTree-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

Game Characterization: Tree-WidthGame 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

Playing the GamePlaying the Game

g

q

ab

f

c

d

p hl

nm

ok

e

i

j

First Move of the CopsFirst Move of the Cops

g

q

ab

f

c

d

p hl

nm

ok

e

i

j

Shrinking the SpaceShrinking the Space

g

q

ab

f

c

d

p hl

nm

ok

e

i

j

The CaptureThe Capture

g

q

ab

f

c

d

p hl

nm

ok

e

i

j

Game Characterization: HT-WidthGame Characterization: HT-Width

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

Playing the GamePlaying the Game

VP R

S

X Y

ZT U

W


VP R

S

X Y

ZT U

W


V

W

P R

S

X Y

T Z U


V

Z

W

P

S

X Y

T U

R

Different Robber’s ChoiceDifferent Robber’s Choice

VP R

S

X Y

ZT U

W


VP R

S

X Y

ZT U

W

Strategies and DecompositionsStrategies and Decompositions

Theorem: A hypergraph has hypertree-width k iff k Cops have a winning strategy

Winning strategies correspond to decompositions and vice versa


V

Z

P R

S

X

T

Y

UW

a(S,X,T,R) ⋈ b(S,Y,U,P)

Possible Choice for the RobberPossible Choice for the Robber

V

Z

P R

S

X

T

Y

UW



V

Z

P R

S

X

T

Y

UW


c(R,P,V,R)

Alternative Choice for the RobberAlternative Choice for the Robber

V

Z

P R

S

X

T

Y

UW


c(R,P,V,R)


V

Z

P R

S

X

T

Y

UW


c(R,P,V,R) d(X,Y) ⋈ e(T,Z,U)


V

Z

P R

S

X

T

Y

UW



d(X,Y) ⋈ f(W,X,Z)

DecompositionDecomposition



d(X,Y) ⋈ f(W,X,Z)

HT-Width 2

Decomposition-based CSP SolvingDecomposition-based CSP Solving

1. Turn CSP into equivalent acyclic instance

2. Solve equivalent acyclic instance (polynomial in width)

S(D,A) R(A,B)

T(B,C)U(E,D)

V(E,B)

W(E,C)

“Compilation”

V(E,B) W(E,C)

T(B,C) ⋈ U(E,D)

R(A,B) ⋈ S(D,A)

Solving Acyclic CSPsSolving Acyclic CSPs

Bottom-Up Phase– Consistency check

Top-Down Phase– Solution extraction

Polynomial complexity Highly parallelizable

Bottom-Up PhaseBottom-Up Phase

Node Ordering Solve(node)

For Each tuple node.relation

For Each child node.children

cons consistentTuples(child.relation,tuple)

If cons = Then

node.relation node.relation \ { tuple }

Exit For

End If

Next child

Next tuple

Semi-join,DAC

ExampleExample

P(x,y,z)

Q(z,c,d)

G(u,z,d) H(c,d)

9 9 98 8 87 7 76 6 6

9 9 28 8 37 8 5 7 5 87 7 58 9 2

9 28 37 5

9 7 29 6 77 5 5 5 4 84 3 5

R(x,a,b)

1 9 21 8 32 7 5 1 7 5

non-empty:satisfiable

Top-Down PhaseTop-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

ExampleExample

P(x,y,z)

Q(z,c,d)

G(u,z,d) H(c,d)

9 9 28 8 37 7 5

9 28 37 5

9 7 29 6 77 5 5 5 4 84 3 5

R(x,a,b)

1 9 21 8 32 7 5 1 7 5

9 9 97 7 7

(xyz = 999)(abxyz = 72999)

(abcdxyz= 7292999)

(abcduxyz = 72921999) (abcduxyz = 72921999)

(True)

ExampleExample(True)

(xyz = 999)

(abxyz = 72999)

(abcdxyz = 7292999)Backtrack-free

Search

…

…

P

R

G

Q

H

(abcduxyz = 72921999)

(abcduxyz = 72921999)

OutlineOutline




Optimal CSPsOptimal CSPs

Domains dom(vi)



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

Optimization for Acyclic CSPsOptimization 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

Tnn

ExampleExample

Dom(vi) = {0,1,2}

E

C

A

B

F

D

r(A,B,C): {(A,B,C) | ABC}

r(A,E,F): {(A,E,F) | AEF}

r(C,D,E): {(C,D,E) | CDE}

r(A,C,E): {(A,C,E) | ACE}

U = 6A+5B+4C+3D+2E+F

ExampleExample

Tree Decomposition

ACE

ABC CDEAEF

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

001012002112

A,C A,E C,E

ExampleExample

Child ABC

ACE

CDEAEF

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

001012002112

U = 8+

A,C A,E C,E

5+

ABC

U=4

U=9

Weight

ExampleExample

Child AEF

ACE

CDEAEF

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

001012002112

U = 8+

A,C A,E C,E

5+

ABC

2+

U=6

ExampleExample

Child CDE

ACE

CDEAEF

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

001012002112

U = 8+

A,C A,E C,E

5+

ABC

2+6 = 21

U=11

U=14

“To-go”

ExampleExample

Best Solution: utility = 27

ACE

CDEAEF

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

001012002112

U = 8+13 = 21

A,C A,E C,E

ABC

U = 1+6 = 7

U = 4+9 = 12U = 14+13 = 27

Solving Acyclic Optimal CSPsSolving Acyclic Optimal CSPs

Bottom-Up Phase– Consistency Check plus Utility Computation

Top-Down Phase– Solution Extraction Best-First

Polynomial complexity Highly parallelizable


Solve(node)For Each tuple node.relation

tuple.weight weight(tuple)

For Each child node.children

cons consistentTuples(child.relation,tuple)

If cons = Then

node.relation node.relation \ { tuple }

Exit For

Else tuple.weight tuple.weight + bestUtilToGo(cons)

End If

Next child

Next tuple


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

Top-Down Phase: Computing UtilityTop-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.

ExampleExample

(True, 0)

(ACE = 112, 27)

(ABCE = 1112, 27)

(ABCEF = 11122, 27)

(ABCDEF =111222, 27)

Backtrack-free A* Search

…

…

ACE

ABC

CDE

AEF

(ABCDEF = 111122, 24)

OCSPs with State VariablesOCSPs with State Variables

Domains dom(vi)



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

ExampleExample

Boolean Polycell

And1

And2

F = 0

Or2

G = 1

Or1

Or3

X

Y

Z

B = 1

D = 1

A = 1

E = 0

C = 1

ExampleExample

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

ExampleExample

Hypertree Decomposition

Or3 ⋈ And1

And2 Or1Or2

Y,Z Y C,X

State Variables: ChallengesState 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.

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

Decision Variables

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0Inconsistent!

IdeaIdea

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

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

U = 9.7E-3

U = 0.01

U = 0.995

U = 0.99

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

U = 0.99

U = 0.005

U = 0.01

U = 9.7E-3U = 4.8E-5

ExampleExample

O3=ok, A1=ok

O3A1CEFXYZ

A2GYZ O1ACXO2BDY

Y,Z Y C,X

ok 1 1 1fty 1 1

1fty 1 0

1fty 1 1

0fty 1 0

0

ok ok 1 0 0 0 0 1ok ok 1 0 0 0 1 1ok ok 1 0 0 1 0 1

…

ok 1 1 1fty 1 1

1fty 1 1

0

ok 1 1 1fty 1 1

1fty 1 1

0

U = 0.01U =

0.005

U = 0.01

U = 9.7E-3U = 4.8E-5

U = 4.9E-7


Solve(node)For Each tuple projDec(node.relation) tuples consistentTuples(node.relation, tuple) tuple.weight weigth(tuple)

Repeat childrenAssign nextBestChildrenAssignment(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 \ tuplesNext tuple

ExampleExample

Root Node Partition

fty ok 1 0 0 0 1 1

fty ok 1 0 0 1 0 1

U=4.9E-9ok ok 1 0 0 0 1 1

ok ok 1 0 0 1 0 1

ok ok 1 0 0 0 0 1

U=9.7E-3U=4.8E-5U=4.9E-7

ok fty 1 0 0 1 1 1

ok fty 1 0 0 0 1 1

ok fty 1 0 0 1 0 1

ok fty 1 0 0 0 0 1

U=4.8E-3U=4.8E-5U=2.4E-7U=2.4E-9

U=4.9E-7U=9.8E-5fty 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

fty fty 1 0 0 1 1 1

fty fty 1 0 0 0 1 1

fty fty 1 0 0 1 1 0

fty fty 1 0 0 0 1 0

fty 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

U=4.8E-5

U=4.9E-7U=2.4E-7U=2.4E-9

U=2.5E-11


Initialization: Queue (True, 0) Ordering on Siblings Expand(entry)

Repeat

siblingsAssign nextBestAssignment(entry.siblings)

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

ExampleExample

Or3 ⋈ And1

And2

Or1

Or2

(True, 0)

(O3A1CXYZ = ok ok 1011, 0.0097)

(O1O2O3A1A2 = fty ok ok ok ok, 0.0097)

…

…

Example: Leading Four SolutionsExample: Leading Four Solutions

Search Queue

(True, 0)


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)


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)


Search Queue


(O3A1CXYZ = ok fty 1111, 4.8E-3)

(O3A1CXYZ = fty ok 1011, 9.8E-5)

(O3A1CXYZ = fty ok 1010, 4.8E-5)

(O1O2O3A1A2 = fty fty ok ok ok, 9.8E-5)


Search Queue

Solutions


(O3A1CXYZ = ok fty 1111, 4.8E-3)

(O3A1CXYZ = fty ok 1011, 9.8E-5)


Solution 1


Search Queue

Solutions

(O1O2O3A1A2 = fty ok ok ok ok, 9.7E-3) Solution 1

(O1O2O3A1A2 = ok ok ok fty ok, 4.8E-3)

(O3A1CXYZ = fty ok 1011, 9.8E-5)



Search Queue

Solutions

(O1O2O3A1A2 = fty ok ok ok ok, 9.7E-3) Solution 1(O1O2O3A1A2 = ok ok ok fty ok, 4.8E-3) Solution 2

(O3A1CXYZ = fty ok 1011, 9.8E-5)



Search Queue

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)



Search Queue

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


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

Prototype for Decomposition-Prototype for Decomposition-based Optimizationbased Optimization

Software Components

OCSP.XML

ConstraintSystemOptkDecomp

(HT-Decomp)BDD

Decomp.-basedOptimization

Tree.XML

OutlineOutline




Methods for Solving CSPsMethods for Solving CSPs

Generate-and-test Backtracking …

“Guessing”

“Decomposition”“Conflicts”

Truth Maintenance Kernels …

Analytic Reduction

Decomposition, Elimination, Decomposition, Elimination, ResolutionResolution

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)

Conditioning, Search, GuessingConditioning, Search, Guessing

Basic Principle– Breaking up the problem into smaller subproblems by

(heuristically) assigning values and testing candidates

E

A B

D C

{1}

{1,2}{1,2}

{1,2} {1,2,3}

E

A B

D C

{1,2}

{1,2}{1,2}

{1,2} {1,2,3}

E = 1

Conditioning, Search, GuessingConditioning, 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

Conflicts, Truth MaintenanceConflicts, 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

ExamplesExamples

Elimination– SAB (Structural Abduction) [Dechter 95]

Conditioning– CBA* (Constraint-based A*) [Williams 0?]

Conflict Generation– GDE (General Diagnosis Engine) [de Kleer Williams 87]

ApproximationsApproximations

Approximate Elimination– Local constraint propagation (incomplete)

Approximate Conditioning– Hill Climbing, Particle Filtering (incomplete)

Approximate Conflict Generation– Focussed ATMS, Sherlock (incomplete)

HybridsHybrids

Elimination + Conditioning– DCDR(b) [Rish Dechter 96]

Conditioning + Conflict Generation– CDA* [Williams 0?]

Elimination + Conflict Generation– XC1 [Mauss Tatar 02]

Challenge: Elimination + Conditioning + Conflicts.

ResourcesResources

Websites– F. Scarcello’s 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, DEXA’99, Florence, Italy, 1999

– Rina Dechter and Judea Pearl, Tree clustering for constraint networks, Artificial Intelligence 38(3), 1989

Documents

Structural Decomposition Methods for Constraint Solving and Optimization