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Practical Tractability of CSPs by Higher Level Consistency and Tree Decomposition

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### Practical Tractability of CSPs by Higher Level Consistency and Tree Decomposition

ShantKarakashian

Dissertation Defense

Collaborations: Bessiere, Geschwender, Hartke, Reeson, Scott, Woodward.

Support: NSF CAREER Award #0133568 & NSF Grant No. RI-111795. Experiments were conducted on the equipment of the Holland Computing Center at UNL.

Context

- Constraint Satisfaction
- General paradigm for modeling & solving combinatorial decision problems
- Applications in Engineering, Management, Computer Science

- Constraint Satisfaction Problems (CSPs)
- NP-complete in general

- Islands of tractability
- Classes of CSPs solvable in polynomial time

Karakashian: Ph.D. Defense

Our Focus

- One tractabilityconditionlinks [Freuder 82]
- Consistency level to
- Width of the constraint network, a structural parameter (treewidth)

- Catch 22
- Computing width is in NP-hard
- Enforcing higher-levels of consistency may require adding constraints, which increases the treewidth

- Goal
- Exploit above condition to achieve practical tractability

Karakashian: Ph.D. Defense

Our Approach

- Exploit a tree decomposition
- Localize application of the consistency algorithm
- Steer constraint propagation along the branches of the tree
- Add redundant constraints at separators to enhance propagation

- Propose consistency properties that
- Enforce a (parameterized) consistency level
- While preserving the width of a problem instance

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Outline- Background
- Contributions
- R(∗,m)C: Consistency property & algorithms [SAC 10, AAAI 10]
- Localized consistency & structure-guided propagation
- Bolstering propagation at separators [CP 12, AAAI 13]
- Counting solutions
- (Appendices include other incidental contributions)

- Conclusions & Future Research

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Constraint Satisfaction Problem (CSP)- Given
- A set of variables, here X={A,B,C,D,E,F,G}
- Their domains, here D={DA,DB,DC,DD,DE,DF,DG}, Di={0,1}
- A set of constraints, here C={C1,C2,C3,C4,C5} where Ci= ⟨Ri,scope(Ri)⟩

- Question: Find one solution (NPC), find all solutions (#P)

R3

R4

F

A

E

B

C

D

G

R1

R2

R5

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Graphical Representations of a CSPD

R4

R5

G

Hypergraph

Primal graph

Dual graph

R3

A

C

B

F

R4

R5

G

D

R1

R2

E

A

B

R3

C

D

B,D

A,D,G

F

E

R1

B

A

R2

A,B,C

A

B

A

B

A,E,F

E,B

E

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Redundant Edges in a Dual Graph- R1R4 forces the value of A in R1 and R4
- The value of A is enforced through R1R3 and R3R4
- R1R4 is redundant

R4

R5

D

B,D

A,D,G

B

A

A

B

R3

A,B,C

A

B

E

A,E,F

E,B

R1

R2

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Solving CSPs- A CSP can be solved by
- Search (conditioning)
- Backtrack search
- Iterative repair (local search)

- Synthesizing & propagating constraints (inference)

- Search (conditioning)
- Backtrack search
- Constructive, exhaustive exploration of search space
- Variable ordering improves the performance
- Constraint propagation prunes the search tree

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Consistency Property: Definition- Example: k-consistency requires that
- For all combinations of k-1 variables
… all combinations of consistent values

… can always be extended to every kth variable

- For all combinations of k-1 variables
- A consistency property
- Guarantees that the values of all combinations of variables of a given size verify some set of constraints
- Is a necessary but not sufficient condition for partial solution to appear in a complete solution

kthvariable

any consistent assignment of length k-1

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Algorithms for Enforcing a Consistency Property- May require adding new implicit or redundant constraints
- For k-consistency: add constraints to eliminate inconsistent (k-1)-tuples
- Which may increase the width of the problem

- We propose a consistency property that
- Never increases the width of the problem
- Allows us to increase & control the level of consistency
- Operates by deleting tuples from relations

kthvariable

any consistent assignment of length k-1

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Tree DecompositionC1

{A,B,C,E} , {R2,R3}

C3

C2

{A,E,F},{R1}

{A,B,D},{R3,R5}

D

C4

- Conditions
- Each constraint appears in at least one cluster with all the variables in the constraint’s scope
- For every variable, the clusters where the variable appears induce a connected subtree

- A tree decomposition:〈T, 𝝌, 𝜓〉
- T: a tree of clusters
- 𝝌: maps variables to clusters
- 𝜓:maps constraints to clusters

R4

R5

{A,D,G},{R4}

G

R3

A

C

B

F

R1

R2

E

𝝌(C1)

𝜓(C1)

Hypergraph

Tree decomposition

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Tree Decomposition: Separators- A separator of two adjacent clusters is the set of variables associated to both clusters
- Width of a decomposition/network
- Treewidth= maximum number of variables in clusters

C1

C

C1

C2

E

{A,B,C,E},{R2,R3}

B

C3

C3

C2

A

F

{A,E,F},{R1}

{A,B,D},{R3,R5}

D

C4

C4

G

{A,D,G},{R4}

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Outline- Background
- Contributions
- R(∗,m)C: Consistency property & algorithms [SAC 10, AAAI 10]
- Localized consistency & structure-guided propagation
- Bolstering propagation at separators [CP 12, AAAI 13]
- Counting solutions
- (Appendices include other incidental contributions)

- Conclusions & Future Research

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Relational Consistency R(∗,m)C[SAC 2010, AAAI 2010]

- A parametrized relational consistency property
- Definition
- For every set of m constraints
- every tuple in a relation can be extended to an assignment
- of variables in the scopes of the other m-1 relations

- R(*,m)C ≡ every m relations form a minimal CSP

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Index-Tree Data Structure- Given: two relations, R1 & R2
- Scope(R1)={X,A,B,C} Scope(R2)={A,B,C,D}

- For a given tuple in R1, find matching tuples in R2

Root

A

0

1

B

0

1

1

C

1

1

1

t1

t2

t4

t3

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Weakening R(∗,m)C- Weaken R(∗,m)C by removing redundant edges [Jégou 89]

R(∗,3)C

wR(∗,3)C

R1 R2 R3

R1 R2 R4

R1 R2 R5

R1 R3 R4

R2 R3 R4

R2 R4 R5

R3 R4 R5

R1 R2 R3

R1 R2 R5

R1 R3 R4

R2 R4 R5

R3 R4 R5

R1

R2

R1

R2

B

B

ABD

BCF

ABD

BCF

CF

CF

A

CFG

CFG

C

AD

AD

R5

R5

ADE

ACEG

ADE

ACEG

CG

CG

R3

R4

AE

R3

R4

AE

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Characterizing R(∗,m)C[Jégou 89]

- GAC [Waltz 75]
- maxRPWC[Bessiere+ 08]
- RmC: Relational m Consistency [Dechter+ 97]

R2C

R3C

R4C

RmC

R(∗,3)C

R(∗,4)C

R(∗,m)C

R(∗,2)C

wR(∗,2)C

GAC

maxRPWC

wR(∗,3)C

wR(∗,4)C

wR(∗,m)C

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations (1)Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations (2)Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Algorithms for Enforcing R(∗,m)C- PerTuple
- For each tuple find a solution for the variables in the m-1 relations
- Many satisfiability searches
- Effective when there are many solutions
- Each search is quick & easy

- AllSol
- Find all solutions of problem induced by mrelations, & keep their tuples
- A single exhaustive search
- Effective when there are few or no solutions

- Hybrid Solvers (portfolio based)[+Scott]

ti

t3

t2

t1

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Hybrid Solver[+Scott]

- Choose between PerTuple & AllSol
- Parameters to characterize the problem
- κpredicts if instance is at the phase transition
- relLinkage approximates the likelihood of a tuple at the overlap two relations to appear in a solution

- Classifier built using Machine Learning
- C4.5
- Random Forests

[Gent+ 96]

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Decision Tree#1 κ

#2 log2(avg(relLinkage))

#3 log2(stDev(relLinkage))

#7 stDev(tupPerVvpNorm)

#10 avg(relPerVar)

No

Yes

#1≤ 0.22

Yes

No

PerTuple

#3≤-2.79

Yes

No

AllSol

#7≤0.03

Yes

No

AllSol

#10≤10.05

No

Yes

PerTuple

#2≤-28.75

Yes

No

#7≤0.23

AllSol

AllSol

PerTuple

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations (3)Task: compute the minimal CSP

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Outline- Background
- Contributions
- R(∗,m)C: Consistency property & algorithms [SAC 10, AAAI 10]
- Localized consistency & structure-guided propagation
- Bolstering propagation at separators [CP 12, AAAI 13]
- Counting solutions
- (Appendices include other incidental contributions)

- Conclusions & Future Research

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Localized Consistency- Consistency property cl-R(∗,m)C
- Restrict R(∗,m)C to the clusters

- Constraint propagation
- Guide along a tree structure

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Generating a Tree Decomposition- Triangulate the primal graph using min-fill
- Identify the maximal cliques using MaxCliques
- Connect the clusters using JoinTree
- Add constraints to clusters where their scopes appear

C1

A,B,C,N

C2

A,I,N

R5

R7

R1

C3

I,M,N

N

A

B

C

C4

A,I,K

C5

I,J,K

R3

D

C6

A,K,L

R6

I

J

K

C1

C7

B,C,D,H

E

C7

C

{A,B,C,N},{R1}

C1

N

B,D,F,H

C8

C2

M

L

R2

C8

Elimination order

R4

H

G

F

B,D,E,F

C3

C9

B

D

C7

C2

A

F,G,H

C10

I

M

C5

K

F

H

C3

C4

C8

C6

L

J

C4

C5

C6

C10

C9

C10

E

G

C9

JoinTree

min-fill

MaxCliques

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Information Transfer Between Clusters- Two clusters communicate via their separator
- Constraints common to the two clusters
- Domains of variables common to the two clusters

E

R2

R1

R3

R4

A

D

R4

B

A

D

C

B

A

D

C

R6

R5

R7

Karakashian: Ph.D. Defense

F

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Characterizing cl-R(∗,m)CRmC

R2C

R3C

R4C

R(∗,3)C

R(∗,4)C

R(∗,m)C

R(∗,2)C ≡

wR(∗,2)C

maxRPWC

wR(∗,3)C

wR(∗,4)C

wR(∗,m)C

- GAC [Waltz 75]
- maxRPWC[Bessiere+ 08]
- RmC: Relational m Consistency [Dechter+ 97]

GAC

cl-R(∗,2)C

cl-R(∗,3)C

cl-R(∗,4)C

cl-R(∗,m)C

cl-w(∗,2)C

cl-w(∗,3)C

cl-w(∗,4)C

cl-w(∗,m)C

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations: LocalizationKarakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Structure-Guided Propagation- Orderings
- Random: FIFO/arbitrary
- Static, Priority, Dynamic: Leaves⟷root

- Structure-based propagation
- Static
- Order of MaxCliques

- Priority:
- Process a cluster once in each direction
- Select most significantly filtered cluster

- Dynamic
- Similar to Priority
- But may process ‘active’ clusters more than once

- Static

C1

A,B,C,N

C2

A,I,N

Root

C3

I,M,N

C4

A,I,K

C5

I,J,K

C6

A,K,L

Leaves

C1

C7

B,C,D,H

{A,B,C,N},{R1}

B,D,F,H

C8

B,D,E,F

C9

C7

C2

F,G,H

C10

MaxCliques

C3

C4

C8

C5

C6

C10

C9

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations: PropagationKarakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Outline- Background
- Contributions
- R(∗,m)C: Consistency property & algorithms [SAC 10, AAAI 10]
- Localized consistency & structure-guided propagation
- Bolstering propagation at separators [CP 12, AAAI 13]
- Counting solutions
- (Appendices include other incidental contributions)

- Conclusions & Future Research

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Bolstering Propagation at Separators[CP 2012, AAAI 2013]

- Localization
- Improves performance
- Reduces the enforced consistency level

- Ideally: add unique constraint
- Space overhead, major bottleneck

- Enhance propagation by bolstering
- Projection of existing constraints
- Adding binary constraints
- Adding clique constraints

E

R2

R1

R3

B

A

D

C

Rsep

R6

R5

R7

F

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Bolstering Schemas: Approximate Unique Separator ConstraintE

Ry

R3

D

C

Ra

Rx

E

E

E

E

E

A

A

D

D

B

B

C

C

R2

R1

R3

A

D

R2

R1

B

C

Ry

Ra

R4

B

A

D

C

B

A

D

C

B

A

D

C

B

B

A

C

R3’

R6

R5

R7

R’3

R’3

R6

R5

Rx

F

F

F

F

F

By projection

Binary constraints

Clique constraints

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Resulting Consistency Propertiescl+clq-R(∗,2)C

cl+clq-R(∗,3)C

cl+clq-R(∗,4)C

cl+clq-R(∗,|ψ(cli)|)C

cl+bin-R(∗,3)C

cl+bin-R(∗,4)C

cl+bin-R(∗,|ψ(cli)|)C

cl+bin-R(∗,2)C

cl+proj-R(∗,2)C

R(∗,2)C

R(∗,4)C

cl+proj-R(∗,3)C

R(∗,3)C

cl+proj-R(∗,4)C

cl+proj-R(∗,|ψ(cli)|)C

maxRPWC

GAC

cl-R(∗,2)C

cl-R(∗,3)C

cl-R(∗,4)C

cl-R(∗,|ψ(cli)|)C

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical EvaluationsKarakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Outline- Background
- Contributions
- R(∗,m)C: Consistency property & algorithms [SAC 10, AAAI 10]
- Localized consistency & structure-guided propagation
- Bolstering propagation at separators [CP 12, AAAI 13]
- Counting solutions
- (Appendices include other incidental contributions)

- Conclusions & Future Research

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Solution Counting- BTD is used to count the number of solutions [Favier+ 09]
- Using Count algorithm on trees [Dechter+ 87]
- Witness-based solution counting
- Find a witness solution before counting

ap

Cp

wasted

no solution

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical Evaluations- Task: Count solutions
- BTD versus WitnessBTD
- GAC: Pre-processing & full look-ahead

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Empirical EvaluationsKarakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Conclusions- Question
- Practical tractability of CSPs exploiting the condition linking
- the level of consistency
- to the width of the constraint graph

- Practical tractability of CSPs exploiting the condition linking
- Solution
- Introduced a parameterized consistency property R(∗,m)C
- Designed algorithms for implementing it
- PerTupleand AllSol
- Hybrid algorithms

- Adapted R(∗,m)C to a tree decomposition of the CSP
- Localizing R(∗,m)C to the clusters
- Strategies for guiding propagation along the structure
- Bolstering separators to strengthen the enforced consistency

- Improved the BTD algorithm for solution counting, WitnessBTD

- Two incidental results in appendices

Karakashian: Ph.D. Defense

- Locallization

- Bolstering

- Sol Counting

- Conclusions

Background

Future Research- Extension to non-table constraints
- Automating the selection of
- a consistency property
- consistency algorithms [Geschwender+ 13]

- Characterizing performance on randomly generated problems
- + much more in dissertation

Karakashian: Ph.D. Defense

Collaborations: Bessiere, Geschwender, Hartke, Reeson, Scott, Woodward.

Support: NSF CAREER Award #0133568 & NSF Grant No. RI-111795. Experiments were conducted on the equipment of the Holland Computing Center at UNL.

Karakashian: Ph.D. Defense

Computing All k-Connected Subgraphs

- Avoids enumerating non-connected k-subgraphs
- Competitive on large graphs with small k

b

a

e

c

d

Karakashian: Ph.D. Defense

Solution Cover Problem is in NP-C

- Solution Cover Problem
- Given a CSPwith global constraints
- is there a set of k solutions
- such that ever tuple in the minimal CSP is covered by at least one solution in the set?

- Set Cover Problem ⟶ Solution Cover Problem
- Set Cover Problem
- Given a finite set U and a collection S of subsets of U
- Are there k elements of S
- Whose union is U?

Karakashian: Ph.D. Defense

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