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Practical Tractability of CSPs by Higher Level Consistency and Tree Decomposition. Shant Karakashian Dissertation Defense. Collaborations : Bessiere , Geschwender , Hartke , Reeson , Scott, Woodward.

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practical tractability of csps by higher level consistency and tree decomposition

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
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
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
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

outline

R(*,m)C

  • 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

constraint satisfaction problem csp

R(*,m)C

  • 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

graphical representations of a csp

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Graphical Representations of a CSP

D

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

redundant edges in a dual graph

R(*,m)C

  • 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

solving csps

R(*,m)C

  • 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)
  • Backtrack search
    • Constructive, exhaustive exploration of search space
    • Variable ordering improves the performance
    • Constraint propagation prunes the search tree

Karakashian: Ph.D. Defense

consistency property definition

R(*,m)C

  • 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

  • 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

algorithms for enforcing a consistency property

R(*,m)C

  • 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

tree decomposition

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Tree Decomposition

C1

{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

tree decomposition separators

R(*,m)C

  • 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

outline1

R(*,m)C

  • 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

relational consistency r m c

R(*,m)C

  • 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

index tree data structure

R(*,m)C

  • 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

weakening r m c

R(*,m)C

  • 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

characterizing r m c

R(*,m)C

  • 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

empirical evaluations 1

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations (1)

Karakashian: Ph.D. Defense

empirical evaluations 2

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations (2)

Karakashian: Ph.D. Defense

algorithms for enforcing r m c

R(*,m)C

  • 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

hybrid solver

R(*,m)C

  • 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

decision tree

R(*,m)C

  • 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

empirical evaluations 3

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations (3)

Task: compute the minimal CSP

Karakashian: Ph.D. Defense

outline2

R(*,m)C

  • 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

localized consistency

R(*,m)C

  • 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

generating a tree decomposition

R(*,m)C

  • 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

information transfer between clusters

R(*,m)C

  • 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

characterizing cl r m c

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Characterizing cl-R(∗,m)C

RmC

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

empirical evaluations localization

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations: Localization

Karakashian: Ph.D. Defense

structure guided propagation

R(*,m)C

  • 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

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

empirical evaluations propagation

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations: Propagation

Karakashian: Ph.D. Defense

outline3

R(*,m)C

  • 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

bolstering propagation at separators

R(*,m)C

  • 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

bolstering schemas approximate unique separator constraint

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Bolstering Schemas: Approximate Unique Separator Constraint

E

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

resulting consistency properties

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Resulting Consistency Properties

cl+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

empirical evaluations

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations

Karakashian: Ph.D. Defense

outline4

R(*,m)C

  • 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

solution counting

R(*,m)C

  • 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

empirical evaluations1

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

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

Karakashian: Ph.D. Defense

empirical evaluations2

R(*,m)C

  • Locallization
  • Bolstering
  • Sol Counting
  • Conclusions

Background

Empirical Evaluations

Karakashian: Ph.D. Defense

conclusions

R(*,m)C

  • 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
  • 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

future research

R(*,m)C

  • 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

slide44

Thank You

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
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 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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