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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

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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


Practical tractability of csps by higher level consistency and tree decomposition

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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