Download - Exploiting tree decomposition within VNS for solving ...conf.laas.fr/jfpc-jiaf2012/jfpc/presentations/fontaine_jfpc12.pdf · Introduction Contributions Results Conclusions Perspectives

IntroductionContributionsResultsConclusionsPerspectives

Exploiting tree decomposition within VNS

for solving constraint optimization problems

Mathieu Fontaine Samir Loudni Patrice Boizumault

GREYC (CNRS - UMR 6072) – Universite de CaenBoulevard du Marechal Juin

14000 Caen

May 22, 2012

M. Fontaine, S. Loudni, P. Boizumault (GREYC)Exploiting tree decomposition within VNS May 22, 2012 1 / 38


1 IntroductionTree DecompositionVNS

2 ContributionsDGVNS

Tightness Dependant Tree Decomposition

3 Results

4 Conclusions

5 Perspectives



Background

Constraint Optimisation Problem

WCSP

Two approaches:Complete search: best solution, optimality proof.

Branch and Bound, constraints filtering...

Local search: good solution, no optimality proof, lower computationtime.

IDWalk [Neveu et al., 2004], VNS/LDS+CP [Loudni et al., 2008]



Exploiting problem’s structure

Tree decomposition:

Using problem structure to solve it

Complete search:

BTD [Terrioux & Jegou, 2003], RDS-BTD [Sanchez et al., 2010] ...

Local search:

?



Motivation

Many real life constraint optimization problems are very large.

Many of them are very structured.

Our goal: Take advantage of this structure to guide a VNS localsearch.

Tree Decomposition Methods allow to reveal the structure of theproblem.



Weighted Constraint Satisfaction Problems

WCSP

(X ,D, C, k) is a WCSP:

X = {X1, . . . ,Xn} set of variables,

D = {D1, . . . ,Dn} set of their finite domains,

C = {c1, . . . , ce} set of constraints,

set of variables (scope) of ci : var(ci ),cost function fci : ⊗j∈var(ci )Dj → [0..k],

k > 0 is an integer value (maximal cost value)

Goal: to find a complete assignment minimizing the sumof the costs of unsatisfied constraints.



Tree Decomposition

Introduced by Robertson and Seymour [Journal of algorithms,1986].

Partition the graph into smaller groups (clusters) of stronglyconnected variables.

Create the acyclic graph of clusters.



Tree Decomposition

Tree Decomposition

A tree decomposition of G = (X ,E ) is a pair (CT ,T ) where:

T = (I ,A) is a tree with nodes set I and edges set A,

CT = {Ci | i ∈ I} is a family of subsets of X such that:

1 ∪i∈I Ci = X ,2 ∀ (u, v) ∈ E , ∃Ci ∈ CT s.t. u, v ∈ Ci ,3 ∀ i , j , k ∈ I , if node j is on the path from i to k in T , then

Ci ∩ Ck ⊆ Cj .



Tree Decomposition

We use a tree decomposition method based on a triangulation of theconstraints graph.

It proceeds in 3 steps:1 Triangulation (MCS),2 Search of maximal cliques (graph of clusters),3 Joint Tree (Maximum spanning tree).



Constraint Graph example

let P = (X ,D, C, k) be a WCSPdefined as:

X = {A,B,C ,D,E ,F}C = {C1(A,B,E ),C2(A,C ),C3(D,C ),C4(B,D),C5(D,F )}

B

A

D

F

C

E

constraint graph associated to P



Triangulation

B

A

D

F

C

E

original graph

B

A

D

F

C

E

triangulated graph



Maximal cliques (graph of clusters)

B

A

D

F

C

E

triangulated graph

B,A

F,D

B,C,D

A,B,C

A,B,E

D

B,C

B

graph of clusters



Joint Tree (Maximal spanning tree)

B,A

F,D

B,C,D

A,B,C

A,B,E

D

B,C

B

graph of clusters

A,B,C

B,C,D

B,C

D

B,A

A,B,E

F,D

tree decomposition



VNS [Mladenovic and Hansen, 1997]

Let Nk be the set of all combinations of k variables and S a currentsolution:

Select in Nk a set X of k variables to unfix in S .

unassign X in S

Rebuild S into S ′

if S ′ better than S

Intensification: S ← S ′ and k ← kinit

else

Diversification: k ← k + 1



Exploiting tree decomposition

Local search:

Guiding the search using Tree Decomposition

Decomposition guided VNS (DGVNS)

Refining tree decomposition using the constraints properties

Tightness Dependent Tree Decomposition (TDTD)

[Fontaine et al., 2011]



Contribution: basic ideas

The graph of clusters constitutes a map of the problem.

Clusters gather very correlated variables.

Covering all the clusters increases the diversification of the search.



Scen08 structure (e = 458, n = 5, 286)

X39

X40

X95

X96

X10

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X11

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X13

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X90

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0

Figure: Scen08 instance’s structureM. Fontaine, S. Loudni, P. Boizumault (GREYC)Exploiting tree decomposition within VNS May 22, 2012 17 / 38


Scen08 decomposition

Figure: Scen08 decomposition (From Simon de Givry)



Contribution: Decomposition Guided VNS

Let the variable neighborhood structures Nk,i consist in all combinaisonsof k variables in cluster Ci . DGVNS consist in:

Center the search on a cluster Ci ,

Select variables in this cluster (and if necessary its neighbors),

Unassign these variables in the current solution

Rebuild the solution



Intensification/Diversification in DGVNS

If solution is improved

Intensify the search by setting k to kinit and moving to anothercluster(Csucc(i)).

Else

Diversify the search by increasing k and moving to another cluster(Csucc(i)).

Moving from cluster Ci to next cluster Csucc(i) provides a betterdiversification.



Refining Tree Decomposition

Tree Decomposition allows to build relevant neighborhoods.

But, hard parts of the problem may still remain hidden.

Removing constraints that are easy to satisfy will lead to morerelevant decompositions.



Constraint Tightness

Let c be a constraint:

Constraint Relation

Its relation Rc is the set of all possible assignations of the scope variables.

Cost function

Its cost function fc : Rc 7→ [0, k>].

Constraint tightness

t(c) =#{tuple | fc(tuple) > 0, tuple ∈ Rc}

#{tuple ∈ Rc}



Tightness Dependant Tree Decomposition

Tightness dependant tree decomposition is performed by:

Removing constraints whose tightness is lower than a fixed treshold,

Applying tree decomposition to this subgraph.



Tightness dependent tree decomposition

λ = 0.3:

A B E f1a a b 10a b b 10a a c 10a b c 100b a b 500b b b 200b a c 0b b c 0

t(c1) = 0.75

A C f2a a 0a c 0b a 1000b c 0

t(c2) = 0.25

D C f3b a 10b c 120c a 0c c 350

t(c3) = 0.75

B D f4a b 1000a c 0b b 0b c 10

t(c4) = 0.5

D F f5b a 1000b c 1000c a 1000c c 0

t(c5) = 0.75

AE

B

D

C

F

original graph




λ = 0.3:


t(c1) = 0.75

A C f2a a 0a c 0b a 1000b c 0t(c2) = 0.25

D C f3b a 10b c 120c a 0c c 350t(c3) = 0.75

B D f4a b 1000a c 0b b 0b c 10t(c4) = 0.5

D F f5b a 1000b c 1000c a 1000c c 0t(c5) = 0.75

AE

B

D

C

F

original graph




λ = 0.3:


t(c1) = 0.75

A C f2a a 0a c 0b a 1000b c 0t(c2) = 0.25

D C f3b a 10b c 120c a 0c c 350t(c3) = 0.75

B D f4a b 1000a c 0b b 0b c 10t(c4) = 0.5

D F f5b a 1000b c 1000c a 1000c c 0t(c5) = 0.75

AE

B

D

C

F

constraint removal




λ = 0.3:


t(c1) = 0.75

A C f2a a 0a c 0b a 1000b c 0t(c2) = 0.25

D C f3b a 10b c 120c a 0c c 350t(c3) = 0.75

B D f4a b 1000a c 0b b 0b c 10t(c4) = 0.5

D F f5b a 1000b c 1000c a 1000c c 0t(c5) = 0.75

AE

B

D

C

F

tightness dependentgraph of constraint




λ = 0.3:


t(c1) = 0.75

A C f2a a 0a c 0b a 1000b c 0t(c2) = 0.25

D C f3b a 10b c 120c a 0c c 350t(c3) = 0.75

B D f4a b 1000a c 0b b 0b c 10t(c4) = 0.5

D F f5b a 1000b c 1000c a 1000c c 0t(c5) = 0.75

ABE

BD

DCDF

tightness dependentdecomposition



Experimental Protocol

Implementation

Toulbar2

C++

VNS:

rebuild: LDS (Limited Discrepancy Search) + CP (ConstraintPropagation)

We compare our approach with:

VNS

select: conflict-var

ID-Walk.

Experimental Protocol:

3 benchmarks, 10 instances

50 runs per instance,

timeout: 1 hour per run.



Instances (n = #variables,e = #constraints)

RLFAP

Radio Link Frequency Allocation Problem,

Real life problem proposed by the CELAR (Centre d’Electronique deL’ARmement),

3 instances:

scen06: n = 100, e = 1, 222,scen07: n = 200, e = 2, 665,scen08: n = 458, e = 5, 286.



Contribution of the Tree Decomposition

Instance Method Success Avg time Avg cost

scen06 DGVNS 50/50 112 3,389S∗ = 3, 389 VNS/LDS+CP 15/50 83 3,399

ID-Walk NA 840 3,447 (3,389)

scen07 DGVNS 40/50 317 345,614S∗ = 343, 592 VNS/LDS+CP 1/50 461 355,982

ID-Walk NA 360 373,334 (343,998)

scen08 DGVNS 3/50 1,811 275S∗ = 262 VNS/LDS+CP 0/50 - 394 (357)

ID-Walk NA 3,000 291 (267)



Contribution of TDTD


scen06 DGVNS 50/50 112 3,389S∗ = 3, 389 λ = 0.2 50/50 58 3,389

λ = 0.3 50/50 61 3,389

scen07 DGVNS 40/50 317 345,614S∗ = 343, 592 λ = 0.3 47/50 229 344,198

λ = 0.4 49/50 221 343,600scen08 DGVNS 3/50 1,811 275S∗ = 262 λ = 0.4 6/50 344 274

λ = 0.5 9/50 442 272



Performance profiles: RLFAP

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

0 50 100 150 200 250

ave

rag

e c

ost

CPU time

scen07

DGVNSSDGVNS-0.4

VNS/LDS+CPoptimum

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600

ave

rag

e c

ost

CPU time

scen08

DGVNSSDGVNS-0.5

VNS/LDS+CPoptimum

Figure: Performance profiles on Scen07 and Scen08



Experiment conclusions

On RLFAP and SPOT5 instances:

Improvement are made in both success rates and computation times.Using DGVNS accelerate the amelioration of the solution quality duringexecution.Tightness dependent tree decomposition successfully allows to refineTree decomposition and enhance DGVNS performances.



GRAPH

Graph benchmark:

random instances,

generated from RLFAP instance.

2 problems:

Graph11: n = 340, e = 3, 417;Graph13: n = 458, e = 4, 915.



Contribution of the Tree Decomposition


Graph11 DGVNS 8/50 3, 046 4, 234S∗ = 3, 080 λ = 0.5 12 / 50 2,818 3,643

VNS/LDS+CP 44/50 2, 403 3, 090ID-Walk 0/50 - 41, 604 (27, 894)

Graph13 DGVNS 0/50 - 22, 489 (18, 639)S∗ = 10110 λ = 0.7 0 / 50 - 11,449 (10,268)

VNS/LDS+CP 3/50 3, 477 14, 522ID-Walk 0/50 - 58, 590 (47, 201)



Experiment conclusions

Experiments on GRAPH instances don’t show similar enhancement,

even using TDTD.

Why?



Impact of treewidth

Width

The width of T = (I ,A), w(T ) = maxi∈I (|Ci |−1).

Treewidth

The treewidth w− of G is the smallest width of all possible treedecompositions of G .



Impact of the decomposition’s method

Instance n w− mc w−

n

Scen06 100 11 10 0.11Scen07 200 18 10 0.09Scen08 458 18 10 0.03

Graph05 100 28 8 0.28Graph06 200 54 8 0.27Graph11 340 102 7 0.3Graph13 458 136 6 0.29



Conclusions

1 Tree decomposition enables to build more relevant neigborhoods.

2 DGVNS clearly outperforms VNS/LDS+CP and ID-Walk on SPOT5instances and CELAR instances.

3 Tightness dependent tree decomposition leads to significantimprovements.

4 The performance of our method relies on the treewidth



Perspectives

Using cost to decompose the problem,

Using problem structure to parallelize DGVNS.



Questions?


Download - Exploiting tree decomposition within VNS for solving ...conf.laas.fr/jfpc-jiaf2012/jfpc/presentations/fontaine_jfpc12.pdf · Introduction Contributions Results Conclusions Perspectives

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