Graph Partitioning with AMPL

26 Pages·2009·0.43 MB·English

by  Antonio Mucherino

GraphPartitioningwithAMPL
Graph Partitioning with AMPL
Antonio Mucherino
Laboratoried’Informatique,ÉcolePolytechnique
CourseonOperationsResearch (ISC610A)
SemesterI- 2008/09
TD5-December4th 2008
GraphPartitioningwithAMPL
Graphpartitioning
Introduction
Recalling some definitions: Clustering
Wealreadyknowwhataclusteringproblemis.
LetX beasetofsampleswhosepartitionisunknown.
Letussupposethatthereisnopreviousknowledgeaboutthe
data(notrainingsetisavailable).
Definition
Clusteringisaimedatfindingapartition{C ,C ,...,C }ofthesetof
1 2 K
data,suchthat
K
X = C, ∀i,j|1≤i <j ≤K C ∩C =∅.
i i j
i[=1
Eachclusterrepresentsasubsetoffeaturesofthesamplesthat
itcontains.
GraphPartitioningwithAMPL
Graphpartitioning
Introduction
Recalling some definitions: Graph
Wealreadyknowwhatagraphis.
Definition
AgraphisanorderedpairG =(V,E)comprisingasetV ofvertices
ornodestogetherwithasetE ofedgesorlinks,whichare2-element
subsetsofV.
Undirectedgraph: agraphinwhichedgeshavenoorientation.
DirectedgraphorDigraph: agraphG =(V,A),whereAisa
setoforderedpairsofvertices,evencalledarcsordirected
edges.
Weightedgraph: agraphinwhichnumbers(weights)are
assignedtoeachedge. Itcanbedirectedandundirected. Itis
denotedbyG =(V,E,w)orG =(V,A,w),wherew represents
theweights.
GraphPartitioningwithAMPL
Graphpartitioning
Introduction
Recalling some definitions: Graph partitioning
Definition
Graphpartitioning istheclusteringproblemoffindingasuitable
partitionofasetofdatarepresentedthroughagraphG.
EachclusterisasubgraphofthegraphG,i.e. asubsetofits
vertices.
Intuitively,thebestpartitionistheonethatseparatessparsely
connecteddensesubgraphsfromeachother.
sparselyconnected: thenumberofedgesbetweenvertices
belongingtodifferentclustersisminimal.
dense: thenumberofedgesbetweenverticesbelongingtothe
sameclusterismaximum.
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Formulating an optimization problem
Howcanwesolveagraphpartitioningproblem?
Weneedtofindapartitioninclustersofaweightedundirected
graphG=(V,E,c),where
V isthesetofverticesofG,
E isthesetofedgesofG,
c isthesetofweightseventuallyassignedtotheedges.
Thisproblemcanbeformulatedasaglobaloptimization
problem.
Wewantthenumberofedgesbetweenverticesbelonging to
differentclusterstobeminimal.
Therefore,weneedtosolveaminimizationproblem,subjecttoa
certainnumberofcontraints.
WewillsolvethisproblembyCPLEX/AMPL.
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Parameters and Variables
Parameters
V,setofverticesofG
E,setofedgesofG
c,setofweightsofG
K,numberofdesiredclustersinthepartition
Variables
x ,binary,indicatesifthevertexu iscontainedinto
uk
theclusterk ≤K:
1 ifu ∈kthcluster
x =
uk (cid:26) 0 otherwise
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Objective function
Whatdoweneedtominimize?
Wewantthetotalweightsoftheedgesbetweendifferent
clusterstobeasminimumaspossible:
Thinkitout: youshouldbeabletogiveananswerwithin1
minute!
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Objective function
Whatdoweneedtominimize?
Wewantthetotalweightsoftheedgesbetweendifferent
clusterstobeasminimumaspossible:
1
min c x x
2 uv uk vl
k6=Xl≤K(uX,v)∈E
Thinkitout: youshouldbeabletogiveananswerwithin1
minute!
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Constraints
ConstraintI
Eachvertexmustbeassignedtoonlyonecluster:
∀u ∈V x =1
uk
kX≤K
ConstraintII
Thetrivialsolution(allthevertecesintoonecluster)mustbe
excluded:
∀k ∈K x ≥1
uk
uX∈V
GraphPartitioningwithAMPL
Graphpartitioning
Mathematicalformulation
Constraints
ConstraintIII (ingeneral,optional)
Eachclustercannotexceedacertaincardinality:
∀k ≤K x ≤C
uk
uX∈V
ConstraintIV (ingeneral,optional)
Verticeshavingdifferentcolorcannotbeclusteredtogether:
∀u 6=v ∈V, k 6=l ≤K, x x ≤γ
uk vl uv
where
1 ifuandvhavethesamecolor
γ =
uv (cid:26) 0 otherwise