Ladies and Gentlemen, Allow Me to Introduce Spectral and Isoperimetric Graph Partitioning

64 Pages·2011·0.5 MB·English

Ladies and Gentlemen, Allow Me to Introduce
Spectral and Isoperimetric Graph Partitioning
DRAFT: PleaseaddyournametotheAcknowledgmentsbysendingjrscommentsand
correctionsatcs.berkeley.edu
JonathanRichardShewchuk
September11,2011
DepartmentofElectricalEngineeringandComputerSciences
UniversityofCaliforniaatBerkeley
Berkeley,California94720
Abstract
Graph partitioning is the problem of cutting a graph into two or more subgraphs of specified sizes while
minimizinganobjectivefunctionsuchasthetotalweightoftheedgescut. Spectralpartitioningalgorithms
find approximate solutions to graph partitioning problems by computing one or more eigenvectors of a
symmetricmatrixthatrepresentstheconnectivityofthegraph,thenusingtheeigenvectorstoinduceacut.
Spectral partitioning has a rich mathematical theory that includes guaranteed bounds on the quality of the
partitions it produces and, at the other extreme, graphs such as expanders that are useful because they do
not have good partitions. Isoperimetric partitioning is an alternative algorithm that has similar foundations
inlinearalgebra, butisfasterandsimplerbecauseitreplacestheeigenvaluecomputationwiththesolution
ofasymmetriclinearsystem. However,itlacksthetheoreticalguaranteesofspectralpartitioningandistoo
newtohavebeentestedinawidevarietyofapplications.
The first half of this report describes the spectral and isoperimetric graph partitioning algorithms and the
virtuesandlimitationsofspectralpartitioningasanapproximationalgorithm. Thesecondhalfisamiscel-
lanyofvariationsandimplicationsofthesealgorithms. TopicsincludeCheeger’sinequality,thenormalized
cut criterion, vertex separators, bipartite graphs, negative edge weights, eigenvector computations, the use
ofmultipleeigenvectors,andpartitioningagraphintomorethantwosubgraphs.
SupportedinpartbytheNationalScienceFoundationunderAwardsCCF-0430065,CCF-0635381,andIIS-0915462,inpartbythe
UniversityofCaliforniaLabFeesResearchProgram,andinpartbyanAlfredP.SloanResearchFellowship. Theclaimsinthis
documentarethoseoftheauthor.TheyarenotendorsedbythesponsorsortheU.S.Government.
Keywords: spectral graph partitioning, isoperimetric graph partitioning, Laplacian matrix, Fiedler vector,
edge separator, vertex separator, sparsest cut, normalized cut, cut ratio, isoperimetric ratio, isoperimetric
number,edgeexpansion,Cheeger’sinequality,spectralclustering,vectorpartitioning
Contents
I SpectralandIsoperimetricGraphPartitioning 1
1 GraphPartitioningandLinearAlgebra 1
1.1 GraphPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 TheLaplacianMatrixandtheWeightofaCut . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 GraphPartitioningasConstrainedQuadraticOptimization . . . . . . . . . . . . . . . . . . 6
1.5 BoundsontheWeightofaBalancedCut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 SpectralGraphPartitioning 9
2.1 TheRelaxedOptimizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 TheSpectralPartitioningAlgorithm(withoutVertexMasses) . . . . . . . . . . . . . . . . . 10
2.3 VertexMasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 TheSpectralPartitioningAlgorithmwithVertexMasses . . . . . . . . . . . . . . . . . . . 14
2.5 TheVibrationAnalogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 UnbalancedCuts: TheSpectralObjectiveFunctionRevisited . . . . . . . . . . . . . . . . . 15
2.7 BoundsontheWeightofanUnbalancedCut . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 HowGoodIsSpectralPartitioning? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 IsoperimetricGraphPartitioning 21
3.1 GraphPartitioningasConstrainedQuadraticOptimization,Revisited . . . . . . . . . . . . . 21
3.2 TheRelaxedOptimizationProblem,Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 SolvingtheRelaxedOptimizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 TheIsoperimetricPartitioningAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 TheCircuitAnalogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Spectralvs. IsoperimetricPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
II VariationsonaTheme 28
A TheNormalizedCut 28
B Cheeger’sInequalityandaGuaranteefortheSweepCut 29
B.1 IsoperimetricNumbersforSomeClassesofGraphs . . . . . . . . . . . . . . . . . . . . . . 30
B.2 ABetterBisectionMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
B.3 ProofofCheeger’sInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
C BetterPartitionsfromMultipleVectors 34
C.1 ExploitingMultipleEigenvectorsthroughVectorPartitioning . . . . . . . . . . . . . . . . . 34
C.2 ExploitingMultipleIsoperimetricSolutionsthroughVectorPartitioning . . . . . . . . . . . 37
D BeyondBisection: k-SubgraphPartitions 39
D.1 VectorPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
D.2 HypercubePartitionsandEigenvectorRotation . . . . . . . . . . . . . . . . . . . . . . . . 40
D.3 SpectralClusteringfromAnglesbetweenSpectralVectors . . . . . . . . . . . . . . . . . . 41
D.4 SpectralClusteringbyEigenvectorRotation(andtheNormalizedMulticut) . . . . . . . . . 42
D.5 ConstrainingtheIndicatorMatrixbyProjection . . . . . . . . . . . . . . . . . . . . . . . . 45
E VertexSeparators 47
F FixedVertexValues,IterativeImprovement,andExplicitBalancing 49
F.1 FixingVerticesinIsoperimetricPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 49
F.2 FixingVerticesinSpectralPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
F.3 ChangingtheDiagonalinSpectralPartitioning . . . . . . . . . . . . . . . . . . . . . . . . 51
F.4 VerifyingtheOptimalityofaDiscreteSolution . . . . . . . . . . . . . . . . . . . . . . . . 52
G GraphswithNegativeEdgeWeights 53
G.1 NegativeEdgeWeightsandSpectralPartitioning . . . . . . . . . . . . . . . . . . . . . . . 53
G.2 NegativeEdgeWeightsandIsoperimetricPartitioning . . . . . . . . . . . . . . . . . . . . . 54
H ComputingEigenvectors 55
I FasterBipartiteGraphPartitioningandDocumentClustering 56
About this Report
These lecture notes follow my An Introduction to the Conjugate Gradient Method Without the Agonizing
Pain(availablefrommywebpage). BythatImeanIdon’treprisematerialherethatIcoverthere. Thereare
afewsectionsofthatreportthatmightmakethisoneeasier. Ifyou,likeme,gotthroughyourundergraduate
linear algebra class with no intuitive or visual understanding of eigenvectors, I suggest Section 5 of that
report. Seeingeigenvectorsasactorswithadistinctivebodylanguagewillopenupspectralgraphpartition-
ing in a way not otherwise possible. Section 3 of that report offers a visual introduction to the quadratic
form. Section 7.1 illustrates visually what M-orthogonality really means. Appendix C2 has a proof that a
symmetric matrix has n orthogonal eigenvectors. Finally, the conjugate gradient method itself is useful for
isoperimetricgraphpartitioning.
I thank Gilbert Bernstein and Darsh Ranjan for proofreading a draft. I thank Inderjit Dhillon, Bruce
Hendrickson, Ravi Kolluri, Gary Miller, James O’Brien, Horst Simon, Shang-Hua Teng, and Stella Yu for
discussions that clarified my understanding of spectral graph partitioners. I thank Dan Spielman, whose
lecture notes filled many gaps, and Luca Trevisan, whose lecture notes provided the proof of Cheeger’s
inequality. And I thank Dylan Scott and Maurya Talisetti, whose Mathematica code rendered some of the
figures.
Part I
Spectral and Isoperimetric Graph Partitioning
1 Graph Partitioning and Linear Algebra
1.1 GraphPartitioning
The goal of graph partitioning is to cut a weighted, undirected graph into two or more subgraphs that are
roughlyequalinsize,sothatthetotalweightofthecutedgesisassmallaspossible.
There are many versions of the graph partitioning problem. The simplest one requests a bisection of a
graph G into two subgraphs G and G that have an equal number of vertices, as illustrated at upper left
1 2
in Figure 1, or differ in size by at most one vertex. A more general version assigns a mass to each vertex,
and asks that the sum of the vertex masses in G be approximately equal to the sum of the vertex masses
1
in G . Most of the uses of graph partitioning are not so demanding that they require the subgraphs to be
2
perfectlybalanced;forinstance,itmightsufficethatthelargerpiecebenomorethantwiceasheavyasthe
smaller. In some graphs, this looser restriction makes it possible to find a much smaller cut. However, the
constraintsshouldnotbetooliberal: ifyousimplyaskforthesmallestcutpossible,apartitionermightslice
justasinglevertexfromtherestofthegraph,asillustratedatupperrightinFigure1. Thus,someversions
ofthegraphpartitioningproblemtradeofftheweightofthecutagainsttheimbalanceofthesubgraphs, as
illustratedatlowerleft.
You can formalize graph partitioning as a constrained optimization problem. You may choose among
severalobjectivefunctions,suchasthetotalcutweightortheratioofcutweighttolightestsubgraph. Most
ofthemareNP-hardtooptimize.
This report summarizes two ways of reducing graph partitioning to problems in linear algebra that can
be solved by effective, well-known algorithms. Spectral graph partitioning reduces graph partitioning to
finding an eigenvector of a symmetric matrix. Isoperimetric graph partitioning reduces graph partitioning
tosolvingasymmetriclinearsystem.
Neitherofthesemethodsisatruereduction,though,becausebothreductionsusethetechiqueofrelaxing
abinary-valuedunknowntoareal-valuedunknown. Ordinarily,abisectionassignseachvertexofG toone
subgraph, either G or G . By contrast, the relaxed graph bisection problem makes vertices divisible—it
1 2
might assign two thirds of one vertex toG , and the other third toG . It might even assign three halves of
1 2
one vertex toG , and minus one half toG . Relaxation turns a combinatorial optimization problem into a
1 2
numericaloptimizationproblem. Therelaxedbisectionproblemsaresolvableinpolynomialtime.
Having solved a relaxed problem, you convert the fractional solution to an approximate solution of the
original graph partitioning problem by rounding the vertex values—assigning each vertex to G or G by
1 2
comparingitwithsomethresholdvalue. YoumightplacethethresholdatavertexthatishalfinG andhalf
1
inG ,butyoudon’thaveto;ifyouwantapartitionthatisasbalancedaspossible,placethethresholdatthe
2
medianvertex.
Andthen...youhopethattheroundedsolutionretainssomesemblanceofnear-optimality. Anattractive
featureofspectralpartitioningisthatinsomecircumstances,aboundcalledCheeger’sinequalityguarantees
that the cut is good. But “good” may fall well short of optimal. In practical applications, we are usually
pleased,buttherearegraphsforwhichtheroundedsolutionisalmostasbadaspossible.
2 JonathanRichardShewchuk
Figure1: Severalwaystocutagraphwhoseedgeweightsandvertexmassesareall1. Theminimumbisection,
atupperleft,comprisesfiveedges. Theminimumcut,atupperright,comprisestwoedgesbutdetachesjustone
vertex. Themaximumcut,atlowerright,comprisesallbutthreeedges,andforthisgraphalsohappenstobethe
maximum bisection. The sparsest cut, at lower left, has a sparsity of 4/(11·9), and for this graph also has the
minimumisoperimetricratio,4/9.
SpectralpartitioningwasintroducedbyKennethHall[27]in1970(inthesettingofclustering;hedidn’t
use the word “partition”), rediscovered by Wilm Donath and Alan Hoffman [13] in 1972 and by Miroslav
Fiedler[19]in1975,andpopularizedbyPothen,Simon,andLiou[49]in1990.1 Ithasbecomeamainstay
ofscientificcomputingandimageprocessing. IsoperimetricpartitioningwasintroducedbyLeoGradyand
Eric Schwartz [25] in 2006, and it appears to be a faster and easier alternative. It will be interesting to see
whetheritwillbecomeaspopularasspectralpartitioning.
Graphpartitionershavemanyapplications. Theyassigncomputationstoprocessorsinparallelcomput-
ers [49, 52, 29, 58]: vertex masses represent the running times of computations, and cut weights represent
communication times between processors. They segment images—that is, they find boundaries separating
objects in photographs or medical images [51, 59, 60]. They are used in divide-and-conquer graph algo-
rithmssuchasnesteddissection[40],whichorderstherowsandcolumnsofasparsematrixtoreducethefill
1Manypublicationscreditthe1973articlesbyDonathandHoffman[14]andFiedler[18],butacarefulreadingofthosepapers
revealsnohintofaspectralpartitioningalgorithm.Sections1.5and2.7mentionafewofthosetwopapers’accomplishments.
GraphPartitioningandLinearAlgebra 3
(nonzero components) in its Cholesky or LU decomposition. They cluster words that are frequently found
togetherindocuments,andclusterdocumentsthathavesimilarwords[11]. Manyoftheearliestalgorithms
weremotivatedbycircuitlayout(oncircuitboardsandinintegratedcircuits),donebyfindingpartitionsof
circuit components that minimize the number of long interconnections between physically distant compo-
nents[34,27,13,2].
Thespectralgraphtheoryintroducedhere,especiallyCheeger’sinequality,hasapplicationsbeyondthe
act of graph partitioning. The mixing times of random walks on graphs and the running times of approx-
imate counting algorithms are governed by the optimal cut (as measured by an objective function called
the isoperimetric ratio) [53]. Graphs with no good partition, called expanders, are used in robust network
design and error-correcting codes. Expanders are central in several results in complexity theory, including
thecomplexityclassequivalenceSL=L(symmetriclog-space=log-space,meaningthatundirectedgraph
reachability is solvable in logarithmic space) [50] and a proof of the famous PCP Theorem [12], which
characterizesNPintermsofprobabilisticallycheckableproofs.
1.2 Definitions
Let G = (V,E) be an undirected graph with vertex set V and edge set E. A partition of G is a set of
subgraphs G , G , ..., G such that every vertex in V appears in exactly one of the subgraphs. Here, the
1 2 j
term subgraph implies that if an edge of G has both its endpoints in the same subgraph G, then G also
i i
containstheedge.
GivenapartitionofG,thecutoredgeseparatoristhesetofedgesinEwhoseendpointslieindifferent
subgraphs. We say that the partition cuts those edges. (Another type of separator, the vertex separator, is
discussed in Section E.) Each edge has a specified numerical weight, which indicates how bad it is to cut
that edge. The weight of a cut is the sum of the weights of all the edges cut. We wish to minimize this
weight.
To influence how a partition should be balanced, we can assign each vertex a mass. The mass of a
subgraph is the sum of the masses of its vertices. We might request that two subgraphs have equal mass
(whichisn’talwayspossible),orthattheirmassesdifferbyatmostafactoroftwo. Becauselinearalgebraic
partitionersoperatebyroundingafractionalsolution,wecanchoosetheroundingthresholdtobewhatever
value minimizes the difference between the subgraphs’ masses. Alternatively, we can look for a partition
thattradescutweightagainstmassimbalanceaccordingtosomeobjectivefunction.
For simplicity, most of this report addresses a bipartition of G into two subgraphs. A bisection is a
perfectly balanced bipartition. Unbalanced bipartitions are discussed starting in Section 2.6, and partitions
intothreeormoresubgraphsearnsomeinkinSectionD.
Two objective functions for judging unbalanced partitions are popular, in part because they emerge
naturallyfromspectralgraphtheory. Theisoperimetricratioofabipartitionis
Cut(G ,G )
1 2
,
min{Mass(G ),Mass(G )}
1 2
whereCut(G ,G )istheweightofthecutandMass(G)isthetotalmassoftheverticesinG. Theminimum
1 2 i i
isoperimetricratioamongallcutsofG iscalledtheisoperimetricnumber. Ifeveryvertexhasmass1,thus
Mass(G) = |V|,theisoperimetricnumberisalsocalledtheedgeexpansion. Ifeachvertexhasmassequal
i i
tothesumoftheadjoiningedgeweights(asdescribedinSectionA),theisoperimetricnumberisalsocalled
the conductance or the Cheeger constant ofG. The graph in Figure 1 has an isoperimetric number of 4/9,
obtainedbythecutatlowerleft.
4 JonathanRichardShewchuk
Thesparsityorcutratioofabipartitionis2
Cut(G ,G )
1 2
.
Mass(G )Mass(G )
1 2
The sparsity is also called the uniform sparsity if every vertex has mass 1, or the normalized cut criterion
ifeachvertexmassisthesumoftheadjoiningedgeweights(seeSectionA). Theoptimalcutsunderthese
criteria are called the sparsest cut, the uniform sparsest cut, and the normalized cut. Figure 1 illustrates a
cut of (uniform) sparsity 1/20 at upper left, and the (uniform) sparsest cut, having sparsity 4/99, at lower
left.
Theaveragecutcriterion
(cid:32) (cid:33)
1 1
Cut(G ,G ) +
1 2
Mass(G ) Mass(G )
1 2
isequaltothesparsitytimesMass(G),sooptimizingoneisequivalenttooptimizingtheother.
Mostgraphpartitioningalgorithmsareformulatedforgraphswithpositiveedgeweights;negativeedge
weights make many algorithms fail. For example, a special case of graph partitioning that is solvable in
O(|V||E|+|V|2log|V|) time is the minimum cut problem [56], which asks for the cut of minimum weight
that disconnects the graph (though it might isolate just a single vertex). However, if some edges have
negative weight, the minimum cut problem becomes NP-hard. Observe that finding the minimum cut of a
graph with negative edge weights is equivalent to finding the maximum cut of a graph with positive edge
weights. Findingthemaximumcut,illustratedatlowerrightinFigure1,isoneofthetwenty-oneproblems
that Karp [33] proved NP-complete in his famous paper that established the importance of the theory of
NP-completeness.
Another partitioning problem solvable in polynomial time if negative edge weights are forbidden, but
NP-hardotherwise,istheclassicsource-targetminimumcutproblem,inwhichthereisoneormoresource
vertices that must be in G , and one or more target vertices that must be in G . The goal is to find the
1 2
minimum cut that separates the source vertices from the target vertices. By the max-flow min-cut theo-
rem [15, 20], which applies only if the edge weights are nonnegative, this problem is equivalent to the
maximumnetworkflowproblem,forwhichthereisarichliteratureofpolynomial-timealgorithms.
Anadvantageofthespectralandisoperimetricpartitionersisthattheycanaccommodatenegativeedge
weights; see Section G. Even though negative edge weights make graph partitioning harder in principle,
theircarefulusemakessomeapplicationsfasterandbetter.
1.3 TheLaplacianMatrixandtheWeightofaCut
Linearalgebraicmethodsforgraphpartitioningbeginwiththeideathatwecanexpresstheweightofacut
inmatrixnotationasaquadraticfunctionofbinaryvariables.
Number the vertices ofG = (V,E) from 1 to n = |V|. Let (i, j) denote an edge connecting vertex i to
vertex j, and let w be its weight, with w = 0 if (i, j) (cid:60) E. Because G is undirected, (i, j) = (j,i) and
ij ij
w = w . Assumetherearenoself-edgesoftheform(i,i).
ij ji
2Thisdefinitionofsparsityisaspecialcaseofthenotionofanonuniformsparsestcutinmulticommodityflows.Thedefinition
hereariseswheneveryvertexiwishestoroutetoeveryvertex jaflowproportionaltomm ,wherem isthemassofvertexi. The
i j i
term“cutratio”referstotheamountofflowthatcancrossthecutdividedbytheamountofdemandacrossthecut.
GraphPartitioningandLinearAlgebra 5
Definethesymmetricn×nLaplacianmatrix Lwhosecomponentsare
(cid:40)
−w i (cid:44) j,
Lij = (cid:80)k(cid:44)iijwik i = j. (1)
L is effectively a matrix representation of G. (For the purpose of partitioning a graph, there is no need to
distinguishedgesofweightzerofromedgesthatarenotinthegraph.)
Suppose we are given a partition of G into subgraphs G and G . Let x be an n-component indicator
1 2
vector that represents the partition as follows. Let c and c be two distinct, arbitrary constants. We set
1 2
x = c ifvertexiisinsubgraphG ,and x = c ifvertexiisinsubgraphG . Thentheweightofthecutis
i 1 1 i 2 2
xTLx
Cut(G ,G ) = , (2)
1 2
(c −c )2
1 2
because
(cid:88) (cid:88)n (cid:88)
xTLx = −2w x x + x2 w
ij i j i ik
(i,(cid:32)(cid:32)j(cid:32))(cid:32)(cid:32)∈(cid:32)(cid:32)E(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) i=(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) k(cid:44)(cid:32)(cid:32)(cid:32)i(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
off-diagonalterms diagonalterms
(cid:88) (cid:16) (cid:17)
= w x2+w x2−2w x x
ij i ij j ij i j
(i,j)∈E
(cid:88)
= w (x −x )2 (3)
ij i j
(i,j)∈E
= (c −c )2Cut(G ,G ). (4)
1 2 1 2
Thelaststepfollowsbecause x − x iszeroiftheedge(i, j)isnotcut,andis±(c −c )if(i, j)iscut. The
i j 1 2
matrix L was chosen deliberately to make this calculation work. (If it seems magical to you, try working
back from Expression (3) to its expansion as a quadratic polynomial. The coefficients of that polynomial
determineL.)
Equation (2) shows that minimizing the weight of the cut is equivalent to minimizing the quadratic
function xTLx. Thisisacoreobservationunderlyingspectralandisoperimetricgraphpartitioners.
Observethatifalltheweightsarenonnegative,itfollowsfromEquation(3)thatxTLx ≥ 0forallx—said
differently,Lispositivesemidefinite(andhasnonegativeeigenvalues). Moreover,ifGisaconnectedgraph
andallitsedgeshavepositiveweight,Expression(3)iszeroifandonlyifallthecomponentsof xareequal
to each other. Therefore, L has exactly one eigenvector with eigenvalue zero,3 namely 1 = [1,1,...1]T.
Thiseigenvectorreflectsthefactthatthecuthaszeroweightifandonlyifeveryvertexisinonesubgraph,
andtheothersubgraphisempty.
Figure2depictsisosurfacesoftheformxTLx = cforathree-vertexgraph. Theisosurfacesareconcentric
ellipticalcylinderswhosecentralaxisisparallelto1.
If G is not connected, but all its edges have positive weight, then L has one zero eigenvalue for each
connected component of G: setting x = 1 for each vertex i in a component and x = 0 for each vertex in
i i
the other components yields an eigenvector with eigenvalue zero. Every eigenvector with eigenvalue zero
(including1)isalinearcombinationoftheseones,becauseExpression(3)canbezeroonlyif xassignsall
3RecallthatvisaneigenvectorofLwitheigenvalueλifLv=λv.
6 JonathanRichardShewchuk
x
2 xTLx = 0
2
1
xTLx = 8
5
1 xTLx = 16
3
3
xTLx = 24
1
x
1
1Tx = 0
x
3
Figure 2: Isosurfaces of the function xTLx for the graph at upper left. The shaded region represents the linear
balanceconstraint1Tx=0.
the vertices in a connected component the same value. The cylindrical axis in Figure 2 is no longer a line,
butratheraplaneorhigher-dimensionalsubspacespannedbytheseeigenvectors.
The foregoing discussion assumes that c and c are constant and distinct, but holds regardless of their
1 2
specific values. Their role will become nebulous in two ways. First, a partitioning problem is relaxed by
permitting the components of x to take on any real value, not just c or c . Second, c and c don’t need
1 2 1 2
to be constant. In Section 2.6, we choose them to be a function of the cut—specifically, a measure of the
balance.
1.4 GraphPartitioningasConstrainedQuadraticOptimization
Thecutweightachievesitsminimum xTLx = 0whenG =G andG = (∅,∅),butthat’snotuseful. Hence,
1 2
weimposeabalanceconstraint.
Supposewechoosetheconstantsc = 1andc = −1tosignifywhichverticesareinG andwhichare
1 2 1
inG ; thus we call x a signed indicator vector. (This name is in contrast to the 0-1 indicator vectors used
2
inisoperimetricpartitioningandsomemultiple-subgraphspectralpartitioningalgorithms.) Wecanspecify
thatexactlyhalftheverticesareinG andhalfareinG byimposingtheconstraint
1 2
1Tx = 0, (5)
which dictates that the number of components of x that are 1 is equal to the number that are −1. Figure 2
illustrates this constraint as a shaded planar cross-section of the cylindrical isosurfaces. Of course, this