Geometric clustering to minimize the sum of cluster sizes
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GeometricClusteringtoMinimize
theSumofClusterSizes
VittorioBil`o1,IoannisCaragiannis2,ChristosKaklamanis2,and
PanagiotisKanellopoulos2
1DipartimentodiMatematica“EnnioDeGiorgi”Universit`adiLecce,ProvincialeLecce-Arnesano,73100Lecce,Italy.2ResearchAcademicComputerTechnologyInstitute&DepartmentofComputerEngineeringandInformaticsUniversityofPatras,26500Rio,Greece
Abstract.Westudygeometricversionsofthemin-sizek-clusteringproblem,aclusteringproblemwhichgeneralizesclusteringtominimizethesumofclusterradiiandhasimportantapplications.Weprovethattheproblemcanbesolvedinpolynomialtimewhenthepointstobeclus-teredarelocatedonaline.ForEuclideanspacesofhigherdimensions,weshowthattheproblemisNP-hardandpresentpolynomialtimeap-proximationschemes.Thelatterresultyieldsanimprovedapproximationalgorithmfortherelatedproblemofk-clusteringtominimizethesumofclusterdiameters.
1Introduction
Clusteringisanareaofcombinatorialproblemswhichisbothalgorithmicallyrich
andpracticallyrelevant.Severalclusteringproblemshavebeenextensivelystud-
iedsincetheyhaveapplicationsinmanyfieldsincludingdatabasesystems,image
processing,datamining,informationretrieval,molecularbiology,andmore.
GivenasetofpointsX,wecallaclusteranynonemptysubsetofX.Aset
ofclustersisaclusteringforXifeachpointofXbelongstosomecluster.A
clusteringiscalledk-clusteringifitconsistsofatmostkclusters.Ingeneral,
clusteringproblemsarestatedasfollows:Aninstanceofsuchaproblemconsists
ofasetXofnpoints,adistancefunctiondist:X×X→Randaninteger
kandtheobjectiveistocomputeak-clusteringofthepointsinXminimiz-
ingf(C1,...,Ck),wherefisafunctiondefinedontheclusters,typicallyusing
thedistancefunctiondist.Dependingonthedefinitionofthefunctionf,many
differentclusteringproblemscanbedefined.Themostlystudiedonesarethe
k-center,k-median,andk-clustering.Theirobjectivesaretoassignthepointsto
atmostkclusterssothatthemaximumdistancefromanypointtoitscluster
center(k-center)orthesumofdistancesfromeachpointtoitsclosestclus-
tercenter(k-median)orthesumofalldistancesbetweenpointsinthesame
cluster(k-clustering)isminimized.TheseproblemsareNP-hardandseveralap-
proximationalgorithmshavebeenproposed[3,5,13]includingpolynomialtime
approximationschemesforgeometricinstancesoftheseproblems[1,2,10,16].Inthispaper,westudyavariationoftheproblemofclusteringasetofpoints
intoaspecificnumberofclusterssoastominimizethesumofclustersizes.The
sizeofaclustermaybeproportionaltotheradius/diameterofthecluster,toits
area,etc.Inparticular,minimizingthesumofclusterradii/diametershasbeen
suggestedasanalternativetothek-centerobjectiveincertainapplicationsso
astoavoidthedissectioneffect[8]:usingthemaximumdiameter/radiusasthe
objectivesometimesresultsinobjectsthatshouldhavebeenplacedinthesame
clustertobeplacedindifferentclusters.
Clusteringtominimizethesumofdiameters/radiihasbeenstudiedforpoints
inmetricspacesin[6]and[8].Anapproximationalgorithmwhichcomputesa
solutionwithatmost10kclustersofcostatmostafactorofO(logn/k)within
theoptimalsolutionforkclusterswaspresentedin[8].Thisresultwasim-
provedbyCharikarandPanigrahyin[6]whereanalgorithmthatcomputesa
constantapproximatesolutionusingatmostkclustersispresented.Inmetric
spaces,ρ-approximationalgorithmsforclusteringtominimizethesumofdi-
ametersgive2ρ-approximationalgorithmsforthecorrespondingradiiproblem
(andviceversa).Negativeresultsincludea2−inapproximabilityboundfor
minimizingthesumofdiametersinmetricspaces[8]whilethecomplexityofthe
correspondingradiiproblemisopen.Fornon-metrics,noapproximationbound
ispossiblefordiametersinpolynomialtimeunlessP=NPevenfork=3
[8].Whenkisfixed,theoptimalsolutionforradii/diameterscanbefoundin
polynomialtimebyenumeratingtheO(nk)possiblesolutions.Thepapers[12]
and[15]presentfastpolynomialtimealgorithmsforthecasek=2,addressing
theEuclideancaseaswell.Capoyleasetal.[7]studyageneralizedversionofthe
problemforpointsontheEuclideanplaneandshowthat,forfixedkandany
functionoftheclusterdiameters,itcanbesolvedinpolynomialtime.
Inthispaper,weconsidergeometricversionsofthemin-sizek-clustering
problem.Formally,aninstance(X,F,d,α)oftheproblemhasasetXofn
pointswithrationalcoordinatesonthed-dimensionalEuclideanspace,acost
functionFthatassociatesafixednon-negativecostwitheachpoint,anda
constantvalueα.Theobjectiveistocomputeak-clusteringCtogetherwith
centerpointsc∈XineachclusterCsuchthat
C∈CCOST(C)isminimized,
whereCOST(C)isdefinedas(maxp∈Cdist(p,c))α+Fcanddist(p,c)denotesthe
Euclideandistancebetweenthepointspandc.Thequantitymaxp∈Cdist(p,c)
istheradiusofclusterCwithcenterc.
Besidesitsimportanceforclusteringoptimization,anothermotivationfor
studyingthemin-sizek-clusteringproblemisthefollowingscenario.Assumethat
atelecommunicationagencywishestogivewirelessaccesstousersscatteredin