Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques: 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2003 and 7th International Workshop on Randomization and Appro

418 Pages·2003·3.75 MB·English

by  Erik D. Demaine

Correlation Clustering with Partial Information
Erik D. Demaine and Nicole Immorlica(cid:2)
Laboratory for Computer Science, MIT, Cambridge, MA 02139, USA.
edemaine, [email protected].
Abstract. Weconsiderthefollowinggeneralcorrelation-clusteringprob-
lem [1]: given a graph with real edge weights (both positive and nega-
tive), partition the vertices into clusters to minimize the total absolute
weight of cut positive edges and uncutnegative edges. Thus, large posi-
tiveweights(representingstrongcorrelationsbetweenendpoints)encour-
agethoseendpointstobelongtoacommoncluster;largenegativeweights
encouragetheendpointstobelongtodifferentclusters;andweightswith
small absolute value represent little information. In contrast to most
clustering problems, correlation clustering specifies neither the desired
numberof clusters nora distance threshold for clustering; both of these
parametersareeffectivelychosentobethebestpossible bytheproblem
definition.
Correlation clusteringwasintroducedbyBansal, Blum,andChawla[1],
motivated by both document clustering and agnostic learning. They
provedNP-hardnessandgaveconstant-factorapproximationalgorithms
forthespecialcaseinwhichthegraphiscomplete(fullinformation)and
every edge has weight +1 or −1. We give an O(logn)-approximation
algorithm for the general case based on a linear-programming rounding
and the “region-growing” technique. Wealso prove that this linear pro-
gram has a gap of Ω(logn), and therefore our approximation is tight
underthisapproach.WealsogiveanO(r3)-approximationalgorithmfor
K -minor-freegraphs.Ontheotherhand,weshowthattheproblemis
r,r
APX-hard,andanyo(logn)-approximationwouldrequireimprovingthe
best approximation algorithms known for minimum multicut.
1 Introduction
Clustering objectsintogroupsisacommontaskthatarisesinmanyapplications
suchas datamining,webanalysis,computationalbiology,facilitylocation,data
compression, marketing, machine learning, pattern recognition, and computer
vision. Clustering algorithms for these and other objectives have been heavily
investigated in the literature. For partial surveys, see e.g. [6, 11, 14, 15, 16, 18].
In a theoretical setting, the objects are usually viewed as points in either a
metric space (typically finite) or a general distance matrix, or as vertices in a
graph. Typical objectives include minimizing the maximum diameter of a clus-
ter(k-clustering)[8],minimizingthe averagedistancebetweenpairsofclustered
(cid:2) Research was supported in part byan NSFGRF.
S.Aroraetal.(Eds.):APPROX2003+RANDOM2003,LNCS2764,pp.1–13,2003.
(cid:2)c Springer-VerlagBerlinHeidelberg2003