TY - JOUR

T1 - Cluster graph modification problems

AU - Shamir, Ron

AU - Sharan, Roded

AU - Tsur, Dekel

N1 - Funding Information:
R. Shamir was supported in part by the Israel Science Foundation (grants number 565/99 and 309/02). R. Sharan was supported by a Fulbright grant and by an Eshkol fellowship from the Ministry of Science, Israel.

PY - 2004/11/30

Y1 - 2004/11/30

N2 - In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making the fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p ≥ 2; Cluster Deletion is polynomial for p = 2 but NP-complete for p > 2; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for a variant of Cluster Editing when p = 2.

AB - In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making the fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p ≥ 2; Cluster Deletion is polynomial for p = 2 but NP-complete for p > 2; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for a variant of Cluster Editing when p = 2.

KW - Approximation

KW - Cluster graph

KW - Clustering

KW - Complexity

KW - Graph modification problem

UR - http://www.scopus.com/inward/record.url?scp=4544382534&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2004.01.007

DO - 10.1016/j.dam.2004.01.007

M3 - Article

AN - SCOPUS:4544382534

VL - 144

SP - 173

EP - 182

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1-2

ER -