Abstract
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 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 NPhard 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 Cluster Editing when p = 2.
Supported in part by the Israel Science Foundation (grant number 565/99).
Supported by an Eshkol fellowship from the Ministry of Science, Israel.
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Shamir, R., Sharan, R., Tsur, D. (2002). Cluster Graph Modification Problems. In: Goos, G., Hartmanis, J., van Leeuwen, J., Kučera, L. (eds) Graph-Theoretic Concepts in Computer Science. WG 2002. Lecture Notes in Computer Science, vol 2573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36379-3_33
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DOI: https://doi.org/10.1007/3-540-36379-3_33
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