Graph-Based Data Clustering with Overlaps
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Cluster editing problem for points on the real line: A polynomial time algorithm
Information Processing Letters
Generalized graph clustering: recognizing (p, q)-cluster graphs
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Alternative parameterizations for cluster editing
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
Even faster parameterized cluster deletion and cluster editing
Information Processing Letters
Graph-based data clustering with overlaps
Discrete Optimization
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Cluster Editing is transforming a graph by at most k edge insertions or deletions into a disjoint union of cliques. This problem is fixed-parameter tractable (FPT). Here we compute concise enumerations of all minimal solutions in O(2.27k +k 2 n+m) time. Such enumerations support efficient inference procedures, but also the optimization of further objectives such as minimizing the number of clusters. In an extended problem version, target graphs may have a limited number of overlaps of cliques, measured by the number t of edges that remain when the twin vertices are merged. This problem is still in FPT, with respect to the combined parameter k and t. The result is based on a property of twin-free graphs. We also give FPT results for problem versions avoiding certain artificial clusterings. Furthermore, we prove that all solutions with minimal edit sequences differ on a so-called full kernel with at most k 2/4+O(k) vertices, that can be found in polynomial time. The size bound is tight. We also get a bound for the number of edges in the full kernel, which is optimal up to a (large) constant factor. Numerous open problems are mentioned.