Fixed-Parameter Enumerability of Cluster Editing and Related Problems

Peter Damaschke
2008 Theory of Computing Systems  
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(2.27 k + 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
more » ... s, 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.
doi:10.1007/s00224-008-9130-1 fatcat:zsxnrpedoba35igvstvsndv46i