Abstract
We introduce a dynamic version of the NP-hard graph modification problem Cluster Editing. The essential point here is to take into account dynamically evolving input graphs: having a cluster graph (that is, a disjoint union of cliques) constituting a solution for a first input graph, can we cost-efficiently transform it into a “similar” cluster graph that is a solution for a second (“subsequent”) input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (three modification scenarios edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as (parameterized) hardness results, thus (except for three open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the two perhaps most natural parameterizations: the distances of the new “similar” cluster graph to (1) the second input graph and to (2) the input cluster graph.
Highlights
The NP-hard Cluster Editing problem[6, 41], known as Correlation Clustering[5], is one of the most popular graph-based data clustering problems in algorithmics
Updating an existing cluster graph Dynamic Cluster Editing can be interpreted to model a smooth transition between cluster graphs, reflecting that “customers” working with clustered data in a dynamic setting may only tolerate a moderate change of the clustering from “one day to another” since “revolutionary” transformations would require too dramatic changes in their work
Editing into a compromise clustering When focusing on the edge-based distance, one may generalize the definition of Dynamic Cluster Editing by allowing Gc to be any graph
Summary
The NP-hard Cluster Editing problem[6, 41], known as Correlation Clustering[5], is one of the most popular graph-based data clustering problems in algorithmics. Motivated by the assumption that the “new” cluster graph should only change moderately but still be a valid representation of the data, we parameterize both on the number of edits necessary to obtain the “new” cluster graph and on the difference between the “old” and the “new” cluster graph. We remark that there have been previous parameterized studies of dynamic (or incremental) graph problems, dealing e.g. with coloring[30], domination[3, 19], and vertex deletion[2, 34] problems
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