Cluster Editing for Multi-Layer and Temporal Graphs

Abstract

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time kO(k+d) sO(1) for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

A New Temporal Interpretation of Cluster Editing

“…Motivated by the fact that many real-world networks of interest are subject to discrete changes over time, there has been much research in recent years into the complexity of graph problems on temporal graphs, which provide a natural model for networks exhibiting these kinds of changes in their edge-sets. A first attempt to generalise Cluster Editing to the temporal setting was made by Chen, Molter, Sorge and Suchý [10], who recently introduced the problem Temporal Cluster Editing: here the goal is to ensure that graph appearing at each timestep is a cluster graph, subject to restrictions on both the number of modifications that can be made at each timestep and the differences between the cluster graphs created at consecutive timesteps. A dynamic version of the problem, Dynamic Cluster Editing, has also recently been studied by Luo, Molter, Nichterlein and Niedermeier [18]: here we are given a solution to a particular instance, together with a second instance (that which will be encountered at the next timestep) and are asked to find a solution for the second instance that does not differ too much from the first.…”