Cluster Editing for Multi-Layer and Temporal Graphs

TL;DR

This paper extends classical cluster editing to multi-layer and temporal graphs, analyzing their parameterized complexity and revealing that multi-layer editing is fixed-parameter tractable while temporal editing is W[1]-hard.

Contribution

It introduces and studies the complexity of Multi-Layer and Temporal Cluster Editing problems, highlighting their differences and providing complexity results.

Findings

Multi-Layer Cluster Editing is fixed-parameter tractable.

Temporal Cluster Editing is W[1]-hard with respect to parameter k.

The two problems exhibit different computational complexities despite similar definitions.

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 $k^{O(k+d)}\cdot s^{O(1)}$ for inputs of size $s$, whereas Temporal Cluster Editing is W[1]-hard with respect to $k$ even if $d = 3$.