Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time

Given an undirected graph, the task in Cluster Editing is to insert and delete a minimum number of edges to obtain a cluster graph, that is, a disjoint union of cliques. In the weighted variant each vertex pair comes with a weight and the edge modifications have to be of minimum overall weight. In this work, we provide the first polynomial-time algorithm to apply the following data reduction rule of Böcker et al. [Algorithmica, 2011] for Weighted Cluster Editing: For a graph $G = (V,E)$, merge a vertex set $S ⊆ V$ into a single vertex if the minimum cut of G[S] is at least the combined cost of inserting all missing edges within G[S] plus the cost of cutting all edges from S to the rest of the graph. Complementing our theoretical findings, we experimentally demonstrate the effectiveness of the data reduction rule, shrinking real-world test instances from the PACE Challenge 2021 by around 24% while previous heuristic implementations of the data reduction rule only achieve 8%

Parameterized Dynamic Cluster Editing

We introduce a dynamic version of the NP-hard Cluster Editing problem. The essential point here is to take into account dynamically evolving input graphs: Having a cluster graph (that is, a disjoint union of cliques) that represents 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 (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 two open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the perhaps two most natural parameterizations: the distance of the new "similar" cluster graph to (i) the second input graph and to (ii) the input cluster graph

(1,1)-Cluster Editing is Polynomial-time Solvable

A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$and$d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether$G$can be turned into a cluster graph by deleting at most$d^*(v)$edges incident to every$v\in V(G)$and adding at most$a^*(v)$edges incident to every$v\in V(G)$. Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of$(a,d)$-{Cluster Editing} for all pairs$a,d$apart from$a=d=1.$Abu-Khzam (2017) conjectured that$(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to$C_3$-free and$C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving$(1,1)$-{Cluster Editing} on$C_3$-free and$C_4$-free graphs of maximum degree at most 3

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.Comment: 26 pages, 2 figures. Extended abstract appeared at IWOCA 202