A New Temporal Interpretation of Cluster Editing

TL;DR

This work introduces Editing to Temporal Cliques (ETC), a novel temporal analogue of Cluster Editing, and analyzes its complexity and structure. ETC is NP-hard even when the underlying graph is a path, but becomes tractable in two restricted settings: (i) when each static edge appears only a bounded number of times, and (ii) for the variant that allows only edge additions (Completion to Temporal Cliques, CTC). A central technical contribution is a five-vertex local characterisation: a temporal graph is a $(\Delta_1,\Delta_2)$-cluster temporal graph if and only if every induced subgraph on at most five vertices satisfies the property (with a 3-vertex analogue for $\Delta_1=1$). This local characterisation underpins an $FPT$ algorithm that is parameterised by the number of modifications $k$ and the lifetime $T$ of the input; it also yields polynomial-time recognition and completion procedures via a decomposition into $\Delta_2$-saturated sets. Overall, the paper advances both hardness results and algorithmic tools for temporal clustering problems, and identifies key directions for extending tractability to broader graph classes and combined-parameter regimes.

Abstract

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.

Figures (2)

• Figure 1: The instance $\mathcal{P}$ to the temporal matching problem is shown on the left and the stretched graph $\mathcal{P}_n'$ on which we solve Editing to Temporal Cliques$\,$ is on the right. Non-filler snapshots are shown in white and filler snapshots are grey. Dotted edges show edges that were removed to leave a $(\Delta_1,\Delta_2)$-cluster temporal graph (which is also a temporal matching $\mathcal{M}'$).
• Figure 2: The counter-example from Lemma \ref{['counter']}. This temporal graph shows that $4$-vertex sub-graphs are not sufficient to characterise $(\Delta_1,\Delta_2)$-cluster temporal graphs.

Theorems & Definitions (70)

• Remark 2.1
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop:NPhard']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4