The Complexity of Transitively Orienting Temporal Graphs

TL;DR

The paper investigates how information flow in temporal graphs can be oriented transitively, introducing temporally transitive orientations and four variants (TTO, Strict TTO, Strong TTO, Strong Strict TTO). It develops a forcing-based framework that reduces feasibility to a mixed Boolean formula $igl( ext{$ ext{3NAE}$}igr)\wedgeigl( ext{$ ext{2SAT}$}igr)$ and yields a polynomial-time algorithm for TTO, while proving NP-hardness for Strict TTO and tractable 2SAT reductions for the strong variants. It further studies Temporal Transitive Completion (TTC) and Multi-layer Transitive Orientation (MTO), showing TTC is NP-hard in general (though FPT in the number of unoriented edges for oriented inputs) and MTO is NP-complete even with limited time-labels. The results delineate a sharp complexity frontier among temporal transitivity problems and propose a concrete SAT-based approach to temporal orientation that can guide practical algorithms and further research in dynamic networks.

Abstract

In a temporal network with discrete time-labels on its edges, entities and information can only ``flow'' along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge $e=\{u,v\}$ with time-label $t$ specifies that ``$u$ communicates with $v$ at time $t$''. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior. An orientation of a temporal graph is called temporally transitive if, whenever $u$ has a directed edge towards $v$ with time-label $t_1$ and $v$ has a directed edge towards $w$ with time-label $t_2\geq t_1$, then $u$ also has a directed edge towards $w$ with some time-label $t_3\geq t_2$. If we just demand that this implication holds whenever $t_2 > t_1$, we call the orientation strictly temporally transitive, as it is based on the strict directed temporal path from $u$ to $w$. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph $\mathcal{G}$ is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether $\mathcal{G}$ is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

Figures (3)

• Figure 3: The orientation $uv$ forces the orientation $wu$ and vice-versa in the examples of (a) a static graph $G$ where $\{u,v\},\{v,w\}\in E(G)$ and $\{u,w\}\notin E(G)$, and of (b) a temporal graph $(G,\lambda )$ where $\lambda (u,w)=3<5=\lambda (u,v)=\lambda (v,w)$.
• Figure 6: Example of a tail-heavy path.
• Figure 7: Temporal graph constructed from the formula $\texttt{NAE}(x_1, x_2, x_2) \land \texttt{NAE}(x_1, x_2, x_3)$ and orientation corresponding to setting $x_1 = \texttt{false}$, $x_2 = \texttt{true}$, and $x_3 = \texttt{false}$. Each attachment vertex is at the clockwise end of its edge.

Theorems & Definitions (23)

• Definition 1: Temporal Graph KKK00
• Theorem 2
• Theorem 3
• Lemma 5
• Definition 6
• Lemma 7
• Definition 8
• Lemma 9: Temporal Triangle Lemma
• Definition 11
• Lemma 12