Menger's Theorem for Temporal Paths (Not Walks)

TL;DR

This work investigates Menger-type connectivity for temporal graphs across variants that differ by walks/paths, vertex-disjointness vs temporal vertex-disjointness, direction, and time semantics (strict vs non-strict). It shows a sharp dichotomy: Menger's theorem holds for temporal vertex-disjoint walks (and thus for strict-to-nonstrict reductions) and also when the maximum number of temporal vertex-disjoint temporal paths $k$ is $1$, but fails in general for paths when $k>1$, with the gap between path- and cut-based metrics being arbitrarily large. The authors provide a time-expanded graph framework to compute walks, a semaphore-based reduction to relate strict and non-strict models, and a set of positive results including a polynomial-time method to find two disjoint temporal vertex-disjoint paths when they exist, plus XP-time parameterized algorithms for cuts. These results illuminate fundamental differences between temporal walks and temporal paths and yield practical algorithms for MAPF-like connectivity and robustness questions in temporal networks.

Abstract

A (directed) temporal graph is a (directed) graph whose edges are available only at specific times during its (discretized) lifetime $τ$. In this setting, we ask that walks respect the temporal aspect by defining $\textit{temporal walks}$ as sequences of adjacent edges whose appearing times are either strictly increasing or non-decreasing (here called non-strict), depending on the scenario. The notion of disjointness between walks is also not unique: two walks are $\textit{vertex-disjoint}$ if they do not share a vertex, and are $\textit{temporal vertex-disjoint}$ if they do not share a vertex at the same time. Thus a $\textit{temporal path}$ is a temporal walk where no repetition of vertices, at any time, is allowed. This is an important distinction that separates the interpretation of our results from those of previous works on the topic. In this paper we focus on various questions regarding connectivity (maximum number of disjoint paths) and robustness (minimum size of a cut) between a given pair of vertices. Such problems are related to the well-known Menger's Theorem on static graphs. We explore all possible interpretations of such problems, according to vertex and temporal vertex-disjointness, strict and non-strict temporal paths, and directed and undirected temporal graphs. We present a number of new results, the main of which states that Menger's Theorem holds when the maximum number of temporal vertex-disjoint temporal paths is equal to 1.

Figures (4)

• Figure 1: The labelling $\lambda$ is represented on top of the edges. Temporal paths are: $P_1=(s,1,u,2,t)$$P_2=(s,1,u,3,t)$, $P_3=(s,2,u,2,t)$, $P_4=(s,2,u,3,t)$. Additionally, $W_1=(s,1,u,1,x,2,y,3,u,3,t)$ is a temporal walk.
• Figure 2: Example given in KKK.00, where $\mathop{\mathrm{\mathsf{w}}}\nolimits(s,t)=1<\mathop{\mathrm{\mathsf{wc}}}\nolimits(s,t)=2$. Observe that all the $s,t$-paths passing by $u$ uses either $x$ or $y$. This leads to the conclusion that there are no 2 vertex-disjoint temporal $s,t$-paths. Additionally, none of $u,x,y$ breaks all $s,t$-paths by itself. Therefore, the minimum size of an $s,t$-cut is equal to 2.
• Figure 3: Example where $\mathop{\mathrm{\mathsf{tp}}}\nolimits(s,t)=2<\mathop{\mathrm{\mathsf{tpc}}}\nolimits(s,t)=3$ in the non-strict context. This is proved in \ref{['prop:menger_counter']}. In fact, \ref{['prop:menger_arbitrary_distance']} tells us that the difference $\mathop{\mathrm{\mathsf{tpc}}}\nolimits(s,t) - \mathop{\mathrm{\mathsf{tp}}}\nolimits(s,t)$ can be arbitrarily large.
• Figure 7: Example of paths in the static expansion (on the right) and temporal walks (on the left). In the static expansion, other edges between occurrences of the same vertex are ommited. For simnplicity, the "expansions" of $s$ and $t$ are also ommited. Notice that all temporal walks are strict.

Theorems & Definitions (28)

• Proposition 1
• Theorem 2
• Theorem 3
• Proposition 4
• Proposition 5
• Corollary 6
• Theorem 7
• Theorem 8
• Theorem 9
• Proposition 10