Realizing temporal transportation trees

TL;DR

This work investigates the periodic upper-bounded temporal tree realization (TTR) problem, asking whether a $\Delta$-periodic edge labeling on a tree can realize prescribed upper bounds on fastest-path durations. It reveals a nuanced complexity: TTR is NP-hard even under strong restrictions (constant $\Delta$, star topology, or trees with small diameter/degree), yet it is fixed-parameter tractable when parameterized by the number of leaves, via a MILP-based reduction and a totally unimodular matrix approach that enables Lenstra-type FPT solving. These results delineate a sharp contrast with static graph realization and exact-time periodic variants, and provide a practical framework for scheduling periodic transportation on tree-like networks. The methods combine combinatorial reductions, MILP encodings, and matrix theory to achieve tractable solutions in the leaf-parameterized regime, with implications for scalable transport scheduling on hierarchical networks.

Abstract

In this paper, we study the complexity of the periodic temporal graph realization problem with respect to upper bounds on the fastest path durations among its vertices. This constraint with respect to upper bounds appears naturally in transportation network design applications where, for example, a road network is given, and the goal is to appropriately schedule periodic travel routes, while not exceeding some desired upper bounds on the travel times. In our work, we focus only on underlying tree topologies, which are fundamental in many transportation network applications. As it turns out, the periodic upper-bounded temporal tree realization problem (TTR) has a very different computational complexity behavior than both (i) the classic graph realization problem with respect to shortest path distances in static graphs and (ii) the periodic temporal graph realization problem with exact given fastest travel times (which was recently introduced). First, we prove that, surprisingly, TTR is NP-hard, even for a constant period $Δ$ and when the input tree $G$ satisfies at least one of the following conditions: (a) $G$ is a star, or (b) $G$ has constant maximum degree. Second, we prove that TTR is fixed-parameter tractable (FPT) with respect to the number of leaves in the input tree $G$, via a novel combination of techniques for totally unimodular matrices and mixed integer linear programming.

Figures (4)

• Figure 1: Visualization of a $\Delta$-periodic temporal graph $(G,\lambda,\Delta)$ with $\Delta = 5$ and the following $\Delta$-periodic labeling $\lambda: E \rightarrow \{1,2,\ldots,5\}$: $\lambda(\{v_1, v_2\})=\lambda(\{v_2, v_3\})=3$, $\lambda(\{v_3, v_4\})=4$, and $\lambda(\{v_4, v_5\})=1$. A fastest temporal path from $v_1$ to $v_5$ first traverses $\{v_1,v_2\}$ at time $3$, then $\{v_2,v_3\}$ a time $8$, then $\{v_3,v_4\}$ at time $9$, and then $\{v_4,v_5\}$ at time $11$, and has duration $9$.
• Figure 2: Visualization of the variable gadget for variable $x$ (left) and the clause gadget for clause $(x,y,z)$ (right). Informally, the label on the bold edge of the variable gadget models whether the variable is set to true or false. In the clause gadget, the red edges are associated with variable $x$, the green ones with variable $y$, and the blue ones with variable $z$. The labels on the bold edges in the clause gadget are forced to have the same label as the bold edge in the variable gadget of the associated variable. In particular, this holds for the three very thick edges. Informally, the clause gadget forbids that all three of those edges obtain the same label, which corresponds to the clause being satisfied if not all three variables are set to the same truth value.
• Figure 3: Illustration of $G^{(1)}$ with three variable gadgets for variables $x,y,z$, a clause gadget for clause $c=(x,y,z)$. A labeling $\lambda$ is illustrated that would be created (see proof of \ref{['lem:corrtd1']}) if $x$ is set to true, $y$ is set to true, and $z$ is set to false.
• Figure 4: Illustration of $G^{(2)}$. Depicted are three variable gadgets for variables $x,y,z$ where $x$ is ordered first, $y$ is ordered second, and $z$ is ordered third. Additionally, a clause gadget for clause $c=(x,y,c)$ is depicted. A labeling $\lambda$ is illustrated that would be created (see proof of \ref{['lem:corrmd1']}) if $x$ is set to true, $y$ is set to true, and $z$ is set to false.

Theorems & Definitions (22)

• Definition 1: temporal graph KKK00
• Definition 2: travel delays
• Lemma 4
• Theorem 5
• Proposition 6
• Lemma 7
• Corollary 8
• Lemma 9
• Lemma 12
• Lemma 13