[Paper Review] Realizing temporal transportation trees

George B. Mertzios, Michail, Othon+1 more|arXiv (Cornell University)|Mar 27, 2024

Transportation Planning and Optimization1 citations

TL;DR

This paper introduces the Periodic Upper-Bounded Temporal Tree Realization (TTR) problem, proving it is NP-hard even for stars or bounded-degree trees with constant period ∆, yet fixed-parameter tractable (FPT) with respect to the number of leaves via mixed integer linear programming (MILP) and totally unimodular matrix techniques. The key contribution is an FPT algorithm parameterized by leaf count, resolving a fundamental complexity gap in temporal graph realization under upper-bound constraints.

ABSTRACT

A temporal graph 𝒢 = (G,λ) can be represented by an underlying graph G = (V,E) together with a function λ that assigns to each edge e ∈ E the set of time steps during which e is present. The reachability graph of 𝒢 is the directed graph D = (V,A) with (u,v) ∈ A if and only if there is a temporal path from u to v. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph D = (V,A) is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on λ (simple, proper, or both). Answering an open question posed by Casteigts et al. (TCS 2024), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is fixed-parameter tractable for the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set number or treedepth, since the problem is W[2]-hard for them.

Motivation & Objective

• To study the computational complexity of realizing periodic temporal trees under upper bounds on fastest path durations, motivated by transportation network design.
• To address the gap between classic graph realization (with exact distances) and periodic realization (with exact delays), focusing on upper-bound constraints.
• To determine whether the problem remains tractable under structural constraints like star topologies or bounded degree.
• To develop an efficient algorithm for solving TTR when the number of leaves is small, a natural parameter in tree-structured networks.
• To establish a novel connection between temporal graph realization and integer programming via totally unimodular matrices.

Proposed method

• Formalize the TTR problem as a ∆-periodic labeling of edges in a tree G such that fastest temporal path durations between all vertex pairs do not exceed given upper bounds D.
• Use a constructive labeling procedure that assigns labels based on vertex degrees and neighbor ordering, ensuring consistent travel delay computation.
• Model the problem using mixed integer linear programming (MILP) by enumerating global label configurations σ, one per starting label choice.
• Leverage totally unimodular matrix properties to ensure integrality and solvability of the MILP instances derived from each configuration.
• Prove correctness by showing that feasible MILP solutions yield valid temporal labelings satisfying all upper-bound constraints.
• Apply FPT techniques by bounding the number of configurations to O(ℓ^ℓ²), where ℓ is the number of leaves, and solve each MILP instance in FPT-time with respect to ℓ.

Experimental results

Research questions

• RQ1 Is the periodic temporal tree realization problem NP-hard when only upper bounds on fastest path durations are given, even under restrictive topologies like stars or bounded-degree trees?
• RQ2 Can the TTR problem be solved efficiently when parameterized by the number of leaves in the tree, despite its NP-hardness in general cases?
• RQ3 Does the use of totally unimodular matrices and MILP enable an FPT algorithm for TTR, and if so, what structural properties of trees make this possible?
• RQ4 How does the complexity of TTR compare to the classic graph realization problem with exact shortest path distances and to the periodic realization with exact delay values?
• RQ5 Can the FPT approach for trees be extended to general graphs, and what parameters (e.g., distance to clique, independent set) might support such generalizations?

Key findings

• TTR is NP-hard even when the input tree G is a star or has constant maximum degree, and the period ∆ is constant.
• TTR is fixed-parameter tractable (FPT) with respect to the number of leaves ℓ in the input tree, with an algorithm running in f(ℓ) · |(G, D, ∆)|^O(1) time for some computable function f.
• The number of global label configurations σ to consider is bounded by O(ℓ^ℓ²), enabling a finite search space for the FPT algorithm.
• Each MILP instance created from a configuration σ has O(ℓ³) integer variables and can be constructed in polynomial time.
• The correctness of the algorithm is guaranteed by showing that feasible MILP solutions produce labelings where all fastest path durations satisfy the upper bounds in D.
• The solution method relies on the structural properties of trees and the unimodularity of the constraint matrix, ensuring integrality and efficient solvability of the MILP instances.