Computing All Shortest Passenger Routes with a Tropical Dijkstra Algorithm

TL;DR

This work tackles the problem of computing all potential shortest passenger routes in public-transport networks when arc costs are given as intervals. It introduces a tropical Dijkstra algorithm that operates on tropical polynomials to compute either the complete set $\mathbf{P}_{s,t}$ or an essential subset $\hat{\mathbf{P}}_{s,t}$ of shortest paths, linking the interval-cost problem to a high-dimensional multi-objective shortest path formulation. The authors formalize the Complete Interval Shortest Path Problem ($C$-ISPP) and the Essential Interval Shortest Path Problem ($E$-ISPP), show their MOSP equivalence, and implement a practical tropical-geometry-based solution. Experiments on Wuppertal and TimPassLib demonstrate feasibility on realistic instances, reveal that the essential set is substantially smaller than the complete set yet often contains most practically relevant routes, and show that restricting transfers enables solving larger instances quickly. The results underscore the value of integrating passenger routing with timetabling, enabling preprocessing and more efficient iterative optimization for integrated scheduling problems.

Abstract

Given a public transportation network, which and how many passenger routes can potentially be shortest paths, when all possible timetables are taken into account? This question leads to shortest path problems on graphs with interval costs on their arcs and is closely linked to multi-objective optimization. We introduce a Dijkstra algorithm based on polynomials over the tropical semiring that computes complete or minimal sets of efficient paths. We demonstrate that this approach is computationally feasible by employing it on the public transport network of the city of Wuppertal and instances of the benchmarking set TimPassLib, and we evaluate the resulting sets of passenger routes.

Figures (6)

• Figure 1:
• Figure 2: Example of efficient vs. weakly efficient paths: The upper path is in $\mathbb{P}_{st}^{<_D}$ but not in $\mathbb{P}_{st}^{\leq_{D}}$, and $(3,3,4,4)$ is a point in $\mathbf{W}(\mathbb{P}_{st}^{<_D})$ but not in $\mathbf{W}(\mathbb{P}_{st}^{\leq_{D}})$.
• Figure 3: Augmented transportation network and derived event-activity network. Stations are yellow, cells are grey.
• Figure 4:
• Figure 5: Progress over time of the tropical Dijkstra algorithm

Theorems & Definitions (50)

• Definition 1: Complete Interval Shortest Path Problem (C-ISPP)
• Example 2
• Definition 3: Best Scenario of a Path
• Lemma 3
• Theorem 4
• proof
• Remark 5
• Definition 6: Strict Dominance
• Lemma 7
• Theorem 8