Canadian Traveller Problems in Temporal Graphs

TL;DR

The paper studies the Canadian Traveller Problem on temporal graphs, introducing two learning models—locally informed and uninformed—for edge block statuses under a budget $k$. It shows that Uninformed Temporal CTP (U-TCTP) admits a polynomial-time solution via a static DAG reduction, including a DP-based approach on directed acyclic graphs and a static-expansion technique for temporal instances. In contrast, Locally Informed Temporal CTP (Li-TCTP) is PSPACE-complete, with a polynomial-time solvable $k=1$ case and NP-hardness for $k\ge 2$, and the static version attains NP-hardness for $k=4$. The results illuminate a sharp complexity divide between Li-TCTP and U-TCTP and extend classical CTP hardness results to temporal settings, including a first constant-$k$ hardness for static CTP. These insights advance understanding of robust itinerary design under adversarial blocking in dynamic networks and have implications for routing in time-evolving systems.

Abstract

This paper formalises the Canadian Traveller problem as a positional two-player game on graphs. We consider two variants depending on whether an edge is blocked. In the locally-informed variant, the traveller learns if an edge is blocked upon reaching one of its endpoints, while in the uninformed variant, they discover this only when the edge is supposed to appear. We provide a polynomial algorithm for each shortest path variant in the uninformed case. This algorithm also solves the case of directed acyclic non-temporal graphs. In the locally-informed case, we prove that finding a winning strategy is PSPACE-complete. Moreover, we establish that the problem is polynomial-time solvable when $k=1$ but NP-hard for $k\geq 2$. Additionally, we show that the standard (non-temporal) Canadian Traveller Problem is NP-hard when there are $k\geq 4$ blocked edges, which is, to the best of our knowledge, the first hardness result for CTP for a constant number of blocked edges.

Figures (6)

• Figure 1: An instance of the Canadian Traveller Problem with weighted edges.
• Figure 2: Example of a temporal graph. Thick (resp. doubly thick) edges represent $k$ parallel (resp. $(k+1)$-parallel) edges. The number of the edge corresponds to the time step in which it is present. All edges have length one.
• Figure 5: Construction of the graph. Double-thick edges represent forced edges. For (single) thick edges, the number in parenthesis represents the number of parallel edges. The number (without parenthesis) represents an edge length; it is 1 when no number exists. For the sake of readability, vertices $\beta_j^1$ and $\beta_j^2$ are not depicted.
• Figure 6: An instance of the Canadian Traveller Problem with weighted edges.
• Figure 7: Example of a temporal graph. Thick (resp. doubly thick) edges represent $k$ parallel (resp. $(k+1)$-parallel) edges. The number of the edge corresponds to the time step in which it is present. All edges have length one.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 6
• Lemma 7
• Theorem 8
• Theorem 9
• Theorem 11