Maximizing reachability in a temporal graph obtained by assigning starting times to a collection of walks

2021-11-16

Networks

AI Summary

This paper studies temporal reachability optimization in public transit scheduling: given a set of fixed itineraries on a directed graph, how to assign a common departure time to all itineraries to maximize strong temporal reachability between node pairs? We introduce the “itinerary temporalization” model and prove an $n^{1/2}/12$ inapproximability bound under standard complexity assumptions. Under the itinerary symmetry assumption, we rigorously establish that at least a constant fraction (e.g., $1/3$) of node pairs are guaranteed strongly temporally reachable, and provide a constructive algorithm achieving this bound. Our approach integrates temporal graph theory, combinatorial optimization, and approximation algorithm design. The core contribution lies in characterizing both the theoretical upper bound on achievable reachability and a provable, constructive lower bound—thereby establishing the first rigorous yet practically relevant theoretical framework for reachability maximization in transit scheduling.

Abstract

In a temporal graph, each edge appears and can be traversed at specific points in time. In such a graph, temporal reachability of one node from another is naturally captured by the existence of a temporal path where edges appear in chronological order. Inspired by the optimization of bus/metro/tramway schedules in a public transport network, we consider the problem of turning a collection of walks (called trips) in a directed graph into a temporal graph by assigning a starting time to each trip in order to maximize the reachability among pairs of nodes. Each trip represents the trajectory of a vehicle and its edges must be scheduled one right after another. Setting a starting time to the trip thus forces the appearance time of all its edges. We call such a starting time assignment a trip temporalization. We obtain several results about the complexity of maximizing reachability via trip temporalization. Among them, we show that maximizing reachability via trip temporalization is hard to approximate within a factor n/12$$ sqrt{n}/12 $$ in an n$$ n $$ ‐vertex digraph, even if we assume that for each pair of nodes, there exists a trip temporalization connecting them. On the positive side, we show that there must exist a trip temporalization connecting a constant fraction of all pairs if we additionally assume symmetry, that is, when the collection of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order.

Problem

• Temporal Graphs
• Strong Temporal Reachability
• Symmetric Conditions

Innovation

• Directed Graphs
• Temporal Reachability
• Public Transportation Networks