Maximizing Reachability via Shifting of Temporal Paths

2026-05-12

AI Summary

This study addresses the problem of maximizing source reachability in a temporal graph composed of $k$ temporal paths, under a total offset budget $b$, by applying time-label shifts that preserve intra-path temporal consistency. Employing parameterized complexity analysis, the work integrates temporal graph modeling with a label-shift propagation mechanism to establish fixed-parameter tractability when parameterized by $(k,b)$ jointly or by $k$ alone. For all other parameterizations, matching complexity lower bounds and XP algorithms are provided. The results comprehensively delineate the computational boundaries of the problem across different parameter settings, yielding either efficient algorithms or hardness evidence accordingly.

Abstract

We examine the problem of maximizing the reachability of a given source in temporal graphs that are given as the union of k temporal paths, i.e., every given path is a sequence of edges with strictly increasing labels that denote availability in time. This type of temporal graphs represent train networks. We consider shifting operations on the labels of the paths that maintain their temporal continuity. This means that we can move the availability of a temporal edge later or earlier in time, and propagate the shifts to all other affected edges of the path in order to preserve its temporal connectivity. We study the parameterized complexity of the problem with respect to the number of paths k, and the total budget b, where b is the maximum number of shifts we are allowed to perform. Our results reveal that fixed parameter tractability can be achieved (1) when parameterized both by k and b, and (2) when parameterized by k, and b is unconstrained. In almost every other case, e.g., parameterized by a single parameter or parameterized by k, while having a bound on b, we establish intractability lower bounds that are matched by XP algorithms.

Problem

• temporal graphs
• reachability
• temporal paths
• shifting operations
• parameterized complexity

Innovation

• temporal graphs
• reachability maximization
• shifting operations
• fixed-parameter tractability
• parameterized complexity