Maximizing Reachability via Shifting of Temporal Paths
This work provides a parameterized complexity analysis for a temporal graph reachability problem motivated by train networks, revealing which parameter combinations yield tractability.
The paper studies the problem of maximizing reachability from a source in temporal graphs composed of k temporal paths, using shifting operations that preserve temporal continuity. It shows fixed-parameter tractability when parameterized by both k and b (budget) or by k alone with unbounded b, and establishes intractability lower bounds with matching XP algorithms for other parameterizations.
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.