Maximizing Reachability via Shifting of Temporal Paths

TL;DR

This paper investigates how shifting the availability times of edges in train network-like temporal graphs can maximize reachability, analyzing the problem's complexity with respect to the number of paths and shift budget.

Contribution

It introduces a novel approach to optimize reachability in temporal graphs through label shifting, providing complexity results and fixed parameter tractability conditions.

Findings

Fixed parameter tractability when parameterized by both k and b

Tractability when parameterized by k with unbounded b

Intractability bounds and XP algorithms for other parameterizations

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.