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.

翻译：我们研究在由k条时间路径并集构成的时间图中，最大化给定源点的可达性问题，其中每条给定路径是严格递增标签（表示时间可用性）的边序列。此类时间图代表铁路网络。我们考虑对路径标签进行移位操作以保持其时间连续性，即可以提前或推迟时间边的可用时间，并将移位传播至路径中所有其他受影响边以保持其时间连通性。我们针对路径数量k和总预算b（允许执行的最大移位次数）研究了问题的参数化复杂度。结果表明，固定参数可解性在以下两种情况下可实现：(1)当以k和b为参数时；(2)当仅以k为参数且b无约束时。在几乎所有其他情况（例如单参数参数化，或带b约束的k参数化）中，我们建立了与XP算法相匹配的难解性下界。