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The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints
arXiv - CS - Discrete Mathematics Pub Date : 2019-09-13 , DOI: arxiv-1909.06437
Arnaud Casteigts; Anne-Sophie Himmel; Hendrik Molter; Philipp Zschoche

Computing a (shortest) path between two vertices in a graph is one of the most fundamental primitives in graph algorithms. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. Such a path is time-respecting, or temporal, if it uses edges over non-decreasing times. In this paper, we investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $\Delta$, referred to as $\Delta$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like infectious diseases and packet routing in communication networks. While the reachability problem for temporal paths in general is known to be polynomial-time solvable, we show that the restless version of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by feedback vertex number or pathwidth of the underlying graph. The main question thus becomes whether the problem is tractable in some natural settings. As of today, no reference set of parameters exist for temporal graphs. We explore several directions in this respect, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.
更新日期:2020-02-20

 

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