Abstract
Temporal graphs equip their directed edges with a departure time and a duration, which allows to model a surprisingly high number of real-world problems. Recently, Wu et al. have shown that a fastest path in a temporal graph G from a given vertex s to a vertex z can be computed in near-linear time, where a fastest path is one that minimizes the arrival time at z minus the departure time at s.
Here, we consider the natural problem of computing a fastest path from s to z that is in addition short, i.e. minimizes the sum of durations of its edges; this maximizes the total amount of spare time at stops during the journey. Using a new dominance relation on paths in combination with lexicographic orders on the departure and arrival times of these paths, we derive a near-linear time algorithm for this problem with running time \(O(n + m \log p(G))\), where \(n := |V(G)|\), \(m := |E(G)|\) and p(G) is upper bounded by both the maximum in-degree and the maximum edge duration of G.
The dominance relation is interesting in its own right, and may be of use for several related problems like fastest paths with minimum fare, fastest paths with minimum number of stops, and other pareto-optimal path problems in temporal graphs.
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Danda, U.S., Ramakrishna, G., Schmidt, J.M., Srikanth, M. (2021). On Short Fastest Paths in Temporal Graphs. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_4
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