Reconfiguring Shortest Paths in Graphs
This addresses practical problems in road networks, data routing, and logistics, but the general case is incremental as it builds on existing graph theory concepts.
The paper tackles the problem of reconfiguring shortest paths in graphs, showing that the general case is intractable, while efficient algorithms are provided for specific applications like rerouting data packets and shipping container stowage.
Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalize the problem to when at most $k$ (for a fixed integer $k\geq 2$) contiguous vertices on a shortest path can be changed at a time.