Reconfiguring Shortest Paths in Graphs

TL;DR

This paper studies the reconfiguration of shortest paths in graphs, proving intractability for general cases and providing efficient algorithms for specific applications like rerouting and shipping, with a generalization for multiple vertex changes.

Contribution

It establishes the computational complexity of reconfiguring shortest paths and offers efficient algorithms for several practical applications, along with a generalization for multiple vertex modifications.

Findings

Reconfiguring shortest paths is intractable in general.

Efficient algorithms are provided for rerouting and related problems.

The problem is generalized to changing up to k contiguous vertices.

Abstract

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.