Exact and approximation algorithms for routing a convoy through a graph

We study routing problems of a convoy in a graph, generalizing the shortest path problem (SPP), the travelling salesperson problem (TSP), and the Chinese postman problem (CPP) which are all well-studied in the classical (non-convoy) setting. We assume that each edge in the graph has a length and a speed at which it can be traversed and that our convoy has a given length. While the convoy moves through the graph, parts of it can be located on different edges. For safety requirements, at all time the whole convoy needs to travel at the same speed which is dictated by the slowest edge on which currently a part of the convoy is located. For Convoy-SPP, we give a strongly polynomial time exact algorithm. For Convoy-TSP, we provide an \(O(\log n)\) $O(logn)$ -approximation algorithm and an O(1)-approximation algorithm for trees. Both results carry over to Convoy-CPP which—maybe surprisingly—we prove to be \(\textsf{NP}\) $\mathsf{NP}$ -hard in the convoy setting. This contrasts the non-convoy setting in which the problem is polynomial time solvable.