  In this paper we study a property of time-dependent graphs, dubbed path
ranking invariance. Broadly speaking, a time-dependent graph is path ranking
invariant if the ordering of its paths (w.r.t. travel time) is independent of
the start time. In this paper we show that, if a graph is path ranking
invariant, the solution of a large class of time-dependent vehicle routing
problems can be obtained by solving suitably defined (and simpler)
time-independent routing problems. We also show how this property can be
checked by solving a linear program. If the check fails, the solution of the
linear program can be used to determine a tight lower bound. In order to assess
the value of these insights, the lower bounds have been embedded into an
enumerative scheme. Computational results on the time-dependent versions of the
\textit{Travelling Salesman Problem} and the \textit{Rural Postman Problem}
show that the new findings allow to outperform state-of-the-art algorithms.

Discrete Mathematics;Optimization and Control