On path ranking in time-dependent graphs

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.Comment: 28 pages, 2 figure

A branch-and-bound algorithm for the time-Dependent rural postman problem

This paper deals with the time-dependent version of the classical Rural Postman Problem in which arc traversal times vary along the planning horizon. The relationship with the time-invariant counterpart is investigated and a branch-and-bound algorithm is developed. Extensive computational results indicate that the algorithm is capable of solving much larger instances than previously reported