On path ranking in time-dependent graphs

TL;DR

This paper introduces the concept of path ranking invariance in time-dependent graphs, showing how it simplifies solving vehicle routing problems and providing methods to verify this property for improved computational efficiency.

Contribution

It defines path ranking invariance, links it to solving time-dependent routing problems via simpler time-independent models, and offers a linear program to check this property.

Findings

Path ranking invariance enables simpler routing solutions.

Linear programming can verify the property efficiently.

Embedding bounds improves algorithm performance on TSP and Postman problems.

Abstract

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.