A Polyhedral Study of the Elementary Shortest Path Problem with Resource Constraints

Abstract

The elementary shortest path problems with resource constraints (ESPPRC) in graphs with negative cycles appear as subproblems in column-generation solution approaches for the well-known vehicle routing problem with time windows (VRPTW). ESPPRC is $\mathcal {NP}$-hard in the strong sense [8]. Most previous approaches alternatively address a relaxed version of the problem where the path does not have to be elementary, and pseudo-polynomial time algorithms based on dynamic programming are successfully applied. However, this method has a significant disadvantage which is a weakening of the lower bound and may induce a malfunction of the algorithm in some applications [9]. Additionally, previous computational studies on variants of VRPs show that labeling algorithms do not outperform polyhedral approaches when the time windows are wide [13] and may not even be applied in some situations [7]. Furthermore, an integer programming approach is more flexible that allows one to easily incorporate general branching decisions or valid inequalities that would change the structure of the pricing subproblem.

In this paper we introduce an ILP formulation of the ESPPRC problem where the capacity and time window constraints are modeled using path inequalities. Path inequalities have been used by Ascheuer et al. [1] and Kallehauge et al. [13], respectively, in solving the asymmetric traveling salesman problem with time windows and the VRPTW. We study the ESPPRC polytope and determine the polytope dimension. We present a new class of strengthened inequalities lifted from the general cutset inequalities and show that they are facet-defining. Computational experiments are performed on the same ESPPRC instances derived from the Solomon’s data sets [9]. Results compared with previous formulations prove the effectiveness of our approach.

This research was supported by the national natural science foundation of China, no. 71501152 and no. 71372202.