The time dependent traveling salesman problem: polyhedra and algorithm

Abstract

The Time Dependent Traveling Salesman Problem (TDTSP) is a generalization of the classical Traveling Salesman Problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.

A Branch-Cut-and-Price Algorithm for the Energy Minimization Vehicle Routing Problem

“…In particular, we show that the arc-load formulation with the new cycle elimination constraints gives the same LP bound as the set partitioning formulation based on 2-cycle-free q-routes. This generalizes a result from Abeledo et al (2013). In addition, these theoretical results can be applied immediately to the analogous formulations for the CVRP.…”

“…(3) The 2-cycle elimination constraints can be seen as a generalization of the 2-cycle elimination constraints for the time dependent traveling salesman problem proposed in Abeledo et al (2013).…”

“…• We derive a family of cycle elimination constraints for the arc-load formulation based on the cycle elimination constraints proposed for the time-dependent traveling salesman problem (Abeledo et al 2013). …”

On multi-robot search for a stationary object

“…TDP, which is NP-hard [2], has been studied from various perspectives during the last few years. Besides exact algorithms introduced by several authors [3], [4], Integer Linear Programming with Branch-and-Cut and Branch-Cutand-Price approaches were proposed [5], [6], [7] for timedependent TSP which generalizes TDP. The best exact algorithm [5], nevertheless, is able to solve instances with up to 107 vertices to optimality in several hours.…”

A customer-centric routing problem with multiple trips of a single vehicle

“…Recently, for the MLP, new and improved mixed integer reformulations have been proposed (Méndez-Díaz et al, 2008;Angel-Bello et al, 2013) and exact methods have been developed based on cutting plane (Méndez-Díaz et al, 2008) and on Branch-and-Cut-and-Price (Abeledo et al, 2013).…”