An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph. The cuts are generated by several identification procedures that have been described in a companion paper. Whenever the cutting-plane procedure does not terminate with an optimal solution the algorithm uses a tree-search strategy that, as opposed to branch-and-bound, keeps on producing cuts after branching. The algorithm has been implemented in FORTRAN. Two different linear programming (LP) packages have been used as the LP solver. The implementation of the algorithm and the interface with one of the LP solvers is described in sufficient detail to permit the replication of our experiments. Computational results are reported with up to 42 STSPs with sizes rangin...

A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems

Routing and scheduling of vehicles and crews: The state of the art

http://web.ist.utl.pt/~ist11038/CD_Casquilho/TSP1992EJOR_Laporte.pdf

The traveling salesman problem: An overview of exact and approximate algorithms

Publisher Summary This chapter describes vehicle routing problem (VRP) as the problem of designing optimal delivery routes from one or more depots to a set of geographically scattered points, cities, or customers. Practical applications of the VRP are numerous and encompass many spheres of activity. The VRP has received the attention of many Operations researchers over the past three decades. This interest is not only because of the practical importance of the problem, but also its intrinsic difficulty. The largest problems of any complexity that had been solved by exact algorithms contained approximately 30 points. The VRP constitutes a generalization of the travelling salesman problem (TSP) that consists of determining the shortest circuit or cycle passing through each of n points only once. The TSP and the VRP are both NP-hard. However, from a practical point of view, VRPs are in general much more difficult to solve than TSPs of the same size. Effort has been devoted to the solution of the TSP by exact methods; most algorithms for the VRP are heuristics, reflecting the relative difficulty of the problem. The chapter presents a survey of exact algorithms for the VRP, emphasizing recent results.

Exact Algorithms for the Vehicle Routing Problem

We report the solution to optimality of ten large-scale symmetric travelling salesman problems. The travelling salesman problem (TSP) is one of the standard problems of the Operations Research/Management Science literature and is cited in virtually every textbook on this subject. The TSP is a hard combinatorial optimization problem for which to date no technically good algorithm is known. All known algorithmic approaches to the TSP that are amenable to a worst-case analysis have yielded nothing for this problem that is as satisfactory as the existing solution procedures for, as an example, flow problems in networks. Furthermore, recent theoretical work has shown that the TSP is algorithmically equivalent to the general zero-one decision problem with linear constraints and objective function. The phrase “This problem is as hard as a travelling salesman problem” reflects this fact and expresses the slim chances of finding an optimal solution to the problem at hand. It is therefore all the more surprising th...

Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality

Two algorithms using cutting planes are developed for solving the Travelling Salesman Problem. In both algorithms the problem is started with a subset of the set of constraints that define the problem (apart from integrality requirements). However, the two algorithms differ in the order in which the omitted constraints and the cutting planes that are required are generated. The computational experience obtained suggests that cutting planes can provide a competitive approach to other efficient methods of solving the problem.

P. Miliotis

Papers

Using cutting planes to solve the symmetric Travelling Salesman problem