Results

**1 - 2**of**2**### Mathematical Programming Approaches to the Traveling Salesman Problem

"... The Traveling Salesman Problem or TSP is probably the best known combinatorial optimisation problem. Informally speaking, one is given a set of cities, along with the cost of traveling between each pair of cities, and one wishes to find a tour of all the cities of minimum cost. The TSP has applicati ..."

Abstract
- Add to MetaCart

(Show Context)
The Traveling Salesman Problem or TSP is probably the best known combinatorial optimisation problem. Informally speaking, one is given a set of cities, along with the cost of traveling between each pair of cities, and one wishes to find a tour of all the cities of minimum cost. The TSP has applications, not only in OR/MS, but in many other fields. In fact, there are no fewer than four books devoted to it [1, 23, 28, 40]. The TSP is notoriously hard to solve, being one of Karp’s original NP-complete problems [27]. Nevertheless, and surprisingly, large-scale in-stances arising in practice can often be solved to proven optimality (or near-optimality) by sophisticated mathematical programming techniques. This article gives an introduction to these techniques. The article is structured as follows. In Section 1, we review some of the basic mathematical programming formulations of the TSP. In Section 2, we discuss polyhedral theory, which is used to derive stronger formulations. In Section 3, we discuss separation routines, which are used to strengthen formulations iteratively. Finally, in Section 4 we discuss exact algorithms for the TSP based on such strengthened formulations. We remark that some other effective techniques for tackling the TSP exist. For an introduction to combinatorial TSP algorithms (i.e., algorithms that work with graph-theoretic relaxations rather than linear programming relaxations), see entry #1.3.5.2. For heuristic and meta-heuristic approaches (which give good, but not necessarily optimal solutions), see entry #1.3.5.3. Throughout this article, we distinguish between the symmetric TSP (STSP), in which the cost of travelling from city A to city B is the same as the cost of travelling in the reverse direction, and the asymmetric TSP (ATSP), in which these costs are permitted to be different.