The Travelling Salesman Problem (TSP) is probably the most well-known member of the wider field of combinatorial optimization (CO) problems. These are difficult optimization problems where the set of feasible solutions (trial solutions which satisfy the constraints of the problem but are not necessarily optimal) is a finite, though usually very large set. The number of feasible solutions grows as some combinatoric factor such as , where N characterizes the size of the problem. We have already commented on the use of neural networks for the TSP in the previous section. Here we show how to combine problem-specific heuristics with simulated annealing, a physical optimization method.
It has often been the case that progress on the TSP has led to progress on many CO problems and more general optimization problems. In this way, the TSP is a playground for the study of CO problems. Though the present work concentrates on the TSP, a number of our ideas are general and apply to all optimization problems.
The most significant issues occur as one tries to find extremely good or exact solutions to the TSP. Many algorithms exist which are fast and find feasible solutions which are within a few percent of the optimum length. Here, we present algorithms which will usually find exact solutions to substantial instances of the TSP. We are limited by space considerations to a brief presentation of the method-more details may be found in [Martin:91a].
In a general instance of the TSP one is given N ``cities'' and a matrix giving the distance or cost function for going from city i to j. Without loss of generality, the distances can be assumed to be positive. A ``tour'' consists of a list of N cities, , where each city appears once and only once. In the TSP, the problem is to find the tour with the minimum ``length,'' where length is defined to be the sum of the lengths along each step of the tour,
and is identified with to make it periodic.
Most common instances of the TSP have a symmetric distance matrix; we will hereafter focus on this case. All CO problems can be formulated as optimizing an objective function (e.g., the length) subject to constraints (e.g., legal tours).