The Traveling Salesman Problem

The first problem discussed in section VIII has only a slight connection with the traveling salesman problem (TSP) in that it also requires finding a minimum-distance solution. This is where the similarities end, however, because there aren’t any tricks for finding perfect solutions to the TSP. The problem is NP-hard [1791: there are no known algorithms for finding perfect solutions that have a complexity that grows as only a polynomial function of the number of cities. As a consequence, we must rely on generating solutions that are less than perfect, but as close as possible within a reasonable amount of time. This is a significant challenge, and for this reason we will devote this entire chapter to the TSP. Furthermore, the TSP is related to other problems in scheduling and partitioning, and its broad generality makes it an important keystone in combinatorial optimization. Some ideas that might be useful for addressing the TSP might also be useful in a variety of other applications. The TSP also serves to illustrate many of the central concepts of this book: the flexibility of an evolutionary approach to problem solving, the ease of incorporating problem-specific knowledge, and the ease of hybridization.