We now have a well-defined local search procedure. How does it measure up? What is its overall running time, and does it always return the best solution?
Embarrassingly, neither of these questions has a satisfactory answer. Each iteration is certainly fast, because a tour has only O (n2) neighbors. However, it is not clear whether there might be an exponential number of iterations required. Likewise, all we can easily say about the final tour is that it is locally optimal - that is, it is superior to the tours in its immediate neighborhood. There might be better solutions further away. For instance, the following picture shows a possible final answer that is clearly suboptimal; the range of local moves is simply too limited to improve upon it.
