Efficiency of Local Search in the Traveling Salesman
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
how many iterations will be needed: whether for instance, there might be an exponential number
of them. 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.
