Many academics, myself included, have a passion for finding special cases that can be solved. The main reason is that they result in publications, which result in tenure, so they pay the rent. Another reason for finding special cases is that we can use them to solve general problems. The accuracy of this approach depends on how closely the special case resembles the general problem.
Again, let's take the TSP. Let's solve the ``Canadian Traveling Salesman Problem.'' In Canada, there is one highway (the TransCanada) that runs from coast to coast. All the major cities are on this highway. Suppose the salesman is restricted to this highway. How would he solve the TSP through the major Canadian cities?
Of course, he would simply travel up and down the highway, visiting the cities in the order he meets them. This is a particularly simple case of the TSP: the cities can be placed on a line so that the distance between any two cities is exactly the distance between them on a line.
If we had a TSP to solve, we might first try to place them on a line so that the above condition holds. Normally, we will fail. We then might ask if there is a way of placing the cities on a line so that the distances on the line are almost the ``real'' distances. Now there are many ways of mapping cities in 2 dimensions onto a line. We could do the sweep heuristic: rotating a line about the center of the plane and mapping the cities in the order the line hits them. It turns out that this approach does not adequately retain distance information.
One approach that does work is that of space-filling curves, which is truly a fascinating subject. The main reason that these curves do a good job is that they tend to keep distance information. Note that this mapping is useful not just for the TSP. It can be used whenever the problem with distances on a line is easier than general distances.
We can also consider the special case where the distances come from a tree (a set of edges which does not contain a cycle). Suppose the cities to be visited are nodes of a tree. One nice property of a tree is that there is a unique path between any two nodes of the tree. Suppose the edges of the tree have lengths and the distance between two nodes is defined to be the length of the unique path between them. How would you solve the TSP for such a problem?
You would simply follow the nodes around the tree, skipping any you have visited before. This turns out to be the optimal route for this type of distance.
How can we use this to solve the general TSP? It strikes me that there are some more clever ways, but here is a straightforward approach. Find the minimum spanning tree for the points and solve the TSP on this tree. The closer the tree distances approximate the tree distances, the better this tour will be.
In all, once there is a solvable special case, it is probably worth while to see if you can approximate the general case with it.