The.Algorithm.Design.Manual.Springer-Verlag.1998

The Euclidean Traveling Salesman
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The Euclidean Traveling Salesman
In most natural applications of the traveling salesman problem, direct routes are inherently shorter than
indirect routes. For example, if the edge weights of the graph are ``as the crow flies'', straight-line
distances between pairs of cities, the shortest path from x to y will always be to fly directly.
Figure: The triangle inequality typically holds in geometric and weighted graph problems.
The edge weights induced by Euclidean geometry satisfy the triangle inequality, which insists that
for all triples of vertices u, v, and w. The reasonableness of this condition is
shown in Figure . Note that the cost of airfares is an example of a distance function that violates the
triangle inequality, since it is sometimes cheaper to fly through an intermediate city than to fly to the
destination directly. TSP remains hard when the distances are Euclidean distances in the plane.
Whenever a graph obeys the triangle inequality, we can approximate the optimal traveling salesman tour
using minimum spanning trees. First, observe that the weight of a minimum spanning tree is a lower
bound on the cost of the optimal tour. Why? Deleting any edge from a tour leaves a path, the total weight
of which must be no greater than that of the original tour. This path has no cycles, and hence is a tree,
which means its weight is at least that of the minimum spanning tree. Thus the minimum spanning tree
cost gives a lower bound on the optimal tour.
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