  The traveling salesman problem is one of the most studied combinatorial
optimization problems, because of the simplicity in its statement and the
difficulty in its solution. We characterize the optimal cycle for every convex
and increasing cost function when the points are thrown independently and with
an identical probability distribution in a compact interval. We compute the
average optimal cost for every number of points when the distance function is
the square of the Euclidean distance. We also show that the average optimal
cost is not a self-averaging quantity by explicitly computing the variance of
its distribution in the thermodynamic limit. Moreover, we prove that the cost
of the optimal cycle is not smaller than twice the cost of the optimal
assignment of the same set of points. Interestingly, this bound is saturated in
the thermodynamic limit.
