  We study a travelling salesman problem where the path is optimized with a
cost function that includes its length $L$ as well as a certain measure $C$ of
its distance from the geometrical center of the graph. Using simulated
annealing (SA) we show that such a problem has a transition point that
separates two phases differing in the scaling behaviour of $L$ and $C$, in
efficiency of SA, and in the shape of minimal paths.
