One thing to note about this specific problem: this is an example of the traveling salesman problem with a metric. This makes efficient (good) approximation algorithms a possibility.

Many NP-hard approximation algorithms classes teach a 1.5 approx known as the Christofides algorithm. This algorithm is guaranteed to provide an approximate solution that is no worse than 1.5 times the optimal total distance, and often much better.

For the ambitious, some of the leading code for solving the TSP to optimality is Concorde (at best it has optimally solved an 85,900 "city" instance): http://www.math.uwaterloo.ca/tsp/concorde/downloads/download...

or solve/run Concorde on Argonne National Laboratory's server here:

http://neos.mcs.anl.gov/neos/solvers/co:concorde/TSP.html

Disclaimer: free for academic use

Ah, cool. So I guess I was wrong that a 5k-city solution was infeasible.