Apologies for resurrecting a dormant thread, but I just got the latest issue of Math Horizons, a magazine published by the Mathematical Association of America, and one of the articles discusses precisely the problem used to set up this Puzzler. Using an algorithm that they go into in the article (and provide links to further resources), they came up with the “most efficient” path through the geographic centers of all 48 contiguous states. The basic idea is that you lay out any random route from one state to another until you return to your starting point, then pick an unvisited state and do likewise, until you have some set of closed circuits that between them hit all the target points. Then you use a rule that they define to “break” pairs of connections in each partial route and join them to another partial route, repeating this until all have been combined into a single route.
There’s additional discussion of similar routes through even larger sets of points, including one that visits hundreds of thousands of points.
The shortest route going through the 48 state capitals is shown here: 
Note that this cannot be the route in the Puzzler because an additional condition there was that you had to start in Delaware.