The RAND team’s guarantee involves some pretty mathematics that we take up later in the book. For now it is best to think of the guarantee as a proof, like those we learned in geometry class. The Dantzig et al. proof establishes that no tour through the 49 cities can have length less than 12,345 miles. Matching the proof with their tour of precisely this length shows that this particular instance of the TSP has been settled, once and for all.
Dantzig and company missed out on the $10,000 contest, but we can report that a computer implementation of their ideas makes easy work of the 33-city TSP. A shortest route for Toody and Muldoon is depicted in Figure 1.3. Although no one in 1962 knew for certain that this was the shortest possible tour, a number of contestants did find and report this same ordering. Among the people tied for first place in the contest were mathematicians Robert Karg and Gerald Thompson, who created a hit- or-miss heuristic strategy that produced the winning solution.6 And the story has a happy ending, at least for the mathematics community. As a tiebreaker, contestants were asked to write a short essay on the virtues of one of Procter & Gamble’s products. Thompson’s prose on soaps took a grand prize.
Figure 1.3: The shortest possible route for the Car 54 contest.