50 USA Landmarks This was brilliant work by George Dantzig, Ray Fulkerson, and Selmer Johnson.
And we should all give them a great round of applause. The technique they crafted to attack the problem continues today to be a focal point of the research and practice of the applied mathematics and engineering fields called operations research and mathematical optimization. The Newsweek article mentions that Dantzig's team harnessed the tool of linear programming (LP) to create a new coach outlet purses to carry solution method. Their technique now goes by the name branch and cut, and it applies much more generally than to the TSP alone. Earlier this year, Martin Groetschel, Lex Schrijver, and I commented on the importance of this optimization methodology, that grew out of the 1954 TSP work. Fast forward to March 2015. The center of so much attention was a tour that visits Staedter's 50 locations and returns to the starting point. It was created by Randy Olson, who recently made a really cool post on a path based method to find Waldo. In his 50 landmarks work, Olson employed one of the many available techniques for finding good quality tours. His result is a good tour; about the quality one can obtain using a pencil and paper to trace out carefully a route by hand. (If you want to try for yourself, here is the point set without the tour; also as a pdf file.) A misleading point in the current articles, however, is a claim that finding the absolute best possible tour is out of the question for modern computers. This claim is, of course, at odds with the 1954 Newsweek story. (Although, to be fair, Dantzig's team actually did all of their work with by hand calculations; electronic computers were not plentiful back then.) The impossibility claim is based on the observation that the number of tours grows extremely fast as the number of cities grows, and thus no computer in the world could examine all discount coach purses canada tours through the 50 locations. This observation is correct: there are indeed a ton of tours through 50 points. But what is wrong is the assumption that to find the best possible route, we have to examine them all, one by one. Think about a similar problem of ordering a group of 50 students from smallest to tallest. There are as many possible orderings of the students as there are tours through the 50 points. Nonetheless, the students can get ordered in a snap. Well, maybe not a snap, but fairly quickly by comparing themselves coach outlet locations vineyard to one another. No one knows how to solve the TSP as quickly as sorting heights, but the mathematics of the TSP gives solution coach outlet atlanta international terminal techniques much better than examining each and every tour. Dantzig, Fulkerson and Johnson told us how to do this, and it has been successfully employed on problems having many thousands of points. Here, for example, is a drawing of the shortest tour (traveling via helicopter, not driving) through all 13,509 cities in the continental USA that had a population of at least 500 people, at the time when it was solved in 1998. Solving a 13,509 city TSP takes some high powered computing, but for Staedter's 50 landmarks, if we are given the travel distance between each pair of locations, we can find the absolute shortest tour through the full collection in a fraction of a second on a MacBook, or even an iPhone.
And not only do we find the tour, we also know it is the best possible. (I should add that every student in my optimization course this semester turned in a TSP computer code for homework 2 in February, and all of their codes easily solve the 50 landmarks problem.).
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