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The problem of pigeons, curves and passenger vendors

Mo Willems-en children’s book Don’t Let the Dove Drive the Bus!, the main character — a dove, an obva — uses all the tricks in the book (literally) to convince the reader that he should be allowed to drive the bus when the usual human driver suddenly has to leave. Willems ’book had an unintended scientific impact in 2012, when the highly respected journal Human Cognition published the highly respected researchers Brett Gibson, Matthew Wilkinson, and Debbie Kelly. Experimentally they showed that pigeons can find solutions, close to the optimum, to simple cases of the famous mathematical curiosity: the problem of traveling salesmen. Their title is ‘Let the pigeon drive the bus: pigeons can plan future routes in a room’.

Let no one claim that scientists have no sense of humor. Or that nice headlines don’t help create advertising.

The problem of the traveling salesman is not just a curiosity. This is a very important example of a class of problems of enormous practical significance, called combination optimization. Mathematicians have a habit of asking deep and meaningful questions about apparent oddities.

The piece of significant curiosities that inspired this article was created in a helpful book you invented for itinerant vendors. Door-to-door vendors. Like any business guy, the German traveling salesman of 1832 (and he was always a man at the time) devoted himself to using his time efficiently and reducing costs. Luckily, help was available, in the form of a manual: an itinerant salesman — a traveling salesman — how to be and what to do, to get orders, and to ensure that he will have a wonderful success in his business.

This elderly peripatetic salesman stated the following:

The business involves a traveling salesman now here, then there, and it will not be possible to properly indicate the travel itineraries that are appropriate for the cases in which they occur; but sometimes, with the right choice and organization of the route, so much time can be achieved that we do not think we will avoid giving some rules … The main point is always to visit as many places as possible without having to touch the same place twice.

The manual did not propose any mathematics to solve this problem, but it did have the best five trajectories.

It was known that the Traveling Salesman Problem or TSP (later changed to the name Traveling Salesperson Problem to avoid sexism, conveniently bearing the same acronym), is a creative example for the mathematical field now known as combination optimization. This means “finding the best option is too big to check one by one.”

Curiously, it seems that the name TSP was not explicitly used until 1984 in any publications on this issue, although it was much earlier in informal discussions among mathematicians.

In the age of the Internet, companies rarely sell their goods from town to town by sending someone with a suitcase full of samples. They put everything online. As usual (reasonable effectiveness) this culture change has not left TSP obsolete. As online shopping grows exponentially, there is a growing demand for effective ways to determine routes and schedules, from packages to supermarket orders to pizza.

The portability of math also comes into play. TSP applications are not limited to towns or city streets. At one time, famous astronomers had telescopes or shared them with some colleagues. Telescopes could be easily redirected to point to new celestial bodies, so it was easy to improvise. No more, when the telescopes used by astronomers are huge, incredibly expensive, and when they enter the network. Focusing the telescope on a new object takes time and cannot be used for observations while the telescope is moving. Aim to visit in the wrong order and waste a lot of time moving the telescope in a long way, and start again near the place where it started again.

When sequencing DNA, the fragmented sequences of DNA bases must be grouped together correctly, and the order in which this is to be done must be optimized to avoid wasting computer time. Other applications range from the efficient routing of aircraft to the design and manufacture of computer microchips and printed circuit boards. Approximate TSP solutions have been used to find effective routes for Meal Wheels and to optimize blood donation to hospitals. A version of TSP also appeared in the movie “Star Wars,” a more appropriate hypothetical President Ronald Reagan’s Strategic Defense Initiative, where a powerful laser orbiting the Earth would be aimed at some incoming nuclear missiles.

In 1956 the pioneer of Merrill Flood operations research argued that TSP could be tough. In 1979, Michael Garey and David Johnson proved that they were right: there is no effective algorithm to solve the problem “in the worst cases.” But the worst cases are often very well-conceived, and are not real-world examples. So the mathematicians in operations research began to see how many cities they could handle for real-world problems.

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