Pigeons, Curve, and the Traveling Salesperson Problem
Sa Mo Willems’ children’s books Don’t Let the Pigeon Drive the Bus!, the main character – a dove, obvs – uses every trick in the book (literally) to convince the reader to be allowed to drive a bus if the regular human driver suddenly leaves. Willems ’book had an unintended scientific consequence in 2012, when the respected journal Human Cognition published a completely respectful paper by the highly respected researchers Brett Gibson, Matthew Wilkinson, and Debbie Kelly. They showed experimentally that pigeons can find solutions, close to the most competent, in simple cases of a famous mathematical curiosity: the Seller’s Problem. Their title was ‘Let the pigeon drive the bus: the pigeons can plan future routes in an instant.’
Don’t let anyone say that scientists aren’t funny. Or those cute titles don’t help to generate publicity.
The Traveling Salesman Problem is not just a greed. This is a very important example of a class of problems that have the greatest practical significance, called combination optimization. Mathematicians have a habit of asking deep and meaningful questions about obvious nonsense.
The piece of essential nonsense that inspires this article has the beginning of a helpful book for-you guessed it-salespeople. Home and home sellers. Like any reasonable business man, the traveling German salesman in 1832 (and in those days it was always a man) set a premium on using his time effectively and minimizing costs. Fortunately, help is already available, in the form of a manual: The traveling salesman – what he needs and what he needs to do, to get orders and to ensure a happy success of his business – to an old salesman.
This old peripatetic vendor points out that:
The business brings the traveling salesman now here, then there, and there are no travel routes that can be properly shown to be appropriate in all the cases that occur; but sometimes, with a suitable choice and arrangement of travel, a lot of time is available, which we think we cannot avoid giving laws also about it… The main point always consists of visiting many places as much as possible, without having to touch the same area twice.
The manual does not suggest any math to solve this problem, but it does contain examples of the five that are said to be the best rounds.
The Traveling Salesman Problem, or TSP, as it is known-which was later modified by the Traveling Salesperson Problem to avoid sexism, which easily has the same acronym-is an example for the mathematics now known combination to optimize. That means ‘finding the best option among the many possibilities that are too numerous to explore each.
Surprisingly, the name TSP did not seem to be used explicitly in any publication about this problem until 1984, even though it was commonly used in the past in informal discussions among mathematicians.
In the Internet age, companies rarely sell their wares by sending anyone from one town to another with a suitcase full of samples. They put everything on the web. As usual (unreasonably effective) this cultural change was not made by the TSP in the past. As online shopping grows, the need for an efficient way to know routes and schedules is even more important for everything from parcels to supermarket orders to pizza.
Bringing math can also be played. TSP applications are not limited to travel between towns or along town streets. In the past, famous astronomers had their own telescopes, or shared them with some colleagues. Telescopes are quick to direct to point at new celestial bodies, so this is easy to do. Not so much, if the telescopes used by astronomers are too large, destructive, and accessible online. Pointing the telescope at a fresh object takes time, and as the telescope is moved, it is not available for observations. Visit targets in the wrong order and many hours are wasted moving the telescope at a distance, and then also returning to a place where it started.
In DNA sequencing, the sequence fragments of DNA bases must be unified, and the sequence in which they are formed must be optimized so as not to waste computer time. Other applications range from efficient aircraft repair to the design and manufacture of computer microchips and printed circuit boards. Estimated solution of TSPs was used to find efficient routes for Foods on the Wheels and to optimize blood transport to hospitals. A version of the TSP is even featured in ‘Star Wars,’ which is more accurately the hypothetical reasoning Strategic Defense Initiative of President Ronald Reagan, in which a powerful laser orbiting Earth is targeted in a series of future missile missiles.
In 1956 the mean operation researcher Merrill Flood argued that the TSP might be difficult. In 1979, Michael Garey and David Johnson proved he was right: there was no efficient algorithm in place to solve the problem of ‘worst cases.’ But the worst-case scenarios in the case have always been very fabricated, and not typical real-world examples. That’s why operations research mathematicians were chosen to see how many cities they could control for the world’s problems.