Bathroom problem, curves, mobile salesperson
In Mo Williams Children’s book Don’t let the pigeons drive the bus!, the main character – a pigeon, obvs – uses every trick in the book (literally) to convince the reader that he should be allowed to drive a bus when the average driver suddenly has to leave. Williams’ book had an unintended scientific consequence in 2012, when the respected journal Human Cognition published a highly respected paper by well-respected researchers Brett Gibson, Matthew Wilkinson, and Debbie Kelly. They have shown experimentally that pigeons can find solutions, close to optimal, to simple cases of a famous mathematical curiosity: the peddler problem. Their title was “Let the Pigeons Drive the Bus: Pigeons Can Plan the Room’s Future Routes”.
No one should claim that scientists lack a sense of humor. Or that cute headlines don’t help spark publicity.
The traveling salesman’s problem isn’t just curiosity. It is a very important example of a class of problems of enormous practical importance, called combinatorial optimization. Mathematicians have a habit of asking deep and important questions in terms of apparent trivia.
The important trivia piece that inspired this article has its origins in a useful—you guessed it—book for mobile salespeople. street vendors. Like any sane businessman, the 1832 German peddler (and always was a man in those days) put a premium on using his time efficiently and cutting costs. Fortunately, help was at hand, in the form of a guide: a peddler – how he should be and what he should do, to obtain orders and ensure happy success in his business – by an old peddler.
This elderly peddler noted the following:
The business now brings the itinerant salesman here, and then there, and the proper routes of travel cannot be properly indicated for all the situations that occur; But sometimes, with proper selection and arrangement of the tour, so much time can be gained, that we do not think that we may avoid giving some rules also about this… The point is always to visit as many places as possible, without having to touch the same place twice.
The guide didn’t suggest any math to solve this problem, but it did contain examples of five allegedly perfect rounds.
The traveling salesman problem, or TSP, as it became known—later changed to the traveling salesperson problem to avoid sexism, which conveniently has the same acronym—is an institutional example of the mathematical area now known as harmonic optimization. Which means “finding the best option out of a set of possibilities that cannot be checked simultaneously.”
Oddly enough, the name TSP appears not to have been used explicitly in any publication relating to this problem until 1984, although it was a common use very early on in informal discussions among mathematicians.
In the age of the internet, companies rarely sell their wares by sending someone from city to city with a bag full of samples. They put everything on the web. As usual (unreasonable efficacy) this culture change did not render the TSP obsolete. With online shopping growing exponentially, the demand for efficient ways to set routes and schedules is more important than ever for everything from packages to supermarket orders to pizza.
Math aptitude also comes into play. TSP applications are not limited to travel between cities or along city streets. Once upon a time, prominent astronomers had their own telescopes, or shared them with a few colleagues. Telescopes could easily be redirected to point at new celestial bodies, so it was easy to improvise. This is no longer the case, when the telescopes used by astronomers are enormous, devastatingly expensive, and accessible online. Pointing the telescope at a new object takes time, and while the telescope is moving, it cannot be used for observation. Visit the targets in the wrong order and a lot of time is wasted moving the telescope a long distance, and then back again to somewhere close to where it started.
In DNA sequencing, the fragmented sequences of DNA bases must be linked together correctly, and the order in which this is done must be optimized to avoid wasting computer time. Other applications range from efficiently guiding aircraft to designing and manufacturing computer chips and printed circuit boards. The approximate solutions of TSPs have been used to find effective methods for meals on wheels and to improve blood delivery to hospitals. A version of the TSP appeared in Star Wars, President Ronald Reagan’s hypothetical strategic defense initiative, in which it was possible to target a series of nuclear missiles coming from a powerful laser orbiting the Earth.
In 1956, operations research pioneer Merrill Flood argued that TSP was likely to be difficult. In 1979, Michael Jarry and David Johnson proved him right: there is no effective “worst-case” problem-solving algorithm. But worst-case scenarios are often very contrived and not typical of real-world examples. So mathematicians in process research set out to see how many cities they could handle to solve real-world problems.