In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

Synopsis

What is the shortest possible route for a traveling salesman seeking to visit each city on a list exactly once and return to his city of origin? It sounds simple enough, yet the traveling salesman problem is one of the most intensely studied puzzles in applied mathematics--and it has defied solution to this day. In this book, William Cook takes readers on a mathematical excursion, picking up the salesman's trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today's state-of-the-art attempts to solve it. He also explores its many important applications, from genome sequencing and designing computer processors to arranging music and hunting for planets.

In Pursuit of the Traveling Salesman travels to the very threshold of our understanding about the nature of complexity, and challenges you yourself to discover the solution to this captivating mathematical problem.

Excerpt

The star of Geoff Mack’s song has been to Reno, Chicago, Fargo, Buffalo, Toronto, Winslow, Sarasota, Wichita, Tulsa, Ottawa, Oklahoma, Tampa, Panama, Mattawa, LaPaloma, Bangor, Baltimore, Salvador, Amarillo, Tocapillo, Barranquilla, and Padilla.

One night in February 1988, my friend Vašek Chvátal and I decided to follow in the footsteps of mathematical giants and take a crack at finding the shortest way around such destinations. Next day we met at Tri-State Camera, a computer vendor in lower Manhattan. When the technician learned we were mathematicians in need of a speedy computer, he looked us in the eye and warned, “You guys aren’t trying to solve that traveling salesman problem, are ya?" Quite a bit of foresight there. This was the first of many computers we ground to a halt, spending the better part of the next twenty years searching for solutions.

The notorious problem is to compute the shortest route to visit each city on a specified list and return to the starting point. in the case of Geoff Mack’s traveler, one could conceivably check all 51,090,942,171,709,440,000 tours through the twenty-two cities. This computation would require even the fastest of supercomputers to roll up its sleeves and prepare for a hard day’s work, but with patience it may be possible to carry out the search. If, however, we had one hundred cities or so, then checking all routes to select the shortest is out of the question, even devoting the entire planet’s computing resources to the task.

This observation on sorting through all possible solutions is by no means a convincing argument that the salesman problem is actually difficult. There are similar problems that are very easy to solve and yet have far more candidate solutions than the number of ways to route a salesman. What sets the traveling salesman problem apart is the fact that despite decades of research by top applied mathematicians around the world, in general it is not known how to significantly improve upon simple bruteforce checking. It is a real possibility that there may never exist an efficient method that is guaranteed to solve every example of the problem. This is a . . .

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