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start:mathematics:travelling_salesman

*
Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?*

Computing algorithms can now tackle the Travelling Salesman Problem with thousands of destinations and give an *almost* exact answer.

“For many other instances with millions of cities, solutions can be found that are guaranteed to be within 1% of an optimal tour.” [source:

Wikipedia, above]

These estimations are sufficiently accurately for logistics and transport companies to plot highly efficient delivery/pickup routes. (Note that similar algorithms are used in the electronics industry to find efficient routing schemes for complex printed-circuit boards and chip layout.)

However, there is *no formal proof* providing a 100% reliable mathematical method to solve any and every situation. (The 'brute force' computing algorithms merely iteratively test a variety of pseudo-random solutions at very high speed, and then choose the best answer.)

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