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A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.

The real-world version of the famous “traveling salesman problem” finally gets a good-enough solution.

Three computer scientists have solved a problem central to a dozen far-flung mathematical fields.

An infinitesimal advance in the traveling salesman problem breathes new life into the search for improved approximate solutions.