When Nathan Klein started graduate school two years ago, his advisers proposed a modest plan: to work together on one of the most famous, long-standing problems in theoretical computer science.

Original story reprinted with permission from *Quanta Magazine*, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

Even if they didn’t manage to solve it, they figured, Klein would learn a lot in the process. He went along with the idea. “I didn’t know to be intimidated,” he said. “I was just a first-year grad student—I don’t know what’s going on.”

Now, in a paper posted online in July, Klein and his advisers at the University of Washington, Anna Karlin and Shayan Oveis Gharan, have finally achieved a goal computer scientists have pursued for nearly half a century: a better way to find approximate solutions to the traveling salesperson problem.

This optimization problem, which seeks the shortest (or least expensive) round trip through a collection of cities, has applications ranging from DNA sequencing to ride-sharing logistics. Over the decades, it has inspired many of the most fundamental advances in computer science, helping to illuminate the power of techniques such as linear programming. But researchers have yet to fully explore its possibilities—and not for want of trying.

The traveling salesperson problem “isn’t a problem, it’s an addiction,” as Christos Papadimitriou, a leading expert in computational complexity, is fond of saying.

Most computer scientists believe that there is no algorithm that can efficiently find the best solutions for all possible combinations of cities. But in 1976, Nicos Christofides came up with an algorithm that efficiently finds approximate solutions—round trips that are at most 50 percent longer than the best round trip. At the time, computer scientists expected that someone would soon improve on Christofides’ simple algorithm and come closer to the true solution. But the anticipated progress did not arrive.

“A lot of people spent countless hours trying to improve this result,” said Amin Saberi of Stanford University.

Now Karlin, Klein and Oveis Gharan have proved that an algorithm devised a decade ago beats Christofides’ 50 percent factor, though they were only able to subtract 0.2 billionth of a trillionth of a trillionth of a percent. Yet this minuscule improvement breaks through both a theoretical logjam and a psychological one. Researchers hope that it will open the floodgates to further improvements.

“This is a result I have wanted all my career,” said David Williamson of Cornell University, who has been studying the traveling salesperson problem since the 1980s.

The traveling salesperson problem is one of a handful of foundational problems that theoretical computer scientists turn to again and again to test the limits of efficient computation. The new result “is the first step towards showing that the frontiers of efficient computation are in fact better than what we thought,” Williamson said.

Fractional Progress

While there is probably no efficient method that always finds the shortest trip, it is possible to find something almost as good: the shortest tree connecting all the cities, meaning a network of connections (or “edges”) with no closed loops. Christofides’ algorithm uses this tree as the backbone for a round-trip tour, adding extra edges to convert it into a round trip.

Any round-trip route must have an even number of edges into each city, since every arrival is followed by a departure. It turns out that the reverse is also true—if every city in a network has an even number of connections then the edges of the network must trace a round trip.

The shortest tree connecting all the cities lacks this evenness property, since any city at the end of a branch has just one connection to another city. So to turn the shortest tree into a round trip, Christofides (who died last year) found the best way to connect pairs of cities that have odd numbers of edges. Then he proved that the resulting round trip will never be more than 50 percent longer than the best possible round trip.

In doing so, he devised perhaps the most famous approximation algorithm in theoretical computer science—one that usually forms the first example in textbooks and courses.

“Everybody knows the simple algorithm,” said Alantha Newman of Grenoble Alpes University and the National Center for Scientific Research in France. And when you know it, she said, “you know the state of the art”—at least, you did until this past July.

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