A Decades-Old Paradox Haunting a Core Global Algorithm Has Finally Been Solved

For nearly 80 years, the simplex method—a foundational algorithm powering global logistics, supply chains, and countless optimization tasks—has harbored a dark secret. While blazingly fast in practice, theoretical analysis warned of a worst-case scenario where it could become exponentially slow. Now, two researchers have finally proven why that theoretical nightmare never materializes in the real world.

In a new paper to be presented this month at the Foundations of Computer Science conference, Sophie Huiberts of the French National Center for Scientific Research (CNRS) and Eleon Bach, a doctoral student at the Technical University of Munich, have closed this long-standing gap between theory and reality. Their work not only provides a rigorous explanation for the algorithm's practical efficiency but also theoretically makes it faster.

An Algorithm Born from Homework

The story of the simplex method begins with a now-famous incident in 1939. George Dantzig, then a graduate student at UC Berkeley, arrived late to class and copied two problems from the blackboard, assuming they were homework. He found them "harder to do than usual." A few weeks later, his professor informed him he had solved two famous open problems in statistics.

After receiving his doctorate in 1946, Dantzig became a mathematical adviser to the US Air Force, which was intensely interested in optimization: how to allocate limited resources across thousands of variables. Drawing on the techniques he'd developed for his "homework," Dantzig invented the simplex method, an algorithm that remains one of the most widely used tools for logistical decision-making today.

The paradox emerged in 1972, when mathematicians proved that the algorithm's runtime could, in theory, grow exponentially with the number of constraints. Geometrically, the simplex method finds the shortest path along the edges of a complex shape called a polyhedron. The worst-case scenario is like getting lost in a labyrinth, taking the longest possible path from start to finish.

The Breakthrough: Taming Complexity with Randomness

A landmark 2001 paper by Daniel Spielman and Shang-Hua Teng provided the first key to solving the paradox. They showed that injecting a tiny bit of randomness into the process could prevent the algorithm from taking the worst possible path, guaranteeing a solution in what's known as polynomial time (e.g., n²)—far better than exponential time (e.g., 2ⁿ).

Bach and Huiberts' new work builds masterfully on that foundation. By incorporating even more randomness, they've established a significantly lower guaranteed runtime and, crucially, proved that their model of the algorithm cannot be made any faster. "This marks a major advance in our understanding of the [simplex] algorithm," said Heiko Röglin, a computer scientist at the University of Bonn, calling it "the first really convincing explanation for the method’s practical efficiency."

While the result is primarily of theoretical interest, it provides powerful mathematical reassurance for a tool that underpins a vast amount of modern infrastructure. "It's now easier to convince those who fear exponential complexity," noted Julian Hall, a mathematician at the University of Edinburgh who designs linear programming software. The work solidifies the foundation of a technology we already trust, replacing intuition with proof.