Original version With this story appeared in Quanta Magazine.
In 1939, after arriving tardy for a statistics course at the University of California, Berkeley, a freshman named George Dantzig copied two problems off the blackboard, thinking they were homework. He later stated that the homework was “harder to do than usual” and apologized to the professor for taking him several extra days to complete it. A few weeks later, his professor told him that he had solved two eminent open statistical problems. Dantzig’s work became the basis for his doctoral dissertation and, several decades later, the inspiration for the film Goodwill hunting.
Dantzig earned his doctorate in 1946, just after World War II, and soon became mathematical adviser to the newly established United States Air Force. As with all newfangled wars, the outcome of World War II depended on the prudent allocation of narrow resources. However, unlike previous wars, this conflict was truly global in scale and was won largely through sheer industrial power. The United States could simply produce more tanks, aircraft carriers, and bombers than its enemies. Knowing this, the military was very interested in optimization problems, that is, the strategic allocation of limited resources in situations that could involve hundreds or thousands of variables.
The Air Force tasked Dantzig with finding up-to-date ways to solve optimization problems like this one. In response, he invented the simplex method, an algorithm that used some of the mathematical techniques he had developed almost a decade earlier while solving blackboard problems.
Nearly 80 years later, the simplex method remains one of the most widely used tools when logistics or supply chain decisions need to be made in the face of convoluted constraints. It is capable and it works. “He was always fast and no one saw him not being fast,” he said Zofia Huiberts French National Center for Scientific Research (CNRS).
At the same time, there is an compelling feature that has long cast a shadow over Dantzig’s method. In 1972, mathematicians proved that the time needed to complete a task can escalate exponentially with the number of constraints. So no matter how quick this method may be in practice, theoretical analyzes consistently present worst-case scenarios that suggest it could take exponentially longer. With the simplex method, “our traditional algorithm study tools don’t work,” Huiberts said.
But in a up-to-date one paper which will be presented in December at the Fundamentals of Computer Science conference, held at Huiberts and Eleon BachPhD student at the Technical University of Munich, seems to have solved this problem. They sped up the algorithm and also provided theoretical reasons why the long-feared exponential runtime does not materialize in practice. Work that is based on groundbreaking result since 2001 according to Daniel Spielman AND Shang Hua Tengis “brilliant [and] beautiful,” says Teng.
“This is a very impressive technical work that masterfully combines many ideas developed in previous research directions, [while adding] some really cool new technical ideas,” he said László Végha mathematician from the University of Bonn who was not involved in these efforts.
Optimal geometry
The simplex method was developed to solve the following problems: Suppose a furniture company produces wardrobes, beds and chairs. Coincidentally, each wardrobe is three times more profitable than each chair, and each bed is twice as profitable. If we wanted to write this as an expression, using AND, BAND C to represent the quantity of furniture produced, we would say that the total profit is proportional to 3AND +2B + C.
To maximize profits, how many units of each item should the company produce? The answer depends on the constraints it faces. Let’s assume that the company can produce a maximum of 50 units per month, the so-called AND + B + C is less than or equal to 50. Cabinets are more hard to make – you can’t make more than 20 – so AND is less than or equal to 20. Chairs require special wood and its supply is narrow, the so-called C must be less than 24.
The simplex method turns such situations – although often involving many more variables – into a geometry problem. Imagine a graph of our constraints for AND, B AND C in three dimensions. If AND is less than or equal to 20, we can imagine a plane on a three-dimensional graph that is perpendicular to AND axis, crossing it at address AND = 20. We would stipulate that our solution must lie somewhere on or below this plane. Similarly, we can create boundaries that are linked to other constraints. Together, these boundaries can divide space into a convoluted three-dimensional shape called a polyhedron.