The Ongoing Story of Gomory Cuts

The story of Gomory cuts is characterized by swings between great acclaim in the early days, near oblivion for decades and an amazing come back in the last 20 years. These cuts have been described as “elegant”, “disappointing” and “the clear winner” at various times over the last 55 years. This essay retraces that roller coaster. Ralph Gomory’s paper “Early Integer Programming” recounts his discovery of fractional cuts. It is a few years after he wrote his doctoral dissertation on nonlinear differential equations that he heard of linear programming for the first time. He was working for the Navy at the time. In one particular instance, it would have been preferable to have solutions in integers. Gomory thought that, somehow, one should be able to accomplish this. Within a few days he had invented fractional cuts. His approach was to first solve the linear program and then, using appropriate integer linear forms, to generate valid linear inequalities cutting off the undesirable fractional solution. By adding these cuts to the linear program, solving again using the simplex algorithm and iterating, Gomory could solve by hand any small integer linear program that he tried. However, he did not have a finiteness proof yet. At this point, he happened to run into Martin Beale in the halls of Princeton University in late 1957 and mentioned that he could solve linear programs in integers. When Beale immediately responded “but that’s impossible”, Gomory realized that he was not the first to think about this problem. As it turns out, Dantzig, Fulkerson, and Johnson had pioneered the cutting plane approach in a seminal paper published in 1954. They devised special-purpose cuts for the traveling salesman problem and, as a result, were able to solve to optimality an instance with 48 cities. However, Gomory’s goal was different and more ambitious. His fractional cuts were general-purpose cuts that applied to all integer linear programs. In his reminiscences “Early Integer Programming”, Gomory recounts the excitement that followed his encounter with Beale.