Revival of the Gomory cuts in the 1990’s

Abstract

In the early 90’s, the research community was unanimous: In order to solve integer programs of meaningful sizes, one had to exploit the structure of the underlying combinatorial problem; Gomory cuts (Gomory, 1960, 1963) made elegant theory (because they did not require knowledge of the underlying structure) but were utterly useless in practice (because they did not use the underlying structure!). I will come back shortly to these widely held beliefs. A practitioner confronted with a general mixed integer linear program (MILP) who did not have the time or skill to “exploit the underlying structure” had to resort to commercial codes that had been perfected in the early 70’s and had not been improved significantly in the following two decades. These branch-and-bound codes could solve small to medium size instances of MILP’s, but many applications were beyond their ability. Because MILP is NP-hard, it was widely accepted that nothing much could be done about this state of affairs. The sentiment was that there was little research left to do on branch-and-bound algorithms for general MILP’s. On the other hand, branch-and-cut algorithms for structured problems generated a tremendous amount of excitement. Padberg and Rinaldi obtained spectacular results for the traveling salesman problem. The same general philosophy was applied with a varying degree of success to numerous other classes of problems. Typically, for each class, a paper or series of papers would first identify facets of the integer polyhedron (i.e. the convex hull of the feasible solutions to the integer program), then present separation algorithms or heuristics for these facets and finally report some computational results. The success of such branch-and-cut algorithms was attributed to the use of facets of the integer polyhedron. By contrast, prominent researchers had a low opinion of the practical usefulness of Gomory cuts, including Gomory himself. Here are a few representative quotes reflecting the general sentiment in the early 90’s.