Comments on Integer Hulls of Two Linear Constraints

Abstract

This paper establishes an intrinsic complexity for the integer-programming problem that goes well beyond the computational complexities of linear programming. To this end, it describes a procedure with the following property: given any two independent linear constraints in two dimensions and any number N however large, the procedure determines two other linear constraints (with integer coefficients) arbitrarily “close” to the given constraints such that the two new constraints have at least N faces in their integer hull. It is possible for the integer-programming optimum to occur at any extreme point of this hull, and the number of extreme points is one less than the number of faces. In contrast, the linear-programming problem consisting of two inequalities in the plane is a triviality. The paper has an expository style and illustrates its highly geometrical arguments with figures.