On the Unlimited Number of Faces in Integer Hulls of Linear Programs with a Single Constraint

Abstract

The convex hull of the feasible integer points to a given integer program is a convex polytope I . The feasible set obtained by relaxing the integrality requirements is another convex polytope L . Cutting-plane algorithms essentially try to remove part of L − I . Hence the more complicated the relationship between L and I , the more difficult (in some sense) the integer program. This paper shows one such complexity: specifically, we construct a series of programs such that I has arbitrarily many faces even though L is a triangle. We also indicate the existence of a large class of problems that exhibit the same behavior.