Cut-Generating Functions for Integer Variables

For an integer linear program, Gomory’s corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation. In this paper, we do not relax these nonnegativity constraints. We generalize a classical result of Gomory and Johnson characterizing minimal cut-generating functions in terms of subadditivity, symmetry, and periodicity. Our result is based on a new concept, the notion of generalized symmetry condition. We also prove a 2-Slope Theorem for extreme cut-generating functions in our setting, in the spirit of the 2-Slope Theorem of Gomory and Johnson.