  The set of all group relaxations of an integer program contains certain
special members called Gomory relaxations. A family of integer programs with a
fixed coefficient matrix and cost vector but varying right hand sides is a
Gomory family if every program in the family can be solved by one of its Gomory
relaxations. In this paper, we characterize Gomory families. Every TDI system
gives a Gomory family, and we construct Gomory families from matrices whose
columns form a Hilbert basis for the cone they generate. The existence of
Gomory families is related to the Hilbert covering problems that arose from the
conjectures of Sebo. Connections to commutative algebra are outlined at the
end.
